Chapter 2
Linear Algebra
Linear algebra is a branch of mathematics that is widely used throughout science
and engineering. However, because linear algebra is a form of continuous rather
than discrete mathematics, many computer scientists have little experience with it.
A good understanding of linear algebra is essential for understanding and working
with many machine learning algorithms, especially deep learning algorithms. We
therefore precede our introduction to deep learning with a focused presentation of
the key linear algebra prerequisites.
If you are already familiar with linear algebra, feel free to skip this chapter. If
you have previous experience with these concepts but need a detailed reference
sheet to review key formulas, we recommend The Matrix Cookbook (Petersen and
Pedersen, 2006). If you have no exposure at all to linear algebra, this chapter
will teach you enough to read this book, but we highly recommend that you also
consult another resource focused exclusively on teaching linear algebra, such as
Shilov (1977). This chapter will completely omit many important linear algebra
topics that are not essential for understanding deep learning.
2.1 Scalars, Vectors, Matrices and Tensors
The study of linear algebra involves several types of mathematical objects:
•
Scalars: A scalar is just a single number, in contrast to most of the other
objects studied in linear algebra, which are usually arrays of multiple numbers.
We write scalars in italics. We usually give scalars lower-case variable names.
When we introduce them, we specify what kind of number they are. For
31
CHAPTER 2. LINEAR ALGEBRA
example, we might say “Let
s ∈ R
be the slope of the line,” while defining a
real-valued scalar, or “Let
n ∈ N
be the number of units,” while defining a
natural number scalar.
•
Vectors: A vector is an array of numbers. The numbers are arranged in
order. We can identify each individual number by its index in that ordering.
Typically we give vectors lower case names written in bold typeface, such
as
x
. The elements of the vector are identified by writing its name in italic
typeface, with a subscript. The first element of
x
is
x
1
, the second element
is
x
2
and so on. We also need to say what kind of numbers are stored in
the vector. If each element is in
R
, and the vector has
n
elements, then the
vector lies in the set formed by taking the Cartesian product of
R n
times,
denoted as
R
n
. When we need to explicitly identify the elements of a vector,
we write them as a column enclosed in square brackets:
x =
x
1
x
2
.
.
.
x
n
. (2.1)
We can think of vectors as identifying points in space, with each element
giving the coordinate along a different axis.
Sometimes we need to index a set of elements of a vector. In this case, we
define a set containing the indices and write the set as a subscript. For
example, to access
x
1
,
x
3
and
x
6
, we define the set
S
=
{
1
,
3
,
6
}
and write
x
S
. We use the
−
sign to index the complement of a set. For example
x
−1
is
the vector containing all elements of
x
except for
x
1
, and
x
−S
is the vector
containing all of the elements of x except for x
1
, x
3
and x
6
.
•
Matrices: A matrix is a 2-D array of numbers, so each element is identified by
two indices instead of just one. We usually give matrices upper-case variable
names with bold typeface, such as
A
. If a real-valued matrix
A
has a height
of
m
and a width of
n
, then we say that
A ∈ R
m×n
. We usually identify
the elements of a matrix using its name in italic but not bold font, and the
indices are listed with separating commas. For example,
A
1,1
is the upper
left entry of
A
and
A
m,n
is the bottom right entry. We can identify all of
the numbers with vertical coordinate
i
by writing a “:” for the horizontal
coordinate. For example,
A
i,:
denotes the horizontal cross section of
A
with
vertical coordinate
i
. This is known as the
i
-th row of
A
. Likewise,
A
:,i
is
32
CHAPTER 2. LINEAR ALGEBRA
A =
2
4
A
1,1
A
1,2
A
2,1
A
2,2
A
3,1
A
3,2
3
5
) A
>
=
A
1,1
A
2,1
A
3,1
A
1,2
A
2,2
A
3,2
Figure 2.1: The transpose of the matrix can be thought of as a mirror image across the
main diagonal.
the
i
-th column of
A
. When we need to explicitly identify the elements of a
matrix, we write them as an array enclosed in square brackets:
A
1,1
A
1,2
A
2,1
A
2,2
. (2.2)
Sometimes we may need to index matrix-valued expressions that are not just
a single letter. In this case, we use subscripts after the expression, but do
not convert anything to lower case. For example,
f
(
A
)
i,j
gives element (
i, j
)
of the matrix computed by applying the function f to A.
•
Tensors: In some cases we will need an array with more than two axes. In
the general case, an array of numbers arranged on a regular grid with a
variable number of axes is known as a tensor. We denote a tensor named “A”
with this typeface:
A
. We identify the element of
A
at coordinates (
i, j, k
)
by writing A
i,j,k
.
One important operation on matrices is the transpose. The transpose of a
matrix is the mirror image of the matrix across a diagonal line, called the main
diagonal, running down and to the right, starting from its upper left corner. See
Fig. 2.1 for a graphical depiction of this operation. We denote the transpose of a
matrix A as A
>
, and it is defined such that
(A
>
)
i,j
= A
j,i
. (2.3)
Vectors can be thought of as matrices that contain only one column. The
transpose of a vector is therefore a matrix with only one row. Sometimes we
33
CHAPTER 2. LINEAR ALGEBRA
define a vector by writing out its elements in the text inline as a row matrix,
then using the transpose operator to turn it into a standard column vector, e.g.,
x = [x
1
, x
2
, x
3
]
>
.
