Mathematics Lecture Notes

Definition3.1Matrix

An $m \times n$ matrix over a field $F$ is a rectangular array $A = (a_{ij})$ with $m$ rows and $n$ columns, where $a_{ij} \in F$ . The entry in row $i$ and column $j$ is denoted $a_{ij}$ or $A_{ij}$ or $(A)_{ij}$ .

Definition3.2Matrix addition and scalar multiplication

For $A, B \in M_{m \times n}(F)$ and $c \in F$ :

$(A + B)_{ij} = a_{ij} + b_{ij}, \quad (cA)_{ij} = c \cdot a_{ij}.$

With these operations, $M_{m \times n}(F)$ is a vector space of dimension $mn$ .

Definition3.3Matrix multiplication

For $A \in M_{m \times n}(F)$ and $B \in M_{n \times p}(F)$ , the product $AB \in M_{m \times p}(F)$ is defined by

$(AB)_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}.$

The $(i, j)$ entry of $AB$ is the dot product of the $i$ -th row of $A$ with the $j$ -th column of $B$ .

ExampleProduct of 2 x 2 matrices

$\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} = \begin{pmatrix} 1 \cdot 5 + 2 \cdot 7 & 1 \cdot 6 + 2 \cdot 8 \\ 3 \cdot 5 + 4 \cdot 7 & 3 \cdot 6 + 4 \cdot 8 \end{pmatrix} = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}.$

ExampleMatrix multiplication is not commutative

$AB = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \quad BA = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}.$

$AB \neq BA$ . In fact, $AB \neq 0$ but $BA = 0$ : the product of two nonzero matrices can be zero.

ExampleMatrix-vector product

$\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} x + 2y + 3z \\ 4x + 5y + 6z \end{pmatrix} = x\begin{pmatrix} 1 \\ 4 \end{pmatrix} + y\begin{pmatrix} 2 \\ 5 \end{pmatrix} + z\begin{pmatrix} 3 \\ 6 \end{pmatrix}.$

$Ax$ is a linear combination of the columns of $A$ with coefficients from $x$ . This is the column picture of matrix-vector multiplication.

ExampleIdentity matrix

The $n \times n$ identity matrix $I_n$ has $1$ 's on the diagonal and $0$ 's elsewhere: $(I_n)_{ij} = \delta_{ij}$ (the Kronecker delta). For any $A \in M_{m \times n}(F)$ :

$I_m A = A = A I_n.$

ExampleDiagonal matrices multiply componentwise

If $D_1 = \text{diag}(d_1, \ldots, d_n)$ and $D_2 = \text{diag}(e_1, \ldots, e_n)$ , then

$D_1 D_2 = \text{diag}(d_1 e_1, \ldots, d_n e_n).$

Diagonal matrices always commute with each other.

ExampleMatrix powers

For a square matrix $A$ , $A^k = \underbrace{A \cdot A \cdots A}_{k \text{ times}}$ and $A^0 = I$ .

$\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}^n = \begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix}$

(proved by induction).

ExampleNilpotent matrix

$N = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ satisfies $N^2 = 0$ . A matrix $N$ with $N^k = 0$ for some $k$ is called nilpotent. The smallest such $k$ is the index of nilpotency.

ExampleIdempotent matrix

$P = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ satisfies $P^2 = P$ . Such a matrix is called idempotent and represents a projection.

ExampleBlock multiplication

Block (partitioned) matrices can be multiplied "as if blocks were scalars," provided dimensions match:

$\begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}\begin{pmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{pmatrix} = \begin{pmatrix} A_{11}B_{11} + A_{12}B_{21} & A_{11}B_{12} + A_{12}B_{22} \\ A_{21}B_{11} + A_{22}B_{21} & A_{21}B_{12} + A_{22}B_{22} \end{pmatrix}.$

ExampleOuter product

For column vectors $u \in F^m$ and $v \in F^n$ , the outer product $uv^T$ is an $m \times n$ matrix of rank at most 1:

$(uv^T)_{ij} = u_i v_j.$

Every rank-1 matrix has this form.

ExampleTranspose properties

For the transpose $A^T$ (with $(A^T)_{ij} = a_{ji}$ ):

$(A + B)^T = A^T + B^T$
$(cA)^T = cA^T$
$(AB)^T = B^T A^T$ (order reverses)
$(A^T)^T = A$

ExampleMatrix product = composition of linear maps

If $T : F^p \to F^n$ has matrix $A$ and $S : F^n \to F^m$ has matrix $B$ , then $S \circ T : F^p \to F^m$ has matrix $BA$ . Matrix multiplication is composition.

Definition3.4Matrix of a linear transformation

Let $T : V \to W$ be linear, $\beta = \{v_1, \ldots, v_n\}$ a basis for $V$ , and $\gamma = \{w_1, \ldots, w_m\}$ a basis for $W$ . The matrix of $T$ with respect to $\beta$ and $\gamma$ is the $m \times n$ matrix $[T]_\beta^\gamma$ whose $j$ -th column is $[T(v_j)]_\gamma$ , the coordinate vector of $T(v_j)$ with respect to $\gamma$ .

ExampleMatrix representation example

$T : \mathbb{R}^2 \to \mathbb{R}^3$ , $T(x, y) = (x + y, x - y, 2x)$ , with standard bases.

$T(e_1) = (1, 1, 2)$ and $T(e_2) = (1, -1, 0)$ , so $[T] = \begin{pmatrix} 1 & 1 \\ 1 & -1 \\ 2 & 0 \end{pmatrix}$ .

RemarkChange of basis

If $Q$ is the change-of-basis matrix from $\beta'$ to $\beta$ and $P$ is the change-of-basis matrix from $\gamma$ to $\gamma'$ , then

$[T]_{\beta'}^{\gamma'} = P [T]_\beta^\gamma Q.$

For a linear operator $T : V \to V$ with two bases $\beta$ and $\beta'$ related by $Q$ :

$[T]_{\beta'} = Q^{-1} [T]_\beta Q.$

Matrices related by $B = Q^{-1}AQ$ are called similar and represent the same linear operator in different bases.

RemarkLooking ahead

The question of finding the "simplest" matrix for a given linear operator (via a good choice of basis) leads to eigenvalues, diagonalization, and ultimately Jordan normal form.

Math Notes

Matrix Operations

Definitions

Examples

Matrix representation of linear maps