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 work-
ing with many machine learning algorithms, especially deep learning algorithms.
We therefore begin the technical content of the book with a focused presentation
of the key linear algebra ideas that are most important in deep learning.
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 num-
bers. 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 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.
27
CHAPTER 2. LINEAR ALGEBRA
• Vectors: A vector is an array of numbers. The numbers have an order to
them, and we can identify each individual number by its index in that or-
dering. 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
.
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
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
.
28
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.
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
.
By convention, we consider that adding (or subtracting) a scalar and a vector
yields a vector with the additive operation performed on each element. The same
thing happens with a matrix or a tensor. This is called a broadcasting operation
in Python’s numpy library.
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
.
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
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
]
>
.
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.
29
CHAPTER 2. LINEAR ALGEBRA
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.
The product operation is defined by
c
i,j
=
X
k
a
i,k
b
k,j
.
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 in this book
1
as AB.
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.
Matrix product operations have many useful properties that make mathemat-
ical analysis of matrices more convenient. For example, matrix multiplication is
distributive:
A(B + C) = AB + AC.
It is also associative:
A(BC) = (AB)C.
Matrix multiplication is not commutative, unlike scalar multiplication.
The transpose of a matrix product also has a simple form:
(AB)
>
= B
>
A
>
.
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 can also multiply matrices and vectors by scalars. To multiply by a scalar,
we just multiply every element of the matrix or vector by that scalar:
sA =
sa
1,1
. . . sa
1,n
.
.
.
.
.
.
.
.
.
sa
m,1
. . . sa
m,n
1
The element-wise product is used relatively rarely, so the notation for it is not as standardized
as the other operations described in this chapter.
30
CHAPTER 2. LINEAR ALGEBRA
We now know enough linear algebra notation to write down a system of linear
equations:
Ax = b (2.1)
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 unknowns to solve for. Each row of A and each element of b provide
another constraint. We can rewrite equation 2.1 as:
A
1,:
x = b
1
A
2,:
x = b
2
. . .
A
m,:
x = b
m
or, even more explicitly, as:
a
1,1
x
1
+ a
1,2
x
2
+ ··· + a
1,n
x
n
= b
1
a
2,1
x
1
+ a
2,2
x
2
+ ··· + a
2,n
x
n
= b
2
. . .
a
m,1
x
1
+ a
m,2
x
2
+ ··· + a
m,n
x
n
= b
m
.
Matrix-vector product notations 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
solve equation 2.1 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 n-dimensional identity
matrix as I
n
. Formally,
∀x ∈ R
n
, I
n
x = x.
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
.
31
CHAPTER 2. LINEAR ALGEBRA
1 0 0
0 1 0
0 0 1
Figure 2.2: Example identity matrix: This is I
3
.
We can now solve equation 2.1 by the following steps:
Ax = b
A
−1
Ax = A
−1
b
I
n
x = A
−1
b
x = A
−1
b.
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 only be represented with 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, Span, and Rank
In order for A
−1
to exist, equation 2.1 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
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
32
CHAPTER 2. LINEAR ALGEBRA
in each of these directions, i.e. x
i
specifies how far to move in the direction of
column i:
Ax =
X
i
x
i
A
:,i
.
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)
.
The span of a set of vectors is the set of all points obtainable by linear combination
of the original vectors.
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. This implies immediately that A must have at least m columns, i.e.,
n ≥ m. Otherwise, the dimensionality of the column space must 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 have at least m linearly independent columns. This condition is both
necessary and sufficient for equation 2.1 to have a solution for every value of b.
In order for the matrix to have an inverse, we additionally need to ensure that
equation 2.1 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.
33
CHAPTER 2. LINEAR ALGEBRA
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
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.
For square matrices, the left inverse and right inverse are the same.
2.5 Norms
Sometimes we need to measure the size of a vector. In machine learning, we
usually measure the size of vectors using an L
p
norm:
||x||
p
=
X
i
|x
i
|
p
!
1
p
for p ∈ R, p ≥ 1.
Norms, including the L
p
norm, are functions mapping vectors to non-negative
values, satisfying these properties that make them behave like distances between
points:
• 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. This is probably
the most common norm used in machine learning. 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
34
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
|.
