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Matrix: Difference between revisions

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== Transposes ==
== Transposes ==


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


<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>.
<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>.

Revision as of 22:05, 4 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 $A=-A^*$.

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