MATH 323 Lecture 4

« previous | Thursday, September 6, 2012 | next »

Matrices (cont'd)

Constructor:

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Mat}_n(\mathbb{R})}
2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{M}_n(\mathbb{R})}
3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \, B = C}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_{ij} = \sum_{t=1}^n a_{it} \, b_{tj} = \vec{a}(i,:) \cdot \vec{b}(:,j)}

NOT commutative: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AB \ne BA}

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

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A \, B) \, C = A \, (B \, C) = A \, B \, C}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{ a_{ij} \mid i = j \right\}} )

A special form of diagonal matrices is when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} entries (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_1} through Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_n} ) can be represented by a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \times n} adjacency matrix:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A : a_{ij} = \begin{cases} 1 & v_i \mbox{ is connected with } v_j \\ 0 & \mbox{otherwise}\end{cases}}

First, let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{(k)}_{ij}} be the (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} ) entry of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A^k} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{(k)}_{ij}} represents the number of walks of length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_i} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_j} .

Theorem 1.3.2

For all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha, \beta \in \mathbb{R}} and for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A,B,C} , the indicated operations are defined:

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A+B = B+A} (commutativity of addition)
2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A+B)+C = A+(B+C)} (associativity of addition)
3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A \, B) \, C = A \, (B \, C)} (associativity of multiplication)
4. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \, (B+C) = A \, B + A \, C} (right distributivity)
5. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A+B) \, C = A \, C + B \, C} (left distributivity)
6. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\alpha \, \beta)A = \alpha(\beta \, A)}
7. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha \, (A \, B) = (\alpha \, A) \, B = A \, (\alpha \, B)}
8. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\alpha + \beta) \, A = \alpha \, A + \beta \, A}
9. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha \, (A+B) = \alpha \, A + \alpha \, B}

Theorem 1.3.3

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B} are nonsingular Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \times n} matrices, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \, B} is also nonsingular and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A \, B)^{-1} = B^{-1} \, A^{-1}}

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

Corollary

For nonsingular matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_1, \ldots, A_k} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_1 \, A_2 \, \dots \, A_k} is also nonsingular and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A_1 \, A_2 \, \dots \, A_k)^{-1} = A_k^{-1} \, \dots \, A_2^{-1} \, A_1^{-1}}