Matrix ~ ApHrOdi$iaC

Sunday, December 25, 2011

Matrix

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Matrix

A matrix is an arrangement of numbers in rows and columns and enclosed by square brackets

$\begin{bmatrix} 1 & 2 & 3\\ 0 & -1 & 2 \end{bmatrix}_{2\times 3}$

If a matrix has m rows and n columns then it is known as matrix of order m x n. A matrix of order m x n is written as

$\begin{pmatrix} a_{11} &a_{12} &a_{13} &. &. &. &. &a_{1n} \\ a_{21} &a_{22} &a_{23} &. &. &. &. &a_{2n} \\ a_{31} &a_{32} &a_{33} &. &. &. &. &a_{3n}\\ .& .& .& .& . &. &. &. \\ .& .& .& .& . &. &. &.\\ .& .& .& .& . &. &. &.\\ a_{m1} &a_{m2} &a_{m3} &. &. &. &. &a_{mn} \end{pmatrix}$
$=\left(a_{ij} \right ),\text{i = 1, 2 .....m, and j = 1,2,....n }$
$=\left(a_{ij} \right )_{m\times n}$

Standard Matrices

i) Raw Matrix:
A matrix having only one row is called a row matrix.
$=\left(1\right )_{1\times 1}, \begin{pmatrix} 2 & 3 & 5 \end{pmatrix}_{1\times 3}$
are the examples of row matrices.

ii)Column Matrix:
A matrix having only one column is known as column matrix.
$\text{Ex.}\left(2 \right )_{1\times 1} , \begin{pmatrix} 2\\ 5\\ 7 \end{pmatrix}_{3\times 1}$
are the examples of Column Matrices.

iii)Square Matrix:
A matrix is said to be square matrix which consists of equal number of rows and columns.
$\left(5 \right )_{1\times 1} , \begin{pmatrix} 1 & 2\\ 4 & 5 \end{pmatrix}_{2\times 2} \text{and}\begin{pmatrix} 0 & 1 & -1\\ 1 & -1 & 0\\ 0 & 1 & -1 \end{pmatrix}_{3\times 3}$
are the examples of Square matrices.

iv) Diagonal matrix:
A square matrix whose non-diagonal elements are zero is called a diagonal matrix.
$\begin{pmatrix} 1 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 3 \end{pmatrix}_{3\times 3} ,\begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix}_{2\times 2}$
are the examples of Diagonal Matrices.

v) Scalar Matrix:
A Diagonal Matrix whose diagonal elements are equal is called a Scalar Matrix.
$\begin{pmatrix} 2 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 2 \end{pmatrix}_{3\times 3} ,\begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}_{2\times 2}$
are the examples of Scalar Matrix.

vi)Unit Matrix or identity Matrix:
A diagonal Matrix whose every diagonal elements are unit is called Unit Matrix. A unit matrix of (n x n) is denoted by I_n
$\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}_{3\times 3} ,\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}_{2\times 2} \text{and} \begin{pmatrix}1 \end{pmatrix}_{1\times 1}$
are the examples of Unit Matrix.

vii)Zero Matrix(or Null Matrix):
A matrix whose every element is zero ir called a zero matrix. It is denoted by O_mn
$\text {O}_{23} =\begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix}_{2\times 3}$
is Null Matrix.

viii)Triangular Matrix:
(a) Upper Triangular Matrix:A square matrix, A = (a_ij), is called an upper triangular matrix if a_ij = 0 for i > j.
$\begin{pmatrix} a_{11} & a_{12} & a_{13}\\ 0 & a_{22} & a_{23}\\ 0 & 0 & a_{33} \end{pmatrix}$
is upper triangular matrix
Or
A square matrix is said to be upper triangular matrix if the elements below the main diagonal is zero.
b)Lower Triangular Matrix:
A square matrix A = (a_ij), is called lower triangular matrix if a_ij = 0 for i < j.
Or
A square matrix is said to be lower triangular matrix if the elements above the main diagonal is zero.
$\begin{pmatrix} 1 & 0 & 0\\ 3 & 3 &0\\ 2 & 1 & 4 \end{pmatrix}_{3\times 3}$
is lower triangular matrix.

ix)Symmetric Matrix:
A square matrix A = (a_ij) is said to be symmetric matric if a_ij = a_ij for all i and j.
$\begin{pmatrix} 1 & 2 & 3\\ 2 & 5 & 7\\ 3 & 7 & 4 \end{pmatrix}_{3\times 3}$
is symmetric matrix.

Skew Symmetric Matrix:
A square matrix A = (a_ij) is called skew symmetric if
a_ij = -a_ij for all i and j. Every diagonal elements of a skew symmetric matrix should be zero.
$\begin{pmatrix} 0 & 2 & 3\\ -2 & 0 & 7\\ -3 & -7 & 0 \end{pmatrix}_{3\times 3}$
is a skew symmetric matrix.

Hermitian Matrix:
A square matrix A is said to be Hermitian Matrix if A* = A.
$\text{i.e. if a}_{ij} = \text {for all i and j}$

Note: The diagonal elements of suh matrices should b purely real numbers.
$\text {A} = \begin{pmatrix} 1 & 1+i & 2-3i \\ 1-i& -2 & 3+4i\\ 2+3i& 3-4i & 5 \end{pmatrix}$
$\text{Then}, \overline{A} =\begin{pmatrix} 1 & 1-i & 2+3i \\ 1+i& -2 & 3-4i\\ 2_3i& 3+4i & 5 \end{pmatrix}$
$\text {A*} = \left (\overline{A} \right)^{T} =\begin{pmatrix} 1 & 1+i & 2-3i \\ 1-i& -2 & 3+4i\\ 2+3i& 3-4i & 5 \end{pmatrix}$

= A
i.e. A is Hermitian Matrix.
Skew Hermitian Matrix
A square matrix A is said to be Skew Hermitian Matrix if A* = -A i.e. if

$\text{a}_{ij} = -\overline{a_{ji}}, \text{for all i and j}$
Note: The diagonalelements of suchmatrices should be purel imaginar number or zero.

$\text {Let A} = \begin{pmatrix} 0 & 1+i & 2-3i \\ -1+i& -2i & 3+4i\\ -2-3i& -3+4i & 3i \end{pmatrix}$

$\text{Then}, \overline{A} =\begin{pmatrix} 0 & 1-i & 2+3i \\ -1-i& 2i & 3-4i\\ -2+3i& -3-4i & -3i \end{pmatrix}$

$\therefore \text{A*} =\left(\overline{A} \right )^{T}$

$=\begin{pmatrix} 0 & -1-i & -2+3i \\ 1-i& 2i & -3-4i\\ 2+3i& 3-4i & -3i \end{pmatrix}$

$=\left(-1 \right)\begin{pmatrix} 0 & 1+i & 2-3i \\ -1+i& -2i & 3+4i\\ -2-3i& -3+4i & 3i \end{pmatrix}$

= -A
i.e. A is Skew Hermitian Matrix.