A scalar can be thought of as a matrix with only a single entry. From this, we
can see that a scalar is its own transpose: a = a
>
.
We can add matrices to each other, as long as they have the same shape, just
by adding their corresponding elements: C = A + B where C
i,j
= A
i,j
+ B
i,j
.
We can also add a scalar to a matrix or multiply a matrix by a scalar, just
by performing that operation on each element of a matrix:
D
=
a · B
+
c
where
D
i,j
= a · B
i,j
+ c.
In the context of deep learning, we also use some less conventional notation.
We allow the addition of matrix and a vector, yielding another matrix:
C
=
A
+
b
,
where
C
i,j
=
A
i,j
+
b
j
. In other words, the vector
b
is added to each row of the
matrix. This shorthand eliminates the need to define a matrix with
b
copied into
each row before doing the addition. This implicit copying of
b
to many locations
is called broadcasting.
2.2 Multiplying Matrices and Vectors
One of the most important operations involving matrices is multiplication of two
matrices. The matrix product of matrices
A
and
B
is a third matrix
C
. In order
for this product to be defined,
A
must have the same number of columns as
B
has
rows. If
A
is of shape
m × n
and
B
is of shape
n × p
, then
C
is of shape
m × p
.
We can write the matrix product just by placing two or more matrices together,
e.g.
C = AB. (2.4)
The product operation is defined by
C
i,j
=
X
k
A
i,k
B
k,j
. (2.5)
Note that the standard product of two matrices is not just a matrix containing
the product of the individual elements. Such an operation exists and is called the
element-wise product or Hadamard product, and is denoted as A B.
The dot product between two vectors
x
and
y
of the same dimensionality is the
matrix product
x
>
y
. We can think of the matrix product
C
=
AB
as computing
C
i,j
as the dot product between row i of A and column j of B.
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CHAPTER 2. LINEAR ALGEBRA
Matrix product operations have many useful properties that make mathematical
analysis of matrices more convenient. For example, matrix multiplication is
distributive:
A(B + C) = AB + AC. (2.6)
It is also associative:
A(BC) = (AB)C. (2.7)
Matrix multiplication is not commutative (the condition
AB
=
BA
does not
always hold), unlike scalar multiplication. However, the dot product between two
vectors is commutative:
x
>
y = y
>
x. (2.8)
The transpose of a matrix product has a simple form:
(AB)
>
= B
>
A
>
. (2.9)
This allows us to demonstrate Eq. 2.8, by exploiting the fact that the value of
such a product is a scalar and therefore equal to its own transpose:
x
>
y =
x
>
y
>
= y
>
x. (2.10)
Since the focus of this textbook is not linear algebra, we do not attempt to
develop a comprehensive list of useful properties of the matrix product here, but
the reader should be aware that many more exist.
We now know enough linear algebra notation to write down a system of linear
equations:
Ax = b (2.11)
where
A ∈ R
m×n
is a known matrix,
b ∈ R
m
is a known vector, and
x ∈ R
n
is a
vector of unknown variables we would like to solve for. Each element
x
i
of
x
is one
of these unknown variables. Each row of
A
and each element of
b
provide another
constraint. We can rewrite Eq. 2.11 as:
A
1,:
x = b
1
(2.12)
A
2,:
x = b
2
(2.13)
. . . (2.14)
A
m,:
x = b
m
(2.15)
or, even more explicitly, as:
A
1,1
x
1
+ A
1,2
x
2
+ ··· + A
1,n
x
n
= b
1
(2.16)
35
CHAPTER 2. LINEAR ALGEBRA
1 0 0
0 1 0
0 0 1
Figure 2.2: Example identity matrix: This is I
3
.
A
2,1
x
1
+ A
2,2
x
2
+ ··· + A
2,n
x
n
= b
2
(2.17)
. . . (2.18)
A
m,1
x
1
+ A
m,2
x
2
+ ··· + A
m,n
x
n
= b
m
. (2.19)
Matrix-vector product notation provides a more compact representation for
equations of this form.
2.3 Identity and Inverse Matrices
Linear algebra offers a powerful tool called matrix inversion that allows us to
analytically solve Eq. 2.11 for many values of A.
To describe matrix inversion, we first need to define the concept of an identity
matrix. An identity matrix is a matrix that does not change any vector when we
multiply that vector by that matrix. We denote the identity matrix that preserves
n-dimensional vectors as I
n
. Formally, I
n
∈ R
n×n
, and
∀x ∈ R
n
, I
n
x = x. (2.20)
The structure of the identity matrix is simple: all of the entries along the main
diagonal are 1, while all of the other entries are zero. See Fig. 2.2 for an example.
