Matrices

Matrix

A matrix is a collection of numbers that can be indexed in a particular way, this indexing mimics a two dimensional grid. The dimensions of a matrix are stated as $r \times c$ where $r, c \in ℕ_{1}$ when it has $m$ rows and $n$ columns.

Suppose that $A$ is a matrix, then we can access the value at the $i$ -th row and $j$ -th column by writing $A_{i, j}$ , we can graphically represent this information by introducing variables for the coefficients and writing them like this:

A = [\begin{matrix} a_{1, 1} & a_{1, 2} & \dots & a_{1, n} \\ a_{2, 1} & a_{2, 2} & \dots & a_{2, n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m, 1} & a_{m, 2} & \dots & a_{m, n} \end{matrix}]

Matrices are denoted by capital letters such as $A, B, M, N$ .

Set of Matrices

Suppose that

r, c \in ℕ_{1}

then we denote the set of all

r \times c

matrices with cofficients from the set

T

M_{r \times c} (T)

Row Matrix

Any matrix

R \in M_{1 \times c} (T)

for some

c \in ℕ_{1}

is known as a row matrix.

Row of a Matrix

Suppose that $A$ is a matrix, then if we want to extract the $i - t h$ row of $A$ we can write $A_{i, -}$ .

Note that this is a function of the form $M_{r \times c} (T) \to M_{1 \times c} (T)$

The notation

A_{i, -}

intends to mimic freezing at a certain row and then sweeping over all columns.

Row Matrix Representation

Given a matrix

A

, there is an associated row matrix representation, of the following form

A = [\begin{matrix} \leftarrow & A_{1, -} & \to \\ \leftarrow & A_{2, -} & \to \\ \leftarrow & ⋮ & \to \\ \leftarrow & A_{m, -} & \to \end{matrix}]

Where the arrows are there to remind you that they expand horizontally. Also observe that this matrix is an element of $M_{r \times 1} (M_{1 \times c} (ℝ))$

Column Matrix

Any matrix

C \in M_{r \times 1} (T)

for some

r \in ℕ_{1}

is known as a row matrix.

We will denote column or row matrices with a lower case boldfaced character, like $𝐱$

Column of a Matrix

Suppose that $A$ is a matrix, then if we want to extract the $j$ -th row of $A$ we can write $A_{|, j}$ .

Note that this is a function of the form $M_{r \times c} (T) \to M_{r \times 1} (T)$

The

|

denotes that the row variable may be varied vertically, while the column index is fixed in place. If we want to represent the matrix as columns graphically, we may do so like this

Column Matrix Representation

Given a matrix

A

, there is an associated column matrix representation, of the following form

A = [\begin{matrix} ↑ & ↑ & ↑ & ↑ \\ A_{|, 1} & A_{|, 2} & \dots & A_{|, n} \\ ↓ & ↓ & ↓ & ↓ \end{matrix}]

Where the arrows are there to remind you that they expand graphically expand vertically. Also observe that this matrix is an element of $M_{1 \times c} (M_{r \times 1} (ℝ))$ .

We can connect some of the previous definitions here to say that

R

is a column matrix whose entries are row matrices.

Matrix Multiplication

Suppose $r_{A}, c_{A}, r_{B}, c_{B} \in ℕ_{1}$ such that $c_{A} = r_{B}$ and that we have two matrices $A, B \in M_{r_{A} \times c_{A}} (ℝ), M_{r_{B} \times c_{B}} (ℝ)$ , defined as

A = [\begin{matrix} a_{1, 1} & a_{1, 2} & \dots & a_{1, c_{A}} \\ a_{2, 1} & a_{2, 2} & \dots & a_{2, c_{A}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{r_{A}, 1} & a_{r_{A}, 2} & \dots & a_{r_{A}, c_{A}} \end{matrix}], B = [\begin{matrix} b_{1, 1} & b_{1, 2} & \dots & b_{1, c_{B}} \\ b_{2, 1} & b_{2, 2} & \dots & b_{2, c_{B}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ b_{r_{B}, 1} & b_{r_{B}, 2} & \dots & b_{r_{B}, c_{B}} \end{matrix}]

Then we define the matrix multiplication of $A$ and $B$ as the matrix $P \in M_{r_{A} \times c_{B}}$

P = [\begin{matrix} c_{1, 1} & c_{1, 2} & \dots & c_{1, c_{P}} \\ c_{2, 1} & c_{2, 2} & \dots & c_{2, c_{P}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ c_{r_{P}, 1} & c_{r_{P}, 2} & \dots & c_{r_{P}, c_{P}} \end{matrix}]

Where:

c_{i, j} : = a_{i, 1} b_{1, j} + a_{i, 2} b_{2, j} + \dots + a_{i, c_{A}} b_{c_{A}, j} = \sum_{k = 1}^{c_{A}} a_{i, k} b_{k, j}

Matrix Functions

Suppose that

𝐌 \in M_{r, c} (T)

and that

f : T \to S

is a function, then we augment this function to

𝐟 : M_{r, c} (T) \to M_{r, c} (S)

as follows:

