MATH 323 Lecture 4

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Matrices (cont'd)

Constructor:

1. $\mathrm {Mat} _{n}(\mathbb {R} )$
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)

$A\,B=C$

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

NOT commutative: $AB\neq BA$

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

• $(A\,B)\,C=A\,(B\,C)=A\,B\,C$
• $A_{1}\,A_{2}\,\dots \,A_{n}$

Exponentiation

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

Identity Matrix

Plays the role of "1" in multiplication:

$I=I_{n}=(\delta _{ij})$, where $\delta _{ij}={\begin{cases}1&i=j\\0&i\neq j\end{cases}}$

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

$A\,I=I\,A=A$

Standard Basis

Acts as an identity vector:

$I=({\vec {e}}_{1},\ldots ,{\vec {e}},n)$, where ${\vec {e}}_{i}$ is a vector of 0s, except the $i$^th element is 1.

Zero Matrix

Plays role of 0 in addition:

$0={\begin{bmatrix}0&\dots &0\\\vdots &\ddots &\vdots \\0&\dots &0\end{bmatrix}}$

...such that

$A+0=0+A=A$

Inverse Matrix

If $A\,B=B\,A=I_{n}$ 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. $(A\,B)^{T}=B^{T}\,A^{T}$ (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

${\begin{aligned}A&={\begin{bmatrix}1&2\\3&0\\5&3\end{bmatrix}}\\A^{T}&={\begin{bmatrix}1&3&5\\2&0&3\end{bmatrix}}\end{aligned}}$

Diagonal Matrix

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

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:

$A:a_{ij}={\begin{cases}1&v_{i}{\mbox{ is connected with }}v_{j}\\0&{\mbox{otherwise}}\end{cases}}$

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. $(A\,B)\,C=A\,(B\,C)$ (associativity of multiplication)
4. $A\,(B+C)=A\,B+A\,C$ (right distributivity)
5. $(A+B)\,C=A\,C+B\,C$ (left distributivity)
6. $(\alpha \,\beta )A=\alpha (\beta \,A)$
7. $\alpha \,(A\,B)=(\alpha \,A)\,B=A\,(\alpha \,B)$
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 $A\,B$ is also nonsingular and $(A\,B)^{-1}=B^{-1}\,A^{-1}$

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

Corollary

For nonsingular matrices $A_{1},\ldots ,A_{k}$, $A_{1}\,A_{2}\,\dots \,A_{k}$ is also nonsingular and $(A_{1}\,A_{2}\,\dots \,A_{k})^{-1}=A_{k}^{-1}\,\dots \,A_{2}^{-1}\,A_{1}^{-1}$