The matrix inverse of
A
is denoted as
A
−1
, and it is defined as the matrix
such that
A
−1
A = I
n
. (2.21)
We can now solve Eq. 2.11 by the following steps:
Ax = b (2.22)
A
−1
Ax = A
−1
b (2.23)
I
n
x = A
−1
b (2.24)
36
CHAPTER 2. LINEAR ALGEBRA
x = A
−1
b. (2.25)
Of course, this depends on it being possible to find
A
−1
. We discuss the
conditions for the existence of A
−1
in the following section.
When
A
−1
exists, several different algorithms exist for finding it in closed form.
In theory, the same inverse matrix can then be used to solve the equation many
times for different values of
b
. However,
A
−1
is primarily useful as a theoretical
tool, and should not actually be used in practice for most software applications.
Because
A
−1
can be represented with only limited precision on a digital computer,
algorithms that make use of the value of
b
can usually obtain more accurate
estimates of x.
2.4 Linear Dependence and Span
In order for
A
−1
to exist, Eq. 2.11 must have exactly one solution for every value
of
b
. However, it is also possible for the system of equations to have no solutions
or infinitely many solutions for some values of
b
. It is not possible to have more
than one but less than infinitely many solutions for a particular
b
; if both
x
and
y
are solutions then
z = αx + (1 − α)y (2.26)
is also a solution for any real α.
To analyze how many solutions the equation has, we can think of the columns
of
A
as specifying different directions we can travel from the origin (the point
specified by the vector of all zeros), and determine how many ways there are of
reaching
b
. In this view, each element of
x
specifies how far we should travel in
each of these directions, with
x
i
specifying how far to move in the direction of
column i:
Ax =
X
i
x
i
A
:,i
. (2.27)
In general, this kind of operation is called a linear combination. Formally, a linear
combination of some set of vectors
{v
(1)
, . . . , v
(n)
}
is given by multiplying each
vector v
(i)
by a corresponding scalar coefficient and adding the results:
X
i
c
i
v
(i)
. (2.28)
The span of a set of vectors is the set of all points obtainable by linear combination
of the original vectors.
37
CHAPTER 2. LINEAR ALGEBRA
Determining whether
Ax
=
b
has a solution thus amounts to testing whether
b
is in the span of the columns of
A
. This particular span is known as the column
space or the range of A.
In order for the system
Ax
=
b
to have a solution for all values of
b ∈ R
m
,
we therefore require that the column space of
A
be all of
R
m
. If any point in
R
m
is excluded from the column space, that point is a potential value of
b
that has
no solution. The requirement that the column space of
A
be all of
R
m
implies
immediately that
A
must have at least
m
columns, i.e.,
n ≥ m
. Otherwise, the
dimensionality of the column space would be less than
m
. For example, consider a
3
×
2 matrix. The target
b
is 3-D, but
x
is only 2-D, so modifying the value of
x
at best allows us to trace out a 2-D plane within
R
3
. The equation has a solution
if and only if b lies on that plane.
Having
n ≥ m
is only a necessary condition for every point to have a solution.
It is not a sufficient condition, because it is possible for some of the columns to be
redundant. Consider a 2
×
2 matrix where both of the columns are equal to each
other. This has the same column space as a 2
×
1 matrix containing only one copy
of the replicated column. In other words, the column space is still just a line, and
fails to encompass all of R
2
, even though there are two columns.
Formally, this kind of redundancy is known as linear dependence. A set of
vectors is linearly independent if no vector in the set is a linear combination of the
other vectors. If we add a vector to a set that is a linear combination of the other
vectors in the set, the new vector does not add any points to the set’s span. This
means that for the column space of the matrix to encompass all of
R
m
, the matrix
must contain at least one set of
m
linearly independent columns. This condition
is both necessary and sufficient for Eq. 2.11 to have a solution for every value of
b
. Note that the requirement is for a set to have exactly
m
linear independent
columns, not at least
m
. No set of
m
-dimensional vectors can have more than
m
mutually linearly independent columns, but a matrix with more than m columns
may have more than one such set.
In order for the matrix to have an inverse, we additionally need to ensure that
Eq. 2.11 has at most one solution for each value of
b
. To do so, we need to ensure
that the matrix has at most
m
columns. Otherwise there is more than one way of
parametrizing each solution.
Together, this means that the matrix must be square, that is, we require that
m
=
n
and that all of the columns must be linearly independent. A square matrix
with linearly dependent columns is known as singular.
If
A
is not square or is square but singular, it can still be possible to solve the
38
CHAPTER 2. LINEAR ALGEBRA
equation. However, we can not use the method of matrix inversion to find the
solution.
So far we have discussed matrix inverses as being multiplied on the left. It is
also possible to define an inverse that is multiplied on the right:
AA
−1
= I. (2.29)
For square matrices, the left inverse and right inverse are equal.
2.5 Norms
Sometimes we need to measure the size of a vector. In machine learning, we usually
measure the size of vectors using a function called a norm. Formally, the
L
p
norm
is given by
||x||
p
=
X
i
|x
i
|
p
!
1
p
(2.30)
for p ∈ R, p ≥ 1.
