Diagonal matrix is a type of square matrix in which all the diagonal elements are present and non-diagonal elements are zero. Diagonal elements are those elements for which row number is equal to the column number i.e. i = j.
So, A3x3 = $\begin{bmatrix} a &0 &0 \\ 0 &b &0 \\ 0 &0 &c \end{bmatrix}$
In the above matrix A, it is clear that all non-diagonal elements are 0 and diagonal elements are present. So, A is a diagonal matrix.

## Block Diagonal Matrix

To know about Block Diagonal Matrix, first we have to know about Block Matrix.

Block Matrix: A block matrix is a matrix which is split into blocks. These blocks are nothing but the smaller matrices and formed by grouping the elements in adjacent rows or columns or if a matrix is define in terms of smaller matrices adjacent rows and columns.
Let E = $\begin{bmatrix} 2 &2 &1 &1 \\ 2 &2 &1 &1 \\ 4 &4 &3 &3 \\ 4 &4 &3 &3 \end{bmatrix}$ be any matrix. It can be partitioned into 4 x 2 blocks as follows:

E11 = $\begin{bmatrix} 2 &2 \\ 2 &2 \end{bmatrix}$

E12 = $\begin{bmatrix} 1 &1 \\ 1 &1 \end{bmatrix}$

E21 = $\begin{bmatrix} 4 &4 \\ 4 &4 \end{bmatrix}$

E22 = $\begin{bmatrix} 3 &3 \\ 3 &3 \end{bmatrix}$

So, E = $\begin{bmatrix} E_{11} &E_{12} \\ E_{21} &E_{22} \end{bmatrix}$ is the block matrix.

So, we can say if A is i x j order matrix and k and l are numbers such that 1 < k < i and 1 < l < j, then we can form a block matrix with its elements C, D, E and F as follows:

A = $\begin{bmatrix} C &D \\ E &F \end{bmatrix}$

Block Diagonal Matrix: A block diagonal matrix is a type of square matrix such that if a block matrix is formed from this square matrix then, its diagonal is an arrangement of square matrices and the non-diagonal blocks or sub-matrices are zero matrices.

Here, Cn x n is a block diagonal matrix of order n x n. Thus, C is a block diagonal matrix if $C_{\alpha \beta }$ = 0, where $\alpha = \beta$.

## Inverse of Diagonal Matrix

Let A = $\begin{bmatrix} a &0 &0 \\ 0 &b &0 \\ 0 &0 &c \end{bmatrix}$ be any diagonal matrix. To find the inverse of A, we use

A-1 = $\frac{adj A}{\left | A \right |}$

Now, for determinant A, we select first row for expansion.

Hence I A I = a $\begin{vmatrix} b&0 \\ 0 &c \end{vmatrix}$ + 0 $\begin{vmatrix} 0 &0 \\ 0 &c \end{vmatrix}$ + 0 $\begin{vmatrix} 0 &b \\ 0 &0 \end{vmatrix}$

= abc

Now, adj A = $\begin{bmatrix} bc &0 &0 \\ 0 &ac &0 \\ 0 &0 &ab \end{bmatrix}$

So, A-1 = $\frac{1}{\left | A \right |}$ x $\begin{bmatrix} bc &0 &0 \\ 0 &ac &0 \\ 0 &0 &ab \end{bmatrix}$

= $\begin{bmatrix} 1/a &0 &0 \\ 0 &1/b&0 \\ 0 &0 &1/c \end{bmatrix}$

Hence, inverse of a diagonal matrix is defined only if the determinant of a diagonal matrix is non zero and the reciprocal of the diagonal elements are present in inverse matrix instead of actual diagonal elements.

## Diagonal Matrix Properties

Listed below are some of the diagonal matrix properties:

Property 1: Any diagonal matrix is also a symmetric matrix. If A is any diagonal matrix then, A = $A^T$Proof:

Let A3x3 = $\begin{bmatrix} -2 &0 &0 \\ 0 & 4 &0 \\ 0 &0 &7 \end{bmatrix}$ is a diagonal matrix. Now, lets calculate AT

AT = $\begin{bmatrix} -2 &0 &0 \\ 0 & 4 &0 \\ 0 &0 &7 \end{bmatrix}$ = A

Hence, A = AT.

Property 2: If two diagonal matrices are of same order, then their addition and multiplication are again a diagonal matrix. So, if C and D are diagonal matrices, then C + D and CD are again diagonal matrices.
Proof:

Let C3x3 = $\begin{bmatrix} 4 &0 &0 \\ 0 &-7 &0 \\ 0 &0 &9 \end{bmatrix}$ and D3x3 = $\begin{bmatrix} -1 &0 &0 \\ 0 & 3 &0 \\ 0 &0 &2 \end{bmatrix}$ are two matrices of same order.

Then, C + D = $\begin{bmatrix} 4 &0 &0 \\ 0 &-7 &0 \\ 0 &0 &9 \end{bmatrix}$ + $\begin{bmatrix} -1 &0 &0 \\ 0 & 3 &0 \\ 0 &0 &2 \end{bmatrix}$

= $\begin{bmatrix} 3 &0 &0 \\ 0 & -4 &0 \\ 0 &0 &11 \end{bmatrix}$ ..........................................(a)

Again, CD = $\begin{bmatrix} 4 &0 &0 \\ 0 &-7 &0 \\ 0 &0 &9 \end{bmatrix}$ . $\begin{bmatrix} -1 &0 &0 \\ 0 & 3 &0 \\ 0 &0 &2 \end{bmatrix}$

= $\begin{bmatrix} -4 &0 &0 \\ 0 &-21 &0 \\ 0 &0 &18 \end{bmatrix}$ .................................................(b)

From (a) and (b), it is clear that C + D and CD are diagonal matrices.

Property 3: If A and B are diagonal matrices, then AB = BA.
Proof:

A = $\begin{bmatrix} 9 &0 \\ 0 &-1 \end{bmatrix}$ and B = $\begin{bmatrix} 1 &0 \\ 0 &7 \end{bmatrix}$ are two matrices.

Then, AB = $\begin{bmatrix} 9 &0 \\ 0 &-1 \end{bmatrix}$. $\begin{bmatrix} 1 &0 \\ 0 &7 \end{bmatrix}$

= $\begin{bmatrix} 9 &0 \\ 0 &-7 \end{bmatrix}$ ...................................... (i)

Now, BA = $\begin{bmatrix} 1 &0 \\ 0 &7 \end{bmatrix}$ . $\begin{bmatrix} 9 &0 \\ 0 &-1 \end{bmatrix}$

= $\begin{bmatrix} 9 &0 \\ 0 &-7 \end{bmatrix}$ ........................................(ii)

Hence, from (i) and (ii), AB = BA which means matrix multiplication is commutative.