Difference between revisions of "Matrix"

Revision as of 14:48, 5 November 2006

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 $A=-A^*$ .

Matrix Product

If $A$ is of order $m_1 \times n$ and $B$ is of order $n \times m_2$ , $C_{m_1 \times m_2}$ is said to be $AB$ if and only if $(C)_{ij}=\displaystyle \sum ^n _{k=1} (A)_{ik} (B)_{kj}$

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 $r(A) + \eta (A) = n$ .

@@ Line 8: / Line 8: @@
 <math>A</math> is said to be symmetric if and only if <math>A=A^T</math>. <math>A</math> is said to be hermitian if and only if <math>A=A^*</math>. <math>A</math> is said to be skew symmetric if and only if <math>A=-A^T</math>. <math>A</math> is said to be skew hermitian if and only if <math>A=-A^*</math>.
+== Matrix Product ==
+If <math>A</math> is of order <math>m_1 \times n</math> and <math>B</math> is of order <math>n \times m_2</math>, <math>C_{m_1 \times m_2}</math> is said to be <math>AB</math> if and only if <math>(C)_{ij}=\displaystyle \sum ^n _{k=1} (A)_{ik} (B)_{kj}</math>
 == Vector spaces associated with a matrix ==

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