Norms, including the
L
p
norm, are functions mapping vectors to non-negative
values. On an intuitive level, the norm of a vector
x
measures the distance from
the origin to the point
x
. More rigorously, a norm is any function
f
that satisfies
the following properties:
• f (x) = 0 ⇒ x = 0
• f (x + y) ≤ f(x) + f(y) (the triangle inequality)
• ∀α ∈ R, f(αx) = |α|f(x)
The
L
2
norm, with
p
= 2, is known as the Euclidean norm. It is simply the
Euclidean distance from the origin to the point identified by
x
. The
L
2
norm is
used so frequently in machine learning that it is often denoted simply as
||x||
, with
the subscript 2 omitted. It is also common to measure the size of a vector using
the squared L
2
norm, which can be calculated simply as x
>
x.
The squared
L
2
norm is more convenient to work with mathematically and
computationally than the
L
2
norm itself. For example, the derivatives of the
squared
L
2
norm with respect to each element of
x
each depend only on the
corresponding element of
x
, while all of the derivatives of the
L
2
norm depend
on the entire vector. In many contexts, the squared
L
2
norm may be undesirable
39
CHAPTER 2. LINEAR ALGEBRA
because it increases very slowly near the origin. In several machine learning
applications, it is important to discriminate between elements that are exactly
zero and elements that are small but nonzero. In these cases, we turn to a function
that grows at the same rate in all locations, but retains mathematical simplicity:
the L
1
norm. The L
1
norm may be simplified to
||x||
1
=
X
i
|x
i
|. (2.31)
The L
1
norm is commonly used in machine learning when the difference between
zero and nonzero elements is very important. Every time an element of
x
moves
away from 0 by , the L
1
norm increases by .
We sometimes measure the size of the vector by counting its number of nonzero
elements. Some authors refer to this function as the “
L
0
norm,” but this is incorrect
terminology. The number of non-zero entries in a vector is not a norm, because
scaling the vector by
α
does not change the number of nonzero entries. The
L
1
norm is often used as a substitute for the number of nonzero entries.
One other norm that commonly arises in machine learning is the
L
∞
norm,
also known as the max norm. This norm simplifies to the absolute value of the
element with the largest magnitude in the vector,
||x||
∞
= max
i
|x
i
|. (2.32)
Sometimes we may also wish to measure the size of a matrix. In the context
of deep learning, the most common way to do this is with the otherwise obscure
Frobenius norm
||A||
F
=
s
X
i,j
A
2
i,j
, (2.33)
which is analogous to the L
2
norm of a vector.
The dot product of two vectors can be rewritten in terms of norms. Specifically,
x
>
y = ||x||
2
||y||
2
cos θ (2.34)
where θ is the angle between x and y.
2.6 Special Kinds of Matrices and Vectors
Some special kinds of matrices and vectors are particularly useful.
40
CHAPTER 2. LINEAR ALGEBRA
Diagonal matrices consist mostly of zeros and have non-zero entries only along
the main diagonal. Formally, a matrix
D
is diagonal if and only if
D
i,j
= 0 for
all
i 6
=
j
. We have already seen one example of a diagonal matrix: the identity
matrix, where all of the diagonal entries are 1. We write
diag
(
v
) to denote a square
diagonal matrix whose diagonal entries are given by the entries of the vector
v
.
Diagonal matrices are of interest in part because multiplying by a diagonal matrix
is very computationally efficient. To compute
diag
(
v
)
x
, we only need to scale each
element
x
i
by
v
i
. In other words,
diag
(
v
)
x
=
v x
. Inverting a square diagonal
matrix is also efficient. The inverse exists only if every diagonal entry is nonzero,
and in that case,
diag
(
v
)
−1
=
diag
([1
/v
1
, . . . ,
1
/v
n
]
>
). In many cases, we may
derive some very general machine learning algorithm in terms of arbitrary matrices,
but obtain a less expensive (and less descriptive) algorithm by restricting some
matrices to be diagonal.
Not all diagonal matrices need be square. It is possible to construct a rectangular
diagonal matrix. Non-square diagonal matrices do not have inverses but it is still
possible to multiply by them cheaply. For a non-square diagonal matrix
D
, the
product
Dx
will involve scaling each element of
x
, and either concatenating some
zeros to the result if
D
is taller than it is wide, or discarding some of the last
elements of the vector if D is wider than it is tall.
A symmetric matrix is any matrix that is equal to its own transpose:
A = A
>
. (2.35)
Symmetric matrices often arise when the entries are generated by some function of
two arguments that does not depend on the order of the arguments. For example,
if
A
is a matrix of distance measurements, with
A
i,j
giving the distance from point
i to point j, then A
i,j
= A
j,i
because distance functions are symmetric.
A unit vector is a vector with unit norm:
||x||
2
= 1. (2.36)
A vector
x
and a vector
y
are orthogonal to each other if
x
>
y
= 0. If both
vectors have nonzero norm, this means that they are at a 90 degree angle to each
other. In
R
n
, at most
n
vectors may be mutually orthogonal with nonzero norm.
If the vectors are not only orthogonal but also have unit norm, we call them
orthonormal.
