A diagonal matrix is a type of square matrix in which all off-diagonal elements are zero. It is noted that the diagonal elements may or may not be zero. In this article, you will learn all the important properties and conditions.
Condition for diagonal matrix
Suppose P be a square matrix having ‘i’ rows and ‘j’ columns. This matrix P is said to be a diagonal matrix if it satisfies the following condition. (pij represents the element in matrix P)
pij = 0 if i ≠ j
and
pij if i = j
Diagonal Matrix Examples
The example of a diagonal matrix is given below.
\[P=\begin{bmatrix} 2 &0 &0 \\ 0& 8 &0 \\ 0&0 & 6 \end{bmatrix}\]
The above matrix P represents a diagonal matrix. The diagonal elements are 2, 8, and 6. This matrix can also be written as P = diag [2, 8, 6].
‘diag‘ represents that it is a diagonal matrix and numbers in the square bracket represents diagonal elements.
Let us take another example.
\[Q=\begin{bmatrix} 7 &0 &0 \\ 0& 0 &0 \\ 0&0 & 0 \end{bmatrix}\]
It is also a diagonal matrix because it has all off-diagonal elements are zero. It can be written as Q = diag [7, 0, 0].
Let us take 2 x 2 matrix.
\[R=\begin{bmatrix} 3 &0 \\ 0& 7 \end{bmatrix}\]
It is also a diagonal matrix. It can be written as R = diag [3, 7].
Diagonal Matrix Properties
There are important properties of this kind of matrix. It helps us to solve complex questions easily. We will discuss some of its properties now. Here we will write the diagonal matrix as diag [].
1. Addition of diagonal matrices
The addition of two diagonal matrices will be a diagonal matrix as shown below. The diagonal elements of the new matrix are the addition of diagonal elements of two matrices of the same position.
diag [p, q, r] + diag [a, b, c] = diag [p+a, q+b, r+c]
2. Multiplication of diagonal matrices
The multiplication of two diagonal matrices will be a diagonal matrix. The diagonal elements of the new matrix are the multiplication of diagonal elements of two matrices of the same position.
diag [p, q, r] × diag [a, b, c] = diag [pa, qb, rc]
3. Inverse of diagonal matrix
The diagonal elements of the inverse of diagonal matrix are the reciprocal of diagonal elements of the original matrix of the same position.
(diag [p, q, r] )-1 = diag [1/p, 1/q,1/r]
4. Transpose of diagonal matrix
The diagonal elements of the transpose of the diagonal matrix are the same as the original matrix have.
(diag [p, q, r] )T = diag [p, q, r]
5. Determinant of diagonal matrix
The determinant of the diagonal matrix is equal to the product of the diagonal elements.
| diag [p, q, r] | = pqr
6. Self Multiplication
If a diagonal matrix is self multiplied ‘n’ times then the diagonal elements of the new matrix will be
(diag [p, q, r] )n = diag [pn, qn, rn]
7. Eigen values of diagonal matrix
The eigen values of a diagonal matrix are its diagonal elements itself i.e.
eigen values of diag [p, q, r] are p, q and r.
Note: In the above properties, we take examples of 3 x 3 matrix. All these properties are also valid for n x n matrix.
Is zero matrix a diagonal matrix?
Yes. Zero matrix is a diagonal matrix because it has all its off-diagonal elements zero. In this matrix, all the diagonal elements are also zero.
