Matrix

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A matrix over a field $F$ is a function from $A\times B$ to $F$, where $A$ and $B$ are the sets $A=\{1,2,\ldots,m\}$ and $B=\{1,2,\ldots,n\}$. A matrix is usually represented as a rectangular array of scalars from the field, such that each column belongs to the vector space $F^m$, where $m$ is the number of rows. If a matrix $A$ has $m$ rows and $n$ columns, its order is said to be $m \times n$, and it is written as $A_{m \times n}$.

The element in the $i^{th}$ row and $j^{th}$ column of $A$ is written as $(A)_{ij}$. It is more often written as $a_{ij}$, in which case $A$ can be written as $[a_{ij}]$.

Determinant

If $A_{m\times n}$ is a matrix over $F$ with $m=n$, a Determinant assigns $A_{m\times n}$ to a member of $F$ and is denoted by $|A|$ or $\begin{vmatrix} a_{11} & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nn}\end{vmatrix}$

It is defined recursively.

$\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix}\dot{=}a_{11} a_{22} - a_{21} a_{12}$ $\begin{vmatrix} a_{11} & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nn}\end{vmatrix}\dot{=}\sum_{k=1}^n (-1)^{k+1} a_{1k} |A'_{1k}|$

where $A'_{cd}$ is the matrix $A$ with the $c^{th}$ row and $d^{th}$ column removed.

Transposes

Let $A$ be $[a_{ij}]$. Then $[a_{ji}]$ is said to be the transpose of $A$, written as $A^T$ or simply $A'$. If A is over the complex field, replacing each element of $A^T$ by its complex conjugate gives us the conjugate transpose $A^*$ of $A$. In other words, $A^*=[\bar {a_{ji}}]$

$A$ is said to be symmetric if and only if $A=A^T$. $A$ is said to be hermitian if and only if $A=A^*$. $A$ is said to be skew symmetric if and only if $A=-A^T$. $A$ is said to be skew hermitian if and only if $A=-A^*$.

Matrix Product

Let $A$ be a matrix of order $m_1 \times m_2$ and $B$ a matrix of order $n_1 \times n_2$. Then the product $AB$ exists if and only if $m_2=n_1$ and in that case we define the product $C=AB$ as the matrix of order $m_1 \times n_2$ for which \[(C)_{ij}=\sum ^{n_1} _{k=1} (A)_{ik} (B)_{kj}\] for all $i$ and $j$ such that $1\le i\le m_1$ and $1\le j\le n_2$.

Vector spaces associated with a matrix

As already stated before, the columns of $A$ form a subset of $F^m$. The subspace of $F^m$ generated by these columns is said to be the column space of $A$, written as $C(A)$. Similarly, the transposes of the rows form a subset of the vector space $F^n$. The subspace of $F^n$ generated by these is known as the row space of $A$, written as $R(A)$.

$y \in C(A)$implies $\exists x$ such that $y_{m \times 1} = A_{m \times n} x_{n \times 1}$

Similarly, $y \in C(A)$implies $\exists x$ such that $y_{n \times 1} = A^T_{n \times m} x_{m \times 1}$

The set $\{x:A_{m \times n}x_{n \times 1} = \phi\}$ forms a subspace of $F^n$, known as the null space $N(A)$ of $A$.

Rank and nullity

The dimension of $C(A)$ is known as the column rank of $A$. The dimension of $R(A)$ is known as the row rank of $A$. These two ranks are found to be equal, and the common value is known as the rank $r(A)$ of $A$.

The dimension of $N(A)$ is known as the nullity $\eta (A)$ of A.

If $A$ is a square matrix of order $n \times n$, then $r(A) + \eta (A) = n$.

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