An orthogonal matrix is a square matrix whose rows are mutually orthonormal
and whose columns are mutually orthonormal:
A
>
A = AA
>
= I. (2.37)
41
CHAPTER 2. LINEAR ALGEBRA
This implies that
A
−1
= A
>
, (2.38)
so orthogonal matrices are of interest because their inverse is very cheap to compute.
Pay careful attention to the definition of orthogonal matrices. Counterintuitively,
their rows are not merely orthogonal but fully orthonormal. There is no special
term for a matrix whose rows or columns are orthogonal but not orthonormal.
2.7 Eigendecomposition
Many mathematical objects can be understood better by breaking them into
constituent parts, or finding some properties of them that are universal, not caused
by the way we choose to represent them.
For example, integers can be decomposed into prime factors. The way we
represent the number 12 will change depending on whether we write it in base ten
or in binary, but it will always be true that 12 = 2
×
2
×
3. From this representation
we can conclude useful properties, such as that 12 is not divisible by 5, or that any
integer multiple of 12 will be divisible by 3.
Much as we can discover something about the true nature of an integer by
decomposing it into prime factors, we can also decompose matrices in ways that
show us information about their functional properties that is not obvious from the
representation of the matrix as an array of elements.
One of the most widely used kinds of matrix decomposition is called eigen-
decomposition, in which we decompose a matrix into a set of eigenvectors and
eigenvalues.
An eigenvector of a square matrix
A
is a non-zero vector
v
such that multipli-
cation by A alters only the scale of v:
Av = λv. (2.39)
The scalar
λ
is known as the eigenvalue corresponding to this eigenvector. (One
can also find a left eigenvector such that
v
>
A
=
λv
>
, but we are usually concerned
with right eigenvectors).
If
v
is an eigenvector of
A
, then so is any rescaled vector
sv
for
s ∈ R, s 6
= 0.
Moreover,
sv
still has the same eigenvalue. For this reason, we usually only look
for unit eigenvectors.
Suppose that a matrix
A
has
n
linearly independent eigenvectors,
{v
(1)
, . . . ,
v
(n)
}
, with corresponding eigenvalues
{λ
1
, . . . , λ
n
}
. We may concatenate all of the
42
CHAPTER 2. LINEAR ALGEBRA
−3 −2 −1 0 1 2 3
x
0
−3
−2
−1
0
1
2
3
x
1
v
(1)
v
(2)
Before multiplication
−3 −2 −1 0 1 2 3
x
0
0
−3
−2
−1
0
1
2
3
x
0
1
v
(1)
¸
1
v
(1)
v
(2)
¸
2
v
(2)
After multiplication
Effect of eigenvectors and eigenvalues
Figure 2.3: An example of the effect of eigenvectors and eigenvalues. Here, we have
a matrix
A
with two orthonormal eigenvectors,
v
(1)
with eigenvalue
λ
1
and
v
(2)
with
eigenvalue
λ
2
. (Left) We plot the set of all unit vectors
u ∈ R
2
as a unit circle. (Right)
We plot the set of all points
Au
. By observing the way that
A
distorts the unit circle, we
can see that it scales space in direction v
(i)
by λ
i
.
eigenvectors to form a matrix
V
with one eigenvector per column:
V
= [
v
(1)
, . . . ,
v
(n)
]. Likewise, we can concatenate the eigenvalues to form a vector
λ
= [
λ
1
, . . . ,
λ
n
]
>
. The eigendecomposition of A is then given by
A = V diag(λ)V
−1
. (2.40)
We have seen that constructing matrices with specific eigenvalues and eigenvec-
tors allows us to stretch space in desired directions. However, we often want to
decompose matrices into their eigenvalues and eigenvectors. Doing so can help us
to analyze certain properties of the matrix, much as decomposing an integer into
its prime factors can help us understand the behavior of that integer.
Not every matrix can be decomposed into eigenvalues and eigenvectors. In some
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CHAPTER 2. LINEAR ALGEBRA
cases, the decomposition exists, but may involve complex rather than real numbers.
Fortunately, in this book, we usually need to decompose only a specific class of
matrices that have a simple decomposition. Specifically, every real symmetric
matrix can be decomposed into an expression using only real-valued eigenvectors
and eigenvalues:
A = QΛQ
>
, (2.41)
where
Q
is an orthogonal matrix composed of eigenvectors of
A
, and
Λ
is a
diagonal matrix. The eigenvalue Λ
i,i
is associated with the eigenvector in column
i
of
Q
, denoted as
Q
:,i
. Because
Q
is an orthogonal matrix, we can think of
A
as
scaling space by λ
i
in direction v
(i)
. See Fig. 2.3 for an example.
While any real symmetric matrix
A
is guaranteed to have an eigendecomposi-
tion, the eigendecomposition may not be unique. If any two or more eigenvectors
share the same eigenvalue, then any set of orthogonal vectors lying in their span
are also eigenvectors with that eigenvalue, and we could equivalently choose a
Q
using those eigenvectors instead. By convention, we usually sort the entries of
Λ
in descending order. Under this convention, the eigendecomposition is unique only
if all of the eigenvalues are unique.
