Inverse of Diagonal Matrix

The inverse of a diagonal matrix is a special case of finding the inverse of a matrix. To find the inverse of diagonal matrix, we use a formula and do not require to find the determinant and adjoint the diagonal matrix. Let us first recall the meaning of a diagonal matrix which is a square matrix with non-zero elements on the main diagonal and the remaining non-diagonal elements are zero. The inverse of a diagonal matrix can be determined by replacing diagonal elements of the matrix with their reciprocals.

In this article, we will explore the concept of finding the inverse of a diagonal matrix, its formula and proof. We will also go through the necessary condition required to find the inverse of the diagonal matrix and solve a few examples based on the concept for a better understanding of the topic.

The inverse of a diagonal matrix is given by replacing the main diagonal elements of the matrix with their reciprocals. The inverse of the diagonal matrix exists if and only if all elements of the main diagonal are non-zero and this is a necessary condition for the inverse of a diagonal matrix to exist. As a diagonal matrix is a square matrix, therefore its inverse is also a square matrix and exists if the diagonal matrix is non-singular, i.e., has a non-zero determinant. Now, let us see the formula to find the inverse of a diagonal matrix.

To generalize the formula for the inverse of diagonal matrix, let us see the specific cases first. First, we will write the formula to find the inverse of 2 × 2 diagonal matrix. For a diagonal matrix A of order 2, we have

\(A = \left[\begin{array}{ccc} a_{11} & 0 \\ \\
0 & a_{22}
\end{array}\right] \), then its inverse is

\(A^{-1} = \left[\begin{array}{ccc} \frac{1}{a_{11}} & 0 \\ \\
0 & \frac{1}{a_{22}}
\end{array}\right] \)

Now, to find the formula of the inverse of a diagonal matrix of order, we have

If \(A = \left[\begin{array}{ccc}
a_{11} & 0 & 0 \\
0 & a_{22} & 0 \\
0 & 0 & a_{33}
\end{array}\right] \), then its inverse is given by,

\(A^{-1} = \left[\begin{array}{ccc}
\frac{1}{a_{11}} & 0 & 0 \\
0 & \frac{1}{a_{22}} & 0 \\
0 & 0 & \frac{1}{a_{33}}
\end{array}\right] \)

Now, the image below shows the general formula for the inverse of diagonal matrix of order n:

Consider a diagonal matrix of order 2 given by \(A = \left[\begin{array}{ccc} a & 0 \\ \\
0 & b
\end{array}\right] \)

Now, assume the inverse of diagonal matrix A to be \(A^{-1} = \left[\begin{array}{ccc} p & q \\ \\
r & s
\end{array}\right] \)

Using the definition of inverse of a matrix, we have

\(\left[\begin{array}{ccc} 1 & 0 \\ \\
0 & 1
\end{array}\right] =I = AA^{-1} = \left[\begin{array}{ccc} a & 0 \\ \\
0 & b
\end{array}\right] \left[\begin{array}{ccc} p & q \\ \\
r & s
\end{array}\right] = \left[\begin{array}{ccc} ap & aq \\ \\
br & bs
\end{array}\right]\)

Therefore, we have

1 = ap ⇒ p = 1/a

0 = aq ⇒ q = 0

1 = bs ⇒ s = 1/b

0 = br ⇒ r = 0

Using the above values, we have the inverse of diagonal matrix A as,

\(A^{-1} = \left[\begin{array}{ccc} \frac{1}{a} & 0 \\ \\
0 & \frac{1}{b}
\end{array}\right] \)

As we can see, the inverse of diagonal matrix A is a diagonal matrix with main diagonal elements as reciprocals of diagonal elements of matrix A. Now, for a diagonal matrix D of order n, we have

\(DD^{-1} = \begin{bmatrix}
d_{11}&0&\cdots &0 \\
0&d_{22}&\cdots &0 \\
\vdots & \vdots & \ddots & \vdots\\
0&0&\cdots &d_{nn}
\end{bmatrix}\) \(\begin{bmatrix}
\frac{1}{d_{11}}&0&\cdots &0 \\
0&\frac{1}{d_{22}}&\cdots &0 \\
\vdots & \vdots & \ddots & \vdots\\
0&0&\cdots &\frac{1}{d_{nn}}
\end{bmatrix}\) \(=\begin{bmatrix}
1&0&\cdots &0 \\
0&1&\cdots &0 \\
\vdots & \vdots & \ddots & \vdots\\
0&0&\cdots &1
\end{bmatrix}\)

Statement: The theorem on the inverse of diagonal matrix states that a diagonal matrix D = diag(d₁, d₂, d₃, ..., d_n) is invertible if and only if all diagonal entries are non-zero, i.e., d_i ≠ 0 for 1 ≤ i ≤ n.

Proof: For a diagonal matrix D = diag(d₁, d₂, d₃, ..., d_n), if all d_i ≠ 0, then

diag(d₁, d₂, d₃, ..., d_n) diag(1/d₁, 1/d₂, 1/d₃, ..., 1/d_n) = diag(1, 1, 1, ..., 1) = I and diag(1/d₁, 1/d₂, 1/d₃, ..., 1/d_n) diag(d₁, d₂, d₃, ..., d_n) = diag(1, 1, 1, ..., 1) = I

Now, assume d_i = 0 for some i, then the i^th row of the matrix D becomes zero, and hence the matrix becomes non-invertible.

Important Notes on Inverse of Diagonal Matrix

• The inverse of a diagonal matrix is a diagonal matrix where the elements of the main diagonal are the reciprocals of the corresponding elements of the original diagonal matrix.
• If any element of the main diagonal of a diagonal matrix is zero, then the entire row of the matrix becomes zero, which makes the matrix singular, and hence, the inverse of the diagonal matrix won't exist.
• The necessary condition of the inverse of the diagonal matrix to exist is that all main diagonal elements must be non-zero.

☛ Related Topics:

• How to Solve Matrices
• Matrix formula

What is Inverse of Diagonal Matrix in Algebra?

The inverse of a diagonal matrix is given by replacing the main diagonal elements of the matrix with their reciprocals. The inverse of a diagonal matrix is a special case of finding the inverse of a matrix.

What is the Formula for Inverse of Diagonal Matrix?

For a diagonal matrix D = diag(d₁, d₂, d₃, ..., d_n) with d_i ≠ 0 for 1 ≤ i ≤ n, the inverse of diagonal matrix D is D^-1 = diag(1/d₁, 1/d₂, 1/d₃, ..., 1/d_n)

When Does the Inverse of Diagonal Matrix Exist?

The inverse of a diagonal matrix exists if and only if all main diagonal elements of the matrix are non-zero.

How to Find Inverse of Diagonal Matrix?

To find the inverse of diagonal matrix, we use a formula and do not require to find the determinant and adjoint the diagonal matrix. We replace the main diagonal elements of the matrix with their reciprocals to find its inverse.

Is the Inverse of Diagonal Matrix a Diagonal Matrix?

Yes, the inverse of diagonal matrix is a diagonal matrix.

Is the Inverse of Diagonal Matrix a Symmetric Matrix?

Yes, the inverse of diagonal matrix is a symmetric matrix.