𝐟 ([\begin{matrix} m_{1, 1} & m_{1, 2} & \dots & m_{1, c_{M}} \\ m_{2, 1} & m_{2, 2} & \dots & m_{2, c_{M}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ m_{r_{M}, 1} & m_{r_{M}, 2} & \dots & m_{r_{M}, c_{M}} \end{matrix}]) = [\begin{matrix} f (m_{1, 1}) & f (m_{1, 2}) & \dots & f (m_{1, c_{M}}) \\ f (m_{2, 1}) & f (m_{2, 2}) & \dots & f (m_{2, c_{M}}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ f (m_{r_{M}, 1}) & f (m_{r_{M}, 2}) & \dots & f (m_{r_{M}, c_{M}}) \end{matrix}]

Packing Function

Suppose that

x \in ℝ

, then we define the following function

pack : ℝ \to M_{1 \times 1} (ℝ)

pack (x) : = [\begin{matrix} x \end{matrix}]

Unpacking Function

Suppose that

[\begin{matrix} x \end{matrix}] \in M_{1 \times 1} (ℝ)

, then we define the following function

unpack : ℝ \to M_{1 \times 1} (ℝ)

unpack ([\begin{matrix} x \end{matrix}]) : = x

The Packing and Unpacking Functions are Inverses

TODO

One by one Matrices are Isomorphic to R

ℝ

and

M_{1, 1} (ℝ)

are isomorphic.

Row and Column Matrices are Isomorphic to

ℝ^{n}

Suppose that

n \in ℕ_{1}

, then

M_{1, n} (ℝ)

and

M_{n, 1} (ℝ)

are isomorphic to

ℝ^{n}

Nested Matrix Multiplication Unpacking

Given two matrices $A \in M_{a \times b} (ℝ) B \in M_{c \times d} (ℝ)$ then product is equal to

unpack ([\begin{matrix} A_{1, -} \cdot B_{|, 1} & A_{1, -} \cdot B_{|, 2} & \dots & A_{1, -} \cdot B_{|, d} \\ A_{2, -} \cdot B_{|, 1} & A_{2, -} \cdot B_{|, 2} & \dots & A_{2, -} \cdot B_{|, d} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ A_{a, -} \cdot B_{|, 1} & A_{a, -} \cdot B_{|, 2} & \dots & A_{a, -} \cdot B_{|, d} \end{matrix}])

With $A_{i, -}$ $B_{|, j}$ and $unpack$ is being used as a matrix function

Matrix Multiplication with Rows and Columns

Suppose that $A, B$ are matrices, and that $A_{R}, B_{C}$ are their row matrix and column matrix representations, then

unpack (A_{R} B_{C}) = A B

where

unpack

is being used as a matrix function

column matrix multiplication

suppose that

C = [\begin{matrix} c_{1} \\ c_{2} \\ ⋮ \\ c_{k} \end{matrix}]

is a column matrix and

M \in M_{n \times k} (ℝ)

, then

M C = \sum_{i = 1}^{k} c_{i} M_{|}, i} \in M_{n \times 1} (ℝ)

By the definition of matrix multiplication the answer should be equal to

[\begin{matrix} m_{1, 1} & m_{1, 2} & \dots & m_{1, k} \\ m_{2, 1} & m_{2, 2,} & \dots & m_{2, k} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ m_{j, 1} & m_{j, 2} & \dots & m_{j, k} \end{matrix}]

[\begin{matrix} c_{1} \\ c_{2} \\ ⋮ \\ c_{k} \end{matrix}]

= [\begin{matrix} c_{1} \cdot m_{1, 1} + c_{2} \cdot m_{1, 2} + \dots + c_{k} \cdot m_{1, k} \\ c_{1} \cdot m_{2, 1} + c_{2} \cdot m_{2, 2} + \dots + c_{k} \cdot m_{2, k} \\ ⋮ \\ c_{1} \cdot m_{j, 1} + c_{2} \cdot m_{j, 2} + \dots + c_{k} \cdot m_{j, k} \end{matrix}]

= c_{1} M_{|}, 1} + c_{2} M_{|}, 2} + \dots + c_{k} M_{|}, k} \in M_{n \times 1} (ℝ)

unit column matrix

e_{i}

is a column matrix who's i-th row is

1

and the rest are

0

column extraction

suppose

M \in M_{j \times k}

then the

n

-th column of

M

is given by

M

multiplied by

e_{n}

(where

1 \leq n \leq k

)

Due to the way that column matrix multiplication works, we can see that $M e_{n} = \sum_{i = 0}^{k} {(e_{n})}_{i} M_{|}, i} = M_{|}, n}$ this is clear since only the $n$ -th component of $e_{n}$ is $1$ and the rest are $0$ .

Complex Transpose

We denote the complex transpose of a matrix as

A^{*} : = {\overline{A}}^{T}

Hermitian Matrix

A matrix

A \in M_{n \times n} (ℂ)

is said to be hermitian diff

A = A^{*}

where we've applied the complex-transpose

Note that the a Hermitian Matrix is also called symmetric or self-adjoint.

Normal

We say that a matrix is normal if

A^{*} A = A A^{*}