What is the Inverse of a Diagonal Matrix?

The inverse of a diagonal matrix is another diagonal matrix formed by taking the reciprocal of each non-zero element on the main diagonal.

Understanding Diagonal Matrices

A diagonal matrix is a square matrix where all the elements outside the main diagonal are zero. The main diagonal runs from the top-left corner to the bottom-right corner.

Definition of Inverse

For a square matrix A, its inverse, denoted as A^-1, satisfies the condition:

A A^-1 = A^-1 A = I

where I is the identity matrix.

Finding the Inverse of a Diagonal Matrix

According to the provided information, the inverse of a diagonal matrix is found by replacing each main diagonal element with its reciprocal. However, it is important to note that the inverse exists only if all the elements on the main diagonal are non-zero. This condition is necessary for the inverse to exist, as division by zero is undefined.

Method

Check for Non-Zero Elements: Ensure every element on the main diagonal is non-zero. If any element is zero, the inverse does not exist.
Reciprocate: Replace each diagonal element d_i with its reciprocal, 1/d_i.
Construct Inverse: The new diagonal matrix with the reciprocated elements is the inverse of the original matrix.

Example

Consider the following diagonal matrix D:

D = | 2  0  0 |
    | 0  3  0 |
    | 0  0  4 |

To find its inverse D^-1:

All diagonal elements (2, 3, and 4) are non-zero.
Find the reciprocals: 1/2, 1/3, and 1/4.
Construct the inverse:

D^-1 = | 1/2  0    0   |
       | 0    1/3  0   |
       | 0    0    1/4 |

D^-1 = | 0.5  0    0   |
       | 0    0.333  0   |
       | 0    0    0.25 |

Existence of Inverse

The reference highlights a crucial point: a diagonal matrix has an inverse if and only if all its diagonal elements are non-zero. If any diagonal element is zero, the matrix is singular, and its inverse does not exist.

Summary

To reiterate, the inverse of a diagonal matrix is obtained by replacing each non-zero diagonal element with its reciprocal. This method is contingent on the existence of the inverse, which requires all diagonal elements to be non-zero.

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