The eigendecomposition of a matrix tells us many useful facts about the
matrix. The matrix is singular if and only if any of the eigenvalues are 0. The
eigendecomposition of a real symmetric matrix can also be used to optimize
quadratic expressions of the form
f
(
x
) =
x
>
Ax
subject to
||x||
2
= 1. Whenever
x
is equal to an eigenvector of
A
,
f
takes on the value of the corresponding eigenvalue.
The maximum value of
f
within the constraint region is the maximum eigenvalue
and its minimum value within the constraint region is the minimum eigenvalue.
A matrix whose eigenvalues are all positive is called positive definite. A matrix
whose eigenvalues are all positive or zero-valued is called positive semidefinite.
Likewise, if all eigenvalues are negative, the matrix is negative definite, and if
all eigenvalues are negative or zero-valued, it is negative semidefinite. Positive
semidefinite matrices are interesting because they guarantee that
∀x, x
>
Ax ≥
0.
Positive definite matrices additionally guarantee that x
>
Ax = 0 ⇒ x = 0.
2.8 Singular Value Decomposition
In Sec. 2.7, we saw how to decompose a matrix into eigenvectors and eigenvalues.
The singular value decomposition (SVD) provides another way to factorize a matrix,
into singular vectors and singular values. The SVD allows us to discover some of
the same kind of information as the eigendecomposition. However, the SVD is
44
CHAPTER 2. LINEAR ALGEBRA
more generally applicable. Every real matrix has a singular value decomposition,
but the same is not true of the eigenvalue decomposition. For example, if a matrix
is not square, the eigendecomposition is not defined, and we must use a singular
value decomposition instead.
Recall that the eigendecomposition involves analyzing a matrix
A
to discover
a matrix
V
of eigenvectors and a vector of eigenvalues
λ
such that we can rewrite
A as
A = V diag(λ)V
−1
. (2.42)
The singular value decomposition is similar, except this time we will write
A
as a product of three matrices:
A = U DV
>
. (2.43)
Suppose that
A
is an
m ×n
matrix. Then
U
is defined to be an
m ×m
matrix,
D to be an m × n matrix, and V to be an n ×n matrix.
Each of these matrices is defined to have a special structure. The matrices
U
and
V
are both defined to be orthogonal matrices. The matrix
D
is defined to be
a diagonal matrix. Note that D is not necessarily square.
The elements along the diagonal of
D
are known as the singular values of the
matrix
A
. The columns of
U
are known as the left-singular vectors. The columns
of V are known as as the right-singular vectors.
We can actually interpret the singular value decomposition of
A
in terms of
the eigendecomposition of functions of
A
. The left-singular vectors of
A
are the
eigenvectors of
AA
>
. The right-singular vectors of
A
are the eigenvectors of
A
>
A
.
The non-zero singular values of
A
are the square roots of the eigenvalues of
A
>
A
.
The same is true for AA
>
.
Perhaps the most useful feature of the SVD is that we can use it to partially
generalize matrix inversion to non-square matrices, as we will see in the next
section.
2.9 The Moore-Penrose Pseudoinverse
Matrix inversion is not defined for matrices that are not square. Suppose we want
to make a left-inverse B of a matrix A, so that we can solve a linear equation
Ax = y (2.44)
45
CHAPTER 2. LINEAR ALGEBRA
by left-multiplying each side to obtain
x = By. (2.45)
Depending on the structure of the problem, it may not be possible to design a
unique mapping from A to B.
If
A
is taller than it is wide, then it is possible for this equation to have
no solution. If
A
is wider than it is tall, then there could be multiple possible
solutions.
The Moore-Penrose pseudoinverse allows us to make some headway in these
cases. The pseudoinverse of A is defined as a matrix
A
+
= lim
α&0
(A
>
A + αI)
−1
A
>
. (2.46)
Practical algorithms for computing the pseudoinverse are not based on this defini-
tion, but rather the formula
A
+
= V D
+
U
>
, (2.47)
where
U
,
D
and
V
are the singular value decomposition of
A
, and the pseudoinverse
D
+
of a diagonal matrix
D
is obtained by taking the reciprocal of its non-zero
elements then taking the transpose of the resulting matrix.
When
A
has more columns than rows, then solving a linear equation using the
pseudoinverse provides one of the many possible solutions. Specifically, it provides
the solution
x
=
A
+
y
with minimal Euclidean norm
||x||
2
among all possible
solutions.
When
A
has more rows than columns, it is possible for there to be no solution.
In this case, using the pseudoinverse gives us the
x
for which
Ax
is as close as
possible to y in terms of Euclidean norm ||Ax − y||
2
.
