Matrix

A matrix is a rectangular array of scalars from any 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 $[a_{ij}]$ .

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 $A=-A^*$ .

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 $y_{n \times 1} = A^T_{n \times m} x_{m \times 1}$

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Vector spaces associated with a matrix

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