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 (and when we use the L
1
norm, we often use it as a proxy for this
function). Some authors refer to this function as the “l
0
norm,” but this is
incorrect terminology, because scaling the vector by α does not change 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
||x||
∞
= max
i
|x
i
|,
e.g., the absolute value of the element with the largest magnitude in the vector.
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
,
which is analogous to the L
2
norm of a vector.
The dot product of two vectors can be rewritten in terms of norms. Specifi-
cally,
x
>
y = ||x||
2
||y||
2
cos θ
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.
Diagonal matrices only have non-zero entries along the main diagonal. For-
mally, a matrix D is diagonal if and only if d
i,j
= 0 for all i 6= j. We’ve already
35
CHAPTER 2. LINEAR ALGEBRA
seen one example of a diagonal matrix: the identity matrix, where all of the di-
agonal entries are 1. In this book
2
, 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 ele-
ment 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 de-
rive 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.
Note that 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 involving 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
>
.
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.
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 90 degree angles 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.
This implies that
A
−1
= A
>
,
2
There is not a standardized notation for constructing a diagonal matrix from a vector.
36
CHAPTER 2. LINEAR ALGEBRA
so orthogonal matrices are of interest because their inverse is very cheap to com-
pute. Pay careful attention to the definition of orthogonal matrices. Counter-
intuitively, 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 multi-
plication by A alters only the scale of v:
Av = λv.
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).
Note that 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.
We can represent the matrix A using an eigendecomposition, with eigenvectors
{v
(1)
, . . . , v
(n)
} and corresponding eigenvalues {λ
1
, . . . , λ
n
} by concatenating the
eigenvectors into a matrix V = [v
(1)
, . . . , v
(n)
], (i.e. one column per eigenvector)
and concatenating the eigenvalues into a vector λ. Then the matrix
A = V diag(λ)V
−1
37
CHAPTER 2. LINEAR ALGEBRA
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
.
has the desired eigenvalues and eigenvectors. If we make V an orthogonal matrix,
then we can think of A as scaling space by λ
i
in direction v
(i)
. See Fig. 2.3 for
an example.
We have seen that constructing matrices with specific eigenvalues and eigen-
vectors 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 cases, the decomposition exists, but may involve complex rather than real
numbers. Fortunately, in this book, we usually need to decompose only a spe-
cific class of matrices that have a simple decomposition. Specifically, every real
symmetric matrix can be decomposed into an expression using only real-valued
38
CHAPTER 2. LINEAR ALGEBRA
eigenvectors and eigenvalues:
A = QΛQ
>
,
where Q is an orthogonal matrix composed of eigenvectors of A, and Λ is a
diagonal matrix, with λ
i,i
being the eigenvalue corresponding to Q
:,i
.
While any real symmetric matrix A is guaranteed to have an eigendecom-
position, the eigendecomposition is not 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 ma-
trix. The matrix is singular if and only if any of the eigenvalues are 0. The
eigendecomposition 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 eigenval-
ues. 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 more generally applicable. Every real matrix has a singular value decom-
position, 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
.
39
CHAPTER 2. LINEAR ALGEBRA
The singular value decomposition is similar, except this time we will write A
as a product of three matrices:
A = U DV
>
.
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 W 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
by left-multiplying each side to obtain
x = By.
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
>
.
40
CHAPTER 2. LINEAR ALGEBRA
Practical algorithms for computing the pseudoinverse are not based on this defi-
nition, but rather the formula
A
+
= V D
+
U
>
,
where U, D, and V are the singular value decomposition of A, and the pseu-
doinverse D
+
of a diagonal matrix D is obtained by taking the reciprocal of all
of its non-zero elements.
When A has more rows than columns, then solving a linear equation using
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 columns than rows, 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
.
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 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(A
>
A).
The trace operator also has many useful properties that make it easy to ma-
nipulate expressions involving the trace operator. For example, the trace operator
is invariant to the transpose operator:
Tr(A) = Tr(A
>
).
The trace of a square matrix composed of many factors is also invariant to
moving the last factor into the first position:
Tr(ABC) = Tr(BCA) = Tr(CAB)
or more generally,
Tr(Π
n
i=1
F
(i)
) = Tr(F
(n)
Π
n−1
i=1
F
(i)
).
