Matrices

Section 1.2 Matrices

Matrices can be viewed in two complementary ways: as arrays of data and as functions acting on vectors. We begin with basic matrix arithmetic, then use matrix-vector multiplication to describe linear maps.

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Subsection Matrices as arrays of numbers

Definition 1.2.1. Matrices.

An $m \times n$ matrix $A$ is a rectangular array of $m \cdot n$ real numbers arranged in $m$ horizontal rows and $n$ vertical columns:

\begin{equation*} A = \begin{bmatrix} a_{11} \amp a_{12} \amp \cdots \amp a_{1n} \\ a_{21} \amp a_{22} \amp \cdots \amp a_{2n} \\ \vdots \amp \vdots \amp \ddots \amp \vdots \\ a_{m1} \amp a_{m2} \amp \cdots \amp a_{mn} \\ \end{bmatrix} \end{equation*}

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The $i$-th row of $A$ is

\begin{equation*} \begin{bmatrix} a_{i1} \amp a_{i2} \amp \cdots \amp a_{in} \end{bmatrix}, \end{equation*}

and the $j$-th column of $A$ is

\begin{equation*} \begin{bmatrix} a_{1j} \\ a_{2j} \\ \vdots \\ a_{mj} \end{bmatrix}. \end{equation*}

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The number $a_{ij}\text{,}$ which is in the $i$-th row and $j$-th column of $A\text{,}$ is the $(i,j)$-entry of $A\text{,}$ and we often write $A = [a_{ij}]\text{.}$ We say $A$ is an "$m$ by $n$" matrix.

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Activity 1.2.1.

Consider the matrix

\begin{equation*} A = \begin{bmatrix} 1 \amp 2 \amp 3 \\ 5 \amp -8 \amp -13 \end{bmatrix} \end{equation*}

$A$ is a $2 \times 3$ matrix. Compute the following:

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(a)

$a_{11}$

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Solution.

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(b)

$a_{12}$

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Solution.

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(c)

$a_{21}$

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Solution.

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(d)

$a_{23}$

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Solution.

$-13$

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(e)

$a_{32}$

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Solution.

No such entry exists.

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Subsection Data matrices

An important application of matrices (but far from the only one!) is to record data. The following table gives a few examples.

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Table 1.2.2. Some data matrices

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Object	Meaning of an entry
Monochrome image	$X_{ij}$ is the pixel value in row $i$ and column $j\text{.}$
Rainfall data	$A_{ij}$ is the rainfall at location $j$ on day $i\text{.}$
Asset returns	$R_{ij}$ is the return of asset $i$ in period $j\text{.}$
Feature matrix	$X_{ij}$ is the value of feature $j$ for entity $i\text{.}$

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Subsection Matrix operations

Definition 1.2.3. Equality of matrices.

Two matrices $A$ and $B$ are equal if they have the same size and all the corresponding entries are equal.

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Activity 1.2.2.

Suppose

\begin{equation*} A = \begin{bmatrix} 3 \amp y \\ 4 \amp -7 \end{bmatrix}\quad\text{and}\quad B = \begin{bmatrix} 6+x \amp -2 \\ 4 \amp -7 \end{bmatrix} \end{equation*}

and $A = B\text{.}$ Find $x$ and $y\text{.}$

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Solution.

Since $A = B\text{,}$ all entries of $A$ must equal the corresponding entries in $B\text{.}$ So it must be true, comparing corresponding entries, that

\begin{align*} 3 \amp = 6 + x\\ y \amp = -2 \end{align*}

Therefore, $x = -3$ and $y = -2\text{.}$

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Definition 1.2.4. Sums of matrices.

If $A = [a_{ij}]$ and $B = [b_{ij}]$ are both $m \times n$ matrices, then their sum $A+B$ is the matrix $C = [c_{ij}]$ where $c_{ij} = a_{ij} + b_{ij}\text{.}$

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Activity 1.2.3.

For the matrices

\begin{equation*} A = \begin{bmatrix} 1 \amp 2 \amp 3 \\ 5 \amp -8 \amp -13 \end{bmatrix} \quad\text{and}\quad B = \begin{bmatrix} 0 \amp 2 \amp 1 \\ 1 \amp 3 \amp -4 \end{bmatrix} \end{equation*}

calculate $A+B\text{.}$

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Solution.

\begin{equation*} \begin{aligned} A+B \amp= \begin{bmatrix} 1+0 \amp 2+2 \amp 3+1\\ 5+1 \amp -8+3 \amp -13+(-4) \end{bmatrix}\\ \amp= \begin{bmatrix} 1 \amp 4 \amp 4\\ 6 \amp -5 \amp -17 \end{bmatrix}. \end{aligned} \end{equation*}

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Warning 1.2.5.

For $A+B$ to be defined, $A$ and $B$ must be the same size. From now on, if we write $A+B\text{,}$ assume that this is the case.

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Definition 1.2.6. Scalar multiples of matrices.