2.10 The Trace Operator
The trace operator gives the sum of all of the diagonal entries of a matrix:
Tr(A) =
X
i
A
i,i
. (2.48)
The trace operator is useful for a variety of reasons. Some operations that are
difficult to specify without resorting to summation notation can be specified using
46
CHAPTER 2. LINEAR ALGEBRA
matrix products and the trace operator. For example, the trace operator provides
an alternative way of writing the Frobenius norm of a matrix:
||A||
F
=
q
Tr(AA
>
). (2.49)
Writing an expression in terms of the trace operator opens up opportunities to
manipulate the expression using many useful identities. For example, the trace
operator is invariant to the transpose operator:
Tr(A) = Tr(A
>
). (2.50)
The trace of a square matrix composed of many factors is also invariant to
moving the last factor into the first position, if the shapes of the corresponding
matrices allow the resulting product to be defined:
Tr(ABC) = Tr(CAB) = Tr(BCA) (2.51)
or more generally,
Tr(
n
Y
i=1
F
(i)
) = Tr(F
(n)
n−1
Y
i=1
F
(i)
). (2.52)
This invariance to cyclic permutation holds even if the resulting product has a
different shape. For example, for A ∈ R
m×n
and B ∈ R
n×m
, we have
Tr(AB) = Tr(BA) (2.53)
even though AB ∈ R
m×m
and BA ∈ R
n×n
.
Another useful fact to keep in mind is that a scalar is its own trace:
a
=
Tr
(
a
).
2.11 The Determinant
The determinant of a square matrix, denoted
det
(
A
), is a function mapping
matrices to real scalars. The determinant is equal to the product of all the
eigenvalues of the matrix. The absolute value of the determinant can be thought
of as a measure of how much multiplication by the matrix expands or contracts
space. If the determinant is 0, then space is contracted completely along at least
one dimension, causing it to lose all of its volume. If the determinant is 1, then
the transformation is volume-preserving.
47
CHAPTER 2. LINEAR ALGEBRA
2.12 Example: Principal Components Analysis
One simple machine learning algorithm, principal components analysis or PCA can
be derived using only knowledge of basic linear algebra.
Suppose we have a collection of
m
points
{x
(1)
, . . . , x
(m)
}
in
R
n
. Suppose we
would like to apply lossy compression to these points. Lossy compression means
storing the points in a way that requires less memory but may lose some precision.
We would like to lose as little precision as possible.
One way we can encode these points is to represent a lower-dimensional version
of them. For each point
x
(i)
∈ R
n
we will find a corresponding code vector
c
(i)
∈ R
l
.
If
l
is smaller than
n
, it will take less memory to store the code points than the
original data. We will want to find some encoding function that produces the code
for an input,
f
(
x
) =
c
, and a decoding function that produces the reconstructed
input given its code, x ≈ g(f (x)).
PCA is defined by our choice of the decoding function. Specifically, to make the
decoder very simple, we choose to use matrix multiplication to map the code back
into R
n
. Let g(c) = Dc, where D ∈ R
n×l
is the matrix defining the decoding.
Computing the optimal code for this decoder could be a difficult problem. To
keep the encoding problem easy, PCA constrains the columns of
D
to be orthogonal
to each other. (Note that
D
is still not technically “an orthogonal matrix” unless
l = n)
With the problem as described so far, many solutions are possible, because we
can increase the scale of
D
:,i
if we decrease
c
i
proportionally for all points. To give
the problem a unique solution, we constrain all of the columns of D to have unit
norm.
In order to turn this basic idea into an algorithm we can implement, the first
thing we need to do is figure out how to generate the optimal code point
c
∗
for
each input point
x
. One way to do this is to minimize the distance between the
input point
x
and its reconstruction,
g
(
c
∗
). We can measure this distance using a
norm. In the principal components algorithm, we use the L
2
norm:
c
∗
= arg min
c
||x − g(c)||
2
. (2.54)
We can switch to the squared
L
2
norm instead of the
L
2
norm itself, because
both are minimized by the same value of
c
. This is because the
L
2
norm is non-
negative and the squaring operation is monotonically increasing for non-negative
48
CHAPTER 2. LINEAR ALGEBRA
arguments.
c
∗
= arg min
c
||x − g(c)||
2
2
. (2.55)
The function being minimized simplifies to
(x − g(c))
>
(x − g(c)) (2.56)
(by the definition of the L
2
norm, Eq. 2.30)
= x
>
x − x
>
g(c) − g(c)
>
x + g(c)
>
g(c) (2.57)
(by the distributive property)
= x
>
x − 2x
>
g(c) + g(c)
>
g(c) (2.58)
(because the scalar g(x)
>
x is equal to the transpose of itself).