41
CHAPTER 2. LINEAR ALGEBRA
2.11 Determinant
The determinant of a square matrix, denoted det(A), is a function mapping ma-
trices to real scalars. The determinant is equal to the product of all the matrix’s
eigenvalues. The absolute value of the determinant can be thought of as a mea-
sure 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 di-
mension, causing it to lose all of its volume. If the determinant is 1, then the
transformation is volume-preserving.
2.12 Example: Principal Components Analysis
One simple machine learning algorithm, principal components analysis (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, i.e. we would like to find a
way of storing the points 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 can use matrix multiplication to map the code back
into R
n
. Let the reconstruction r(c) = Dc, where D ∈ R
n×l
is the matrix defining
the decoding.
To simplify the computation of the optimal encoding, we constrain 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, r(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 −r(c)||
2
.
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-
42
CHAPTER 2. LINEAR ALGEBRA
negative and the squaring operation is monotonically increasing for non-negative
arguments.
c
∗
= arg min
c
||x −r(c)||
2
2
The function being minimized simplifies to
(x −r(c))
>
(x −r(c))
(by the definition of the L
2
norm)
= x
>
x −x
>
r(c) −r(c)
>
x + r(c)
>
r(c)
(by the distributive property)
= x
>
x −2x
>
r(c) + r(c)
>
r(c)
(because a scalar 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
>
r(c) + r(c)
>
r(c).
To make further progress, we must substitute in the definition of r(c):
c
∗
= arg min
c
−2x
>
Dc + c
>
D
>
Dc
= arg min
c
−2x
>
Dc + c
>
I
l
c
(by the orthogonality and unit norm constraints on D)
= arg min
c
−2x
>
Dc + c
>
c
We can solve this optimization problem using vector calculus (see section 4.3
if you do not know how to do this):
∇
c
(−2x
>
Dc + c
>
c) = 0
−2D
>
x + 2c = 0
c = D
>
x. (2.2)
This is good news: we can optimally encode x just using a vector-product
operation.
43
CHAPTER 2. LINEAR ALGEBRA
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
(i)
j
)
2
subject to D
>
D = I
l
(2.3)
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 equation 2.2
into equation 2.3 and simplifying D into d, the problem reduces to
d
∗
= arg min
d
X
i
||x
(i)
− x
(i)>
dd||
2
2
subject to ||d||
2
= 1.
At this point, it can be helpful to rewrite the problem in terms of matrices.
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
Disregarding the constraint for the moment, we can simplify the Frobenius norm
portion as follows:
arg min
d
||X −Xdd
>
||
2
F
= arg min
d
Tr
X −Xdd
>
>
X −Xdd
>
(by the alternate definition of the Frobenius norm)
= arg min
d
Tr(X
>
X −X
>
Xdd
>
− dd
>
X
>
X + dd
>
X
>
Xdd
>
)
= arg min
d
Tr(X
>
X) − Tr(X
>
Xdd
>
) −Tr(dd
>
X
>
X) + Tr(dd
>
X
>
Xdd
>
)
= arg min
d
−Tr(X
>
Xdd
>
) −Tr(dd
>
X
>
X) + Tr(dd
>
X
>
Xdd
>
)
(because terms not involving d do not affect the arg min)
= arg min
d
−2 Tr(X
>
Xdd
>
) + Tr(dd
>
X
>
Xdd
>
)
44
CHAPTER 2. LINEAR ALGEBRA
(because the trace is invariant to transpose)
= arg min
d
−2 Tr(X
>
Xdd
>
) + Tr(X
>
Xdd
>
dd
>
)
(because we can cycle the order of the matrices inside a trace)
At this point, we re-introduce the constraint:
arg min
d
−2 Tr(X
>
Xdd
>
) + Tr(X
>
Xdd
>
dd
>
) subject to d
>
d = 1
= arg min
d
−2 Tr(X
>
Xdd
>
) + Tr(X
>
Xdd
>
) subject to d
>
d = 1
(due to the constraint)
= arg min
d
−Tr(X
>
Xdd
>
) subject to d
>
d = 1
= arg max
d
Tr(X
>
Xdd
>
) subject to d
>
d = 1
= arg max
d
Tr(d
>
X
>
Xd) subject to d
>
d = 1
This optimization problem may be solved using eigendecomposition. Specif-
ically, the optimal d is given by the eigenvector of X
>
X corresponding to the
largest eigenvalue.
In the general case, where l > 1, 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.
45
