Matrix
A matrix is an arrangement of numbers in rows and columns and enclosed by square brackets
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
,\text{i = 1, 2 .....m, and j = 1,2,....n } "=\left(a_{ij} \right ),\text{i = 1, 2 .....m, and j = 1,2,....n }")
_{m\times n} "=\left(a_{ij} \right )_{m\times n}")
Standard Matrices
i) Raw Matrix:
A matrix having only one row is called a row matrix.
_{1\times 1}, \begin{pmatrix} 2 & 3 & 5 \end{pmatrix}_{1\times 3} "=\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.
_{1\times 1} , \begin{pmatrix} 2\\ 5\\ 7 \end{pmatrix}_{3\times 1} "\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.
_{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} "\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.
are the examples of Diagonal Matrices.
v) Scalar Matrix:
A Diagonal Matrix whose diagonal elements are equal is called a Scalar Matrix.
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 In
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 Omn
is Null Matrix.
viii)Triangular Matrix:
(a) Upper Triangular Matrix:A square matrix, A = (aij), is called an upper triangular matrix if aij = 0 for i > j.
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 = (aij), is called lower triangular matrix if aij = 0 for i < j.
Or
A square matrix is said to be lower triangular matrix if the elements above the main diagonal is zero.
is lower triangular matrix.
ix)Symmetric Matrix:
A square matrix A = (aij) is said to be symmetric matric if aij = aij for all i and j.
is symmetric matrix.
Skew Symmetric Matrix:
A square matrix A = (aij) is called skew symmetric if
aij = -aij for all i and j. Every diagonal elements of a skew symmetric matrix should be zero.
is a skew symmetric matrix.
Hermitian Matrix:
A square matrix A is said to be Hermitian Matrix if A* = A.
Note: The diagonal elements of suh matrices should b purely real numbers.
^{T} =\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
Note: The diagonalelements of suchmatrices should be purel imaginar number or zero.
^{T} "\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.
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A matrix is an arrangement of numbers in rows and columns and enclosed by square brackets
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
Standard Matrices
i) Raw Matrix:
A matrix having only one row is called a row matrix.
are the examples of row matrices.
ii)Column Matrix:
A matrix having only one column is known as column matrix.
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.
are the examples of Square matrices.
iv) Diagonal matrix:
A square matrix whose non-diagonal elements are zero is called a diagonal matrix.
are the examples of Diagonal Matrices.
v) Scalar Matrix:
A Diagonal Matrix whose diagonal elements are equal is called a Scalar Matrix.
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 In
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 Omn
is Null Matrix.
viii)Triangular Matrix:
(a) Upper Triangular Matrix:A square matrix, A = (aij), is called an upper triangular matrix if aij = 0 for i > j.
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 = (aij), is called lower triangular matrix if aij = 0 for i < j.
Or
A square matrix is said to be lower triangular matrix if the elements above the main diagonal is zero.
is lower triangular matrix.
ix)Symmetric Matrix:
A square matrix A = (aij) is said to be symmetric matric if aij = aij for all i and j.
is symmetric matrix.
Skew Symmetric Matrix:
A square matrix A = (aij) is called skew symmetric if
aij = -aij for all i and j. Every diagonal elements of a skew symmetric matrix should be zero.
is a skew symmetric matrix.
Hermitian Matrix:
A square matrix A is said to be Hermitian Matrix if A* = A.
Note: The diagonal elements of suh matrices should b purely real numbers.
= 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
Note: The diagonalelements of suchmatrices should be purel imaginar number or zero.
= -A
i.e. A is Skew Hermitian Matrix.
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