**Matrix**
([edit](http://github.com/nicebyte/wiki/edit/master/Wiki/Matrix.md))
# Definition
A _matrix_ is a rectangular array of values. The values can be of any nature:
real or complex numbers, or something else entirely - as long as they belong to
an algebraic Ring. The values are referred to as the _elements_ or
_coefficients_ of the matrix.
# Matrix Operations
## Addition
If $\boldsymbol{A}$ and $\boldsymbol{B}$ are both $m \times n$ matrices, their
sum $\boldsymbol{C} = \boldsymbol{A} + \boldsymbol{B}$ is defined as a matrix
where each element is equal to the sum of the corresponding elements of
$\boldsymbol{A}$ and $\boldsymbol{B}$.
## Multiplication
Given an $n \times m$ matrix $\boldsymbol{A}$ and a scalar value $\alpha$,
$\boldsymbol{C} = \alpha \boldsymbol{A}$ is the matrix where each element is
equal to the corresponding element of $\boldsymbol{A}$ multiplied by $\alpha$.
Given an $n \times m$ matrix $\boldsymbol{A}$ and an $m \times k$ matrix
$\boldsymbol{B}$, their product is defined as the matrix
$\boldsymbol{C} = \boldsymbol{A}\boldsymbol{B}$ where each coefficient is equal
to the inner product of the corresponding row from $\boldsymbol{A}$ and column
from $\boldsymbol{B}$. Note that matrix multiplication is noncommutative, and
requires the second matrix to have as many columns as the first one has rows.
## Transpose
Given a matrix $\boldsymbol{A}$, its _transpose_ $\boldsymbol{A}^T$ is defined
as a matrix obtained from $\boldsymbol{A}$ by rewriting its rows as columns.
## Inverse
### The Identity Matrix
An identity matrix is any $n \times n$ matrix that has unit values along its
main diagonal and zeroes everywhere else. An example of a $2 \times 2$ identity
matrix:
$$
\begin{matrix}
1 && 0 \\
0 && 1
\end{matrix}
$$
Identity matrices are denoted as $\boldsymbol{I}$.
### Definition of the Inverse Matrix
A matrix $\boldsymbol{B}$ is said to be the _inverse_ of $\boldsymbol{A}$ if
$\boldsymbol{AB} = \boldsymbol{BA} = \boldsymbol{I}$.
The inverse of $\boldsymbol{A}$ is denoted $\boldsymbol{A}^{-1}$.
Only square matrices can have an inverse.
### Computing the Inverse
#### Cofactor Matrix
The _cofactor matrix_ of $\boldsymbol{A}$ is the matrix
$$
\begin{matrix}
(-1)^{1+1}\boldsymbol{A}_{11} && (-1)^{1+2}\boldsymbol{A}_{12} &&
(-1)^{1+3}\boldsymbol{A}_{13} && ... \\
(-1)^{2+1}\boldsymbol{A}_{21} && (-1)^{2+2}\boldsymbol{A}_{22} && ... \\
... \\
(-1)^{n+1}\boldsymbol{A}_{n1} && ... && (-1)^{n+n}\boldsymbol{A}_{nn}
\end{matrix}
$$
where $\boldsymbol{A}_{ij}$ are the first-order Minors of $\boldsymbol{A}$
corresponding to the given row and column.
#### Adjoint Matrix
The _adjoint matrix_ of $\boldsymbol{A}$, denoted $adj(\boldsymbol{A})$ is
the transpose of the cofactor matrix.
$$
\boldsymbol{A}^{-1} = \frac{1}{\det \boldsymbol{A}}adj(\boldsymbol{A})
$$
When $\det \boldsymbol{A} = 0$, an inverse does not exist.
# Matrices As Representations of Linear Transforms
A matrix represents a transformation from one linear space to another. The
columns of the matrix are the basis vectors of the source space expressed
in the destination space.
If $\overline{a}$ is a vector in the source space, and the matrix
$\boldsymbol{M}$ represents a transformation from the source space to the
destination space, then $\overline{b}=\boldsymbol{M}\overline{a}$ is
$\overline{a}$ expressed in destination space.
If $\boldsymbol{A}$ represents a transform form space $1$ to space $2$, and
$\boldsymbol{B}$ represents a transform from space $2$ to space $3$, then
$\boldsymbol{C}=\boldsymbol{B}\boldsymbol{A}$ represents a transform from space
$1$ to space $3$.
## A Note on Inverse Matrices
If $\boldsymbol{A}$ represents a transform from space $1$ to space $2$, then its inverse,
$\boldsymbol{A}^{-1}$ represents the transform from space $2$ to space $1$.
## Orthogonal Matrices
If the column vectors of a matrix are all unit length and orthogonal to each
other, the matrix is called _orthogonal_.
If a matrix is orthogonal, its row vectors are also necessarily unit length and
orthogonal to each other.
The transpose of an orthogonal matrix is its inverse.
The Determinant of an orthogonal matrix is $1$. Transforms specified by
orthogonal matrices preserve volume.
Categories: Mathematics, LinearAlgebra