If $A = [a_{ij}]$ is an $m \times n$ matrix and $r$ is a real number, then the scalar multiple of $A$ by $r\text{,}$ written $rA\text{,}$ is the $m \times n$ matrix $C = [c_{ij}]\text{,}$ where $c_{ij} = r a_{ij}\text{,}$ that is, $C$ is the matrix obtained by multiplying every entry of $A$ by $r\text{.}$

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Activity 1.2.4.

For $A$ as in Activity 1.2.3, calculate $-2A\text{.}$

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Solution.

\begin{equation*} -2A = \begin{bmatrix} -2 \amp -4 \amp -6 \\ -10 \amp 16 \amp 26 \end{bmatrix} \end{equation*}

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Definition 1.2.7. Linear combinations of matrices.

If $A_1, A_2, \ldots, A_k$ are $m \times n$ matrices and $c_1, c_2, \ldots, c_k$ are real numbers, then an expression of the form

\begin{equation*} c_1 A_1 + c_2 A_2 + \cdots + c_k A_k \end{equation*}

is called a linear combination of $A_1, A_2, \ldots, A_k\text{.}$ The scalars $c_1, \ldots, c_k$ are called the coefficients of the linear combination.

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This is the same idea as a linear combination of vectors from Vectors. The only difference is that the objects being combined are matrices of the same size.

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Activity 1.2.5.

Compute the following linear combination of matrices:

\begin{equation*} 4 \begin{bmatrix} 0 \amp 2 \\ -3 \amp 3 \end{bmatrix} - \frac{1}{2} \begin{bmatrix} 4 \amp 2 \\ 6 \amp 2 \end{bmatrix} \end{equation*}

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Solution.

\begin{align*} \amp= \begin{bmatrix} 0 \amp 8 \\ -12 \amp 12 \end{bmatrix} - \begin{bmatrix} 2 \amp 1 \\ 3 \amp 1 \end{bmatrix} \amp= \begin{bmatrix} -2 \amp 7 \\ -15 \amp 11 \end{bmatrix} \end{align*}

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Theorem 1.2.8.

Let $A\text{,}$ $B\text{,}$ and $C$ be $m \times n$ matrices.

$A + B = B + A\text{,}$ i.e., matrix addition is commutative.
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$A + (B + C) = (A + B) + C\text{,}$ i.e., matrix addition is associative.
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There is a unique $m \times n$ matrix $O$ such that $A + O = A$ for any $m \times n$ matrix $A\text{.}$ The matrix $O$ is called the $m \times n$ zero matrix, and is the matrix with zeros in every entry.
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For each $m \times n$ matrix $A\text{,}$ there is a unique $m \times n$ matrix $D$ such that $A + D = O\text{.}$ The matrix $D$ must be the matrix $-A = (-1)A\text{.}$ The matrix $-A$ is called the negative of $A\text{.}$
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Let $r$ and $s$ be real numbers. Then

$r(sA) = (rs)A\text{.}$
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$(r+s)A = rA + sA\text{.}$
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$r(A+B) = rA + rB\text{.}$
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Why is this true?.

We prove Property 1 only, i.e., the commutativity of addition. Let $A = [a_{ij}]$ and $B = [b_{ij}]\text{.}$ Then:

\begin{align*} A + B \amp = [a_{ij} + b_{ij}]\\ \amp = [b_{ij} + a_{ij}] \amp \text{(since real numbers are commutative)}\\ \amp= B + A \end{align*}

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Definition 1.2.9.

If $A = [a_{ij}]$ is an $m \times n$ matrix, then the transpose of $A\text{,}$ denoted $A^T = [a_{ij}^T]\text{,}$ is the $n \times m$ matrix defined by

\begin{equation*} a_{ij}^T = a_{ji} \end{equation*}

In other words, the transpose of $A$ is obtained by interchanging the rows and the columns of $A\text{.}$

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Activity 1.2.6.

Compute the transpose for each of the given matrices:

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(a)

\begin{equation*} A = \begin{bmatrix} 1 \amp 2 \amp 3 \\ 5 \amp -8 \amp -13 \end{bmatrix}. \end{equation*}

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Solution.

\begin{equation*} A^T = \begin{bmatrix} 1 \amp 5 \\ 2 \amp -8 \\ 3 \amp -13 \end{bmatrix} \end{equation*}

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(b)

\begin{equation*} B = \begin{bmatrix} 5 \amp 2 \amp 3 \\ 6 \amp 2 \amp 3 \\ -1 \amp -2 \amp 3 \end{bmatrix} \end{equation*}

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Solution.

\begin{equation*} B^T = \begin{bmatrix} 5 \amp 6 \amp -1 \\ 2 \amp 2 \amp -2 \\ 3 \amp 3 \amp 3 \end{bmatrix} \end{equation*}

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(c)

\begin{equation*} C = \begin{bmatrix} 10 \\ 20 \\ 30 \end{bmatrix}. \end{equation*}

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Solution.

\begin{equation*} C^T = \begin{bmatrix} 10 \amp 20 \amp 30 \end{bmatrix} \end{equation*}

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Note 1.2.10.

