MATH 323 Lecture 4

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Matrices (cont'd)

Constructor:

1. ${\mathrm{M}\mathrm{a}\mathrm{t}}_n(\mathbb{ℝ})$
2. ${\mathrm{M}}_n(\mathbb{ℝ})$
3. ${\mathrm{M}}_{m\times n}(\mathbb{ℝ})$

1 and 2 are called a ring because they have the + and · operations defined on them since they are square matrices.

Multiplication (cont'd)

$AB=C$

$c_{ij}=\sum _{t=1}^n{a}_{it}{b}_{tj}=\overrightarrow{a}(i,:)\cdot \overrightarrow{b}(:,j)$

NOT commutative: $AB\ne BA$

Due to associativity of multiplication in #Theorem 1.3.2, we do not write parentheses around multiplication:

• $(AB)C=A(BC)=ABC$
• $A_1{A}_2\dots A_n$

Exponentiation

$A^n=\underset{n}{\underbrace{A\cdots A}}$ for integer $n$

Identity Matrix

Plays the role of "1" in multiplication:

$I=I_n=({\delta }_{ij})$, where ${\delta }_{ij}=\{\begin{array}{ll}1& i=j\\ 0& i\ne j\end{array}$

(In other words, 1s along diagonal, 0s everywhere else) ...such that

$AI=IA=A$

Standard Basis

Acts as an identity vector:

$I=({\overrightarrow{e}}_1,\dots ,\overrightarrow{e},n)$, where ${\overrightarrow{e}}_i$ is a vector of 0s, except the $i$^th element is 1.

Zero Matrix

Plays role of 0 in addition:

$0=\left[\begin{array}{ccc}0& \dots & 0\\ ⋮& \ddots & ⋮\\ 0& \dots & 0\end{array}\right]$

...such that

$A+0=0+A=A$

Inverse Matrix

If $AB=BA=I_n$ for $n\times n$ matrices $A$ and $B$, we say than $B$ is the inverse of $A$: $B=A^{-1}$.

Not every matrix is invertible. A matrix that does not have a multiplicative inverse is said to be singular.

Transpose of a Matrix

For a $m\times n$ matrix $A=(a_{ij})$, the transpose of $A$, written $A^T$, will be $n\times m$ and is defined as follows:

$A^T=(a_{ij}^T)$, where $a_{ij}^T=a_{ji}$

Geometrically, the first row becomes the first column, second row becomes second column, etc. The matrix is simply "reflected" about its "diagonal"

Rules:

1. $(A^T{)}^T=A$
2. $(\alpha A{)}^T=\alpha A^T$, $\alpha \in \mathbb{ℝ}$
3. $(A+B{)}^T=A^T+B^T$
4. $(AB{)}^T=B^T{A}^T$ (opposite order)

A $n\times n$ matrix $A$ is said to be symmetric iff $A^T=A$. This means that $a_{ij}=a_{ji}$

Example

$\begin{array}{rl}A& =\left[\begin{array}{cc}1& 2\\ 3& 0\\ 5& 3\end{array}\right]\\ A^T& =\left[\begin{array}{ccc}1& 3& 5\\ 2& 0& 3\end{array}\right]\end{array}$

Diagonal Matrix

A diagonal matrix is a square ($n\times n$) matrix that has values along its diagonal ($\{a_{ij}\mid i=j\}$)

A special form of diagonal matrices is when $D=\alpha I$ (i.e. all numbers along diagonal are the same): these special diagonal matrices commute with arbitrary matrices

Matrices and Graphs

A graph consists of vertices (data points) and edges that connect them.

A graph with $n$ entries ($v_1$ through $v_n$) can be represented by a $n\times n$ adjacency matrix:

$A:a_{ij}=\{\begin{array}{ll}1& v_i\text{ is connected with }{v}_j\\ 0& \text{otherwise}\end{array}$

First, let $a_{ij}^{(k)}$ be the ($i$, $j$) entry of $A^k$.

$a_{ij}^{(k)}$ represents the number of walks of length $k$ from $v_i$ to $v_j$.

Theorem 1.3.2

For all $\alpha ,\beta \in \mathbb{ℝ}$ and for all $A,B,C$, the indicated operations are defined:

1. $A+B=B+A$ (commutativity of addition)
2. $(A+B)+C=A+(B+C)$ (associativity of addition)
3. $(AB)C=A(BC)$ (associativity of multiplication)
4. $A(B+C)=AB+AC$ (right distributivity)
5. $(A+B)C=AC+BC$ (left distributivity)
6. $(\alpha \beta )A=\alpha (\beta A)$
7. $\alpha (AB)=(\alpha A)B=A(\alpha B)$
8. $(\alpha +\beta )A=\alpha A+\beta A$
9. $\alpha (A+B)=\alpha A+\alpha B$

Theorem 1.3.3

If $A$ and $B$ are nonsingular $n\times n$ matrices, then $AB$ is also nonsingular and $(AB{)}^{-1}=B^{-1}{A}^{-1}$

(note the opposite order on the right-hand side)

Corollary

For nonsingular matrices $A_1,\dots ,A_k$, $A_1{A}_2\dots A_k$ is also nonsingular and $(A_1{A}_2\dots A_k{)}^{-1}=A_k^{-1}\dots A_2^{-1}{A}_1^{-1}$