Difference between revisions of "Matrix"

Revision as of 22:02, 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 $A$ .

$A$ 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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 A matrix is a rectangular array of scalars from any [[field]], such that each column belongs to the [[vector space]] <math>F^m</math>, where <math>m</math> is the number of rows. If a matrix <math>A</math> has <math>m</math> rows and <math>n</math> columns, its order is said to be <math>m \times n</math>, and it is written as <math>A_{m \times n}</math>.
+The element in the <math>i^{th}</math> row and <math>j^{th}</math> column of <math>A</math> is written as <math>(A)_{ij}</math>. It is more often written as <math>a_{ij}</math>, in which case <math>A</math> can be written as <math>[a_{ij}]</math>.
+== 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>.
+<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>.

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Difference between revisions of "Matrix"

Revision as of 22:02, 4 November 2006

Transposes