Inverse of Diagonal Matrix

A square matrix is said to be a square matrix with zeros everywhere except on the main diagonal. The main diagonal of a matrix is the line of elements from the top left corner to the bottom right corner. A square matrix A = [aij] is called a diagonal matrix if: aij = 0, for all i ≠ j. This means that only the elements a11, a22, a33, …, ann can be non-zero. For example,

It is a diagonal matrix.

What is the Inverse of a Diagonal Matrix?

The inverse of a diagonal matrix is another diagonal matrix where each element in the diagonal is replaced by its reciprocal. The main condition of the inverse of the matrix is that all elements of the main diagonal should be non-zero. The inverse of a diagonal matrix exists only if none of its diagonal elements are zero. It is a non-singular matrix, meaning its determinant is not zero. 

For writing the formula for the inverse of a diagonal matrix, first we will write the diagonal matrix and then its inverse using the reciprocals of the diagonal elements.
For a diagonal matrix A of order 2, the formula is: 
A = 0a11 a220, then the inverse of matrix A is
A-1 = 01a11 1a220
The formula for a diagonal matrix A of order 3:

The inverse is:

What is the Proof for Inverse of Diagonal Matrix?

The inverse of a matrix is said to be another matrix that, when multiplied with the original matrix, gives the identity matrix. Now, consider a diagonal matrix,
A = 0x y0

Assume that the inverse of a diagonal matrix as:
A-1 = ca db

We know that, A × A-1 = I
The identity matrix I is,
I = 01 10

A × A-1 = I

0x y0 × ca db = 01 10

ycxa ydxb = 01 10

Therefore, we have:
xa = 1, a = 1/x
xb = 0, b = 0
yc = 1, c = 1/y
yd = 0, d = 0

By using the above values, we can get the inverse of a diagonal matrix,
A-1 = 01x 1y0

The inverse of a diagonal matrix is found by taking the reciprocal of each non-zero diagonal element. For a diagonal matrix of order n, we have:

Inverse of Diagonal Matrix Theorem

Statement: A diagonal matrix can have an inverse only if all the numbers on its main diagonal are not zero. If the matrix is, D = diag(d1, d2, d3,…, dn) then each diagonal element di must not be equal to zero. 

Proof: Let’s take a diagonal matrix as:
D = diag(d1, d2, d3,…, dn)
If no element from this matrix is zero, we can find its inverse by taking the reciprocal of each diagonal element. 
The inverse matrix would look like:
D-1 = diag 1d1,1d2, 1d3, ..., 1dn 
If we multiply the original matrix D by its inverse D-1, we get
D × D-1 = diag (1, 1, 1,..., 1), which is an identity matrix.
This proves that the inverse exists when all the diagonal values are not zero. 

If one of the diagonal elements is 0, then the matrix becomes singular and its inverse does not exist. If an entire row of a matrix is zero, the matrix is singular and cannot be inverted, because its determinant is zero. Therefore, the matrix becomes non-invertible.
 

In real life, we deal with large amounts of data, systems, and calculations that involve matrices. Finding the inverse of a diagonal matrix is especially useful because it’s a straightforward process that involves taking the reciprocals of the diagonal numbers. Here are some real-life examples where a matrix is used.

• Computer graphics: In computer graphics, diagonal matrices are used to scale images to shrink or stretch them in height, width, or depth; to reverse this scaling, their inverse is applied.
• Electrical engineering: In electrical systems with multiple independent components, the matrix that shows resistance to current flow is often diagonal. The inverse of the diagonal matrix is used to find admittance, which is essential in designing stable and efficient power systems.
• Economics: Diagonal matrices are used to show the cost or productivity of each sector, like agriculture, manufacturing, etc. To find how much output is needed to meet demand, the inverse of the diagonal matrix is used to reverse the effects of cost or efficiency changes.
• Control Systems: In automatic control systems, diagonal matrices are used to model systems that operate independently. The inverse helps to determine how much input is needed to get the desired output.
• Machine Learning and Data Science: In data science, diagonal matrices are used in scaling features, like adjusting different data inputs like age, income, height, etc. The inverse is used to return the data to its original scale for interpretation or visualization.