Row Matrix:
A matrix having a single row is called a row matrix.
Example: \(\left[ \begin{matrix} 1 & 2 & 3 \\\end{matrix} \right]\)
Column Matrix:
A matrix having a single column is called a column matrix.
Example: \(\left[ \begin{matrix} 1 \\ 2 \\ 3 \\\end{matrix} \right]\).
Square Matrix:
An m x n matrix A is said to be a square matrix if m = n i.e. number of rows = number of columns.
For Example: \(A\,=\,\left[ \begin{matrix} 1 & 2 & 3 \\ 3 & 2 & 2 \\ 1 & 2 & 1 \\\end{matrix} \right]\) is a square matrix of order 3 × 3.
Traces of a Matrix:
The sum of the elements of a square matrix A lying along the principal diagonal is called the trace of A i.e. tr(A). Thus if A = [aij]mxn.
Then \({{t}_{r}}\left( A \right)=\underset{i=1}{\overset{n}{\mathop{\sum }}}\,{{a}_{ii}}~={{a}_{11}}~+\text{ }{{a}_{22}}~+\,…\,+\text{ }{{a}_{mn}}\).
Diagonal Matrix:
A square matrix all whose elements except those in the leading diagonal, are zero is called a diagonal matrix. For a square matrix A = [aij]mxn to be a diagonal matrix, aij = 0, whenever i ≠ j.
For Example: \(A\,=\,\left[ \begin{matrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 2 \\\end{matrix} \right]\) is a diagonal matrix of order 3 × 3.
Here A can also be represented as diag (2 3 2).
Scalar Matrix:
A diagonal matrix whose all the elements are equal is called a scalar matrix.
For a square matrix A = [aij]mxn to be a scalar matrix, \({{a}_{ij}}=\left\{ \begin{align} & 0,\,\,i\ne j \\ & m,\,\,i=i \\\end{align} \right.\), where m ≠ 0.
For example: \(A\,=\,\left[ \begin{matrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \\\end{matrix} \right]\) is a scalar matrix.
Unit Matrix or Identity Matrix:
A diagonal matrix of order n which has unity for all its elements, is called a unit matrix of order n and is denoted by In.
For example: \({{I}_{3}}\,=\,\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{matrix} \right]\).
Triangular Matrix:
A square matrix in which all the elements below the principal diagonal are zero is called Upper Triangular matrix and a square matrix in which all the elements above the principal diagonal are zero is called Lower Triangular matrix.
Given a square matrix A = [aij]mxn, for upper triangular matrix, aij = 0, i > j and for lower triangular matrix, aij = 0, i < j.
Diagonal matrix:
Diagonal matrix is both upper and lower triangular
A triangular matrix A = [aij]mxn is called strictly triangular if aii = 0 for 1 < i < n.
For Example: \(\left[ \begin{matrix} 2 & 2 & 1 \\ 0 & 3 & 1 \\ 0 & 0 & 2 \\\end{matrix} \right]\) and \(\left[ \begin{matrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 3 & 1 \\\end{matrix} \right]\)are respectively upper and lower triangular matrices.
Null Matrix:
If all the elements of a matrix (square or rectangular) are zero, it is called a null or zero matrix.
For A = [aij] to be null matrix, aij = 0 > ” i, j.
For Example: \(\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\\end{matrix} \right]\) is a zero matrix.
Transpose of a Matrix:
The matrix obtained from any given matrix A, by interchanging its rows and columns, is called the transpose of A and is denoted by A’.
If A = [aij]mxn and A’ = [bij]mxn then bij = aij ” i, j.
For Example: If \(A\,=\,\left[ \begin{matrix} 1 & 2 & 3 \\ 3 & 2 & 2 \\ 1 & 2 & 1 \\\end{matrix} \right]\), then \(A’\,=\,\left[ \begin{matrix} 1 & 3 & 1 \\ 2 & 2 & 2 \\ 3 & 2 & 1 \\\end{matrix} \right]\)
Properties of Transpose:
(i) (A’)’ = A
(ii) (A + B)’ = A’ + B’, A and B being conformable matrices
(iii) (kA)’ = kA’, α being scalar
(iv) (AB)’ = B’A’, A and B being conformable for multiplication
Conjugate of a Matrix:
The matrix obtained from any given matrix A containing complex number as its elements, on replacing its elements by the corresponding conjugate complex numbers is called conjugate of A and is denoted by A̅.
For Example: \(A\,=\,\left[ \begin{matrix} 1+2i & 2-3i & 03+4i \\ 4-5i & 5+6i & 6-7i \\ 8 & 7+8i & 7 \\\end{matrix} \right]\) then \(\overline{A}\,=\,\left[ \begin{matrix} 1-2i & 2+3i & 3-4i \\ 4+5i & 5-6i & 6+7i \\ 8 & 7-8i & 7 \\\end{matrix} \right]\)
Properties of Conjugate:
\(i)\ \overline{\left( \overline{A} \right)}=A\).
\(ii)\ \left( \overline{A+B} \right)=\overline{A}+\overline{B}\).
\(iii)\ \overline{\left( \alpha A \right)}=\overline{\alpha }\overline{A}\) , α being any number real or complex.
\(iv)\ \overline{\left( AB \right)}=\overline{A}\ \overline{B}\), A and B being conformable for multiplication.
Transpose Conjugate of a Matrix:
The transpose of the conjugate of a matrix is called transposed conjugate of A and is denoted by Aq. The conjugate of the transpose of A is the same as the transpose of the conjugate of A i.e. \(\overline{\left( {{A}’} \right)}\ =\ {{\overline{\left( A \right)}}^{\prime }}\ =\mathop{A}^{\Theta }\).
If A = [aij]mxn, then AΘ = [bij]mxn where \({{b}_{ji}}~=~\overline{{{a}_{ij}}}\)
i.e. the (j, i)th element of AΘ = the conjugate of (i, j)th element of A.
For Example: A = aij = 0 then \(\left[ \begin{matrix} 1-2i & 4+5i & 8 \\ 2+3i & 5-6i & 7-8i \\ 3-4i & 6+7i & 7 \\\end{matrix} \right]\)
Properties of Transpose Conjugate:
(i) (AΘ)Θ = A
(ii) (A + B)Θ = AΘ + BΘ
(iii) (kA)Θ = AΘ, k being any number
(iv) (AB)Θ = BΘAΘ