# Classification of Matrices

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