Observe from the previous activity that, when transposed, a column vector becomes a row vector. And vice versa.

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Definition 1.2.11. Main diagonal.

If $A=[a_{ij}]$ is an $m\times n$ matrix, the elements $a_{11}, a_{22}, a_{33},\ldots$ are called the main diagonal of $A\text{.}$ A matrix $A$ is called diagonal if its only nonzero entries occur on its main diagonal.

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Below are four matrices of various dimensions, with the main diagonal written in bold font.

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\begin{equation*} \begin{bmatrix} \mathbf{a_{11}} \amp a_{12} \\ a_{21} \amp \mathbf{a_{22}} \\ a_{31} \amp a_{32} \end{bmatrix} \end{equation*}

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\begin{equation*} \begin{bmatrix} \mathbf{a_{11}} \amp a_{12} \amp a_{13} \\ a_{21} \amp \mathbf{a_{22}} \amp a_{23} \end{bmatrix} \end{equation*}

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\begin{equation*} \begin{bmatrix} \mathbf{a_{11}} \amp a_{12} \amp a_{13} \\ a_{21} \amp \mathbf{a_{22}} \amp a_{23} \\ a_{31} \amp a_{32} \amp \mathbf{a_{33}} \end{bmatrix} \end{equation*}

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\begin{equation*} \begin{bmatrix} \mathbf{a_{11}} \\ a_{21} \end{bmatrix} \end{equation*}

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Forming the transpose of a matrix $A$ can be viewed as flipping $A$ about its main diagonal.

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Theorem 1.2.12.

If $r$ is a scalar and $A$ and $B$ are matrices of the appropriate sizes, then:

$(A^T)^T = A\text{.}$
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$(A + B)^T = A^T + B^T\text{.}$
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$(rA)^T = rA^T\text{.}$
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Why is this true?.

Proof of property 2:

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Let $A = [a_{ij}]$ and $B = [b_{ij}]\text{.}$ Then $A + B = [c_{ij}]$ where $c_{ij} = a_{ij} + b_{ij}\text{.}$

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Then

\begin{align*} (A + B)^T \amp = [c_{ij}^T]\\ \amp = [c_{ji}] \amp \text{By definition of transpose}\\ \amp = [a_{ji} + b_{ji}] \amp \text{Since $c_{ij} = a_{ij} + b_{ij}$}\\ \amp = [a_{ji}] + [b_{ji}] \amp \text{By definition of matrix addition}\\ \amp = A^T + B^T \amp \text{By definition of transpose} \end{align*}

Therefore, $(A + B)^T = A^T + B^T\text{.}$

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Definition 1.2.13. Symmetry and skew-symmetry.

A matrix $A$ with real entries is called:

Symmetric if $A^T = A\text{.}$

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Skew-symmetric if $A^T = -A\text{.}$

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Warning 1.2.14.

The previous definition only makes sense if the matrix $A$ is square, i.e., if it has the same number of rows and columns.

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Activity 1.2.7.

Determine whether the following matrices are symmetric, skew symmetric, or neither:

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(a)

$A = \begin{bmatrix} 0 \amp 2 \amp -3 \\ -2 \amp 0 \amp 5 \\ 3 \amp -5 \amp 0 \end{bmatrix}$

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Solution.

$A$ is skew symmetric since $A^T = -A\text{.}$

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(b)

$B = \begin{bmatrix} 3 \amp 5 \amp 2 \\ 5 \amp 1 \amp 4 \\ 2 \amp 4 \amp -1 \end{bmatrix}$

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Solution.

$B$ is symmetric since $B^T = B\text{.}$

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(c)

$C = \begin{bmatrix} 1 \amp 2 \amp -3 \\ -2 \amp 0 \amp 5 \\ 3 \amp 5 \amp 0 \end{bmatrix}$

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Solution.

$C$ is neither symmetric nor skew symmetric.

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(d)

$D = \begin{bmatrix} 0 \amp 0 \amp 0 \\ 0 \amp 0 \amp 0 \\ 0 \amp 0 \amp 0 \end{bmatrix}$

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Solution.

$D$ is both symmetric and skew symmetric.

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Note 1.2.15.

If a matrix $A$ is both symmetric and skew symmetric, then $A$ must be a zero matrix!

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Note 1.2.16. Shape habit.

Before adding, multiplying, transposing, or applying a matrix, first identify its shape. A $2\times 3$ matrix has two rows and three columns. Matrix addition requires the same shape. Matrix-vector multiplication and matrix multiplication require compatible dimensions.

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Prev Top Next

Object	Meaning of an entry
Monochrome image	\(X_{ij}\) is the pixel value in row \(i\) and column \(j\text{.}\)
Rainfall data	\(A_{ij}\) is the rainfall at location \(j\) on day \(i\text{.}\)
Asset returns	\(R_{ij}\) is the return of asset \(i\) in period \(j\text{.}\)
Feature matrix	\(X_{ij}\) is the value of feature \(j\) for entity \(i\text{.}\)