We can now change the function being minimized again, to omit the first term,
since this term does not depend on c:
c
∗
= arg min
c
−2x
>
g(c) + g(c)
>
g(c). (2.59)
To make further progress, we must substitute in the definition of g(c):
c
∗
= arg min
c
−2x
>
Dc + c
>
D
>
Dc (2.60)
= arg min
c
−2x
>
Dc + c
>
I
l
c (2.61)
(by the orthogonality and unit norm constraints on D)
= arg min
c
−2x
>
Dc + c
>
c (2.62)
We can solve this optimization problem using vector calculus (see Sec. 4.3 if
you do not know how to do this):
∇
c
(−2x
>
Dc + c
>
c) = 0 (2.63)
− 2D
>
x + 2c = 0 (2.64)
c = D
>
x. (2.65)
This makes the algorithm efficient: we can optimally encode
x
just using a
matrix-vector operation. To encode a vector, we apply the encoder function
f(x) = D
>
x. (2.66)
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CHAPTER 2. LINEAR ALGEBRA
Using a further matrix multiplication, we can also define the PCA reconstruction
operation:
r(x) = g (f (x)) = DD
>
x. (2.67)
Next, we need to choose the encoding matrix
D
. To do so, we revisit the
idea of minimizing the
L
2
distance between inputs and reconstructions. However,
since we will use the same matrix
D
to decode all of the points, we can no longer
consider the points in isolation. Instead, we must minimize the Frobenius norm of
the matrix of errors computed over all dimensions and all points:
D
∗
= arg min
D
s
X
i,j
x
(i)
j
− r(x
(i)
)
j
2
subject to D
>
D = I
l
(2.68)
To derive the algorithm for finding
D
∗
, we will start by considering the case
where
l
= 1. In this case,
D
is just a single vector,
d
. Substituting Eq. 2.67 into
Eq. 2.68 and simplifying D into d, the problem reduces to
d
∗
= arg min
d
X
i
||x
(i)
− dd
>
x
(i)
||
2
2
subject to ||d||
2
= 1. (2.69)
The above formulation is the most direct way of performing the substitution,
but is not the most stylistically pleasing way to write the equation. It places the
scalar value
d
>
x
(i)
on the right of the vector
d
. It is more conventional to write
scalar coefficients on the left of vector they operate on. We therefore usually write
such a formula as
d
∗
= arg min
d
X
i
||x
(i)
− d
>
x
(i)
d||
2
2
subject to ||d||
2
= 1, (2.70)
or, exploiting the fact that a scalar is its own transpose, as
d
∗
= arg min
d
X
i
||x
(i)
− x
(i)>
dd||
2
2
subject to ||d||
2
= 1. (2.71)
The reader should aim to become familiar with such cosmetic rearrangements.
At this point, it can be helpful to rewrite the problem in terms of a single
design matrix of examples, rather than as a sum over separate example vectors.
This will allow us to use more compact notation. Let
X ∈ R
m×n
be the matrix
defined by stacking all of the vectors describing the points, such that
X
i,:
=
x
(i)
>
.
We can now rewrite the problem as
d
∗
= arg min
d
||X −Xdd
>
||
2
F
subject to d
>
d = 1. (2.72)
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CHAPTER 2. LINEAR ALGEBRA
Disregarding the constraint for the moment, we can simplify the Frobenius norm
portion as follows:
arg min
d
||X −Xdd
>
||
2
F
(2.73)
= arg min
d
Tr
X −Xdd
>
>
X −Xdd
>
(2.74)
(by Eq. 2.49)
= arg min
d
Tr(X
>
X −X
>
Xdd
>
− dd
>
X
>
X + dd
>
X
>
Xdd
>
) (2.75)
= arg min
d
Tr(X
>
X) − Tr(X
>
Xdd
>
) − Tr(dd
>
X
>
X) + Tr(dd
>
X
>
Xdd
>
)
(2.76)
= arg min
d
−Tr(X
>
Xdd
>
) − Tr(dd
>
X
>
X) + Tr(dd
>
X
>
Xdd
>
) (2.77)
(because terms not involving d do not affect the arg min)
= arg min
d
−2 Tr(X
>
Xdd
>
) + Tr(dd
>
X
>
Xdd
>
) (2.78)
(because we can cycle the order of the matrices inside a trace, Eq. 2.52)
= arg min
d
−2 Tr(X
>
Xdd
>
) + Tr(X
>
Xdd
>
dd
>
) (2.79)
(using the same property again)
At this point, we re-introduce the constraint:
arg min
d
−2 Tr(X
>
Xdd
>
) + Tr(X
>
Xdd
>
dd
>
) subject to d
>
d = 1 (2.80)
= arg min
d
−2 Tr(X
>
Xdd
>
) + Tr(X
>
Xdd
>
) subject to d
>
d = 1 (2.81)
(due to the constraint)
= arg min
d
−Tr(X
>
Xdd
>
) subject to d
>
d = 1 (2.82)
= arg max
d
Tr(X
>
Xdd
>
) subject to d
>
d = 1 (2.83)
= arg max
d
Tr(d
>
X
>
Xd) subject to d
>
d = 1 (2.84)
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CHAPTER 2. LINEAR ALGEBRA
This optimization problem may be solved using eigendecomposition. Specifically,
the optimal
d
is given by the eigenvector of
X
>
X
corresponding to the largest
eigenvalue.
In the general case, where
l >
1, the matrix
D
is given by the
l
eigenvectors
corresponding to the largest eigenvalues. This may be shown using proof by
induction. We recommend writing this proof as an exercise.
Linear algebra is one of the fundamental mathematical disciplines that is
necessary to understand deep learning. Another key area of mathematics that is
ubiquitous in machine learning is probability theory, presented next.
52
