Diagonal Matrix

The arrangement of numbers organized into rows and columns, forming a square or a rectangular shape, is known as a matrix.

A diagonal matrix is a square matrix in which all the elements except the diagonal elements are zero. We will learn about the concept of diagonal matrices in this article. 

A diagonal matrix is a square matrix in which all the elements outside the main diagonal are zero. Only the elements on the main diagonal, from the top-left to the bottom-right, may be non-zero. The example of a diagonal matrix is:

\( \begin{bmatrix} 5 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 3 \end{bmatrix} \)

Here, all the elements are 0 except the diagonal elements.

Lower triangular matrix: A lower triangular matrix is a square matrix in which every element above the main diagonal is 0. 
Example:

\( \begin{bmatrix} 3 & 0 & 0 \\ 5 & 4 & 0 \\ 6 & 5 & 7 \end{bmatrix} \)
 

Upper triangular matrix: The upper triangular matrix is a square matrix where every element below the diagonal is 0. 
Example:

\( \begin{bmatrix} 5 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 3 \end{bmatrix} \)
 

A diagonal matrix is considered as both an upper and a lower triangular matrix as it contains zeros in all positions except the main diagonal. 

In this section, we will learn about the main properties of a diagonal matrix, including how they behave in addition, multiplication, and transposition. The properties of a diagonal matrix help us to solve problems and understand the applications of diagonal matrices. 

1. A diagonal matrix is always a square matrix

A diagonal matrix must have an equal number of rows and columns, meaning that it is always a square matrix. 

Example: 

\( \begin{bmatrix} 5 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 3 \end{bmatrix} \)
 

The above matrix has 3 rows and 3 columns, so it is called a square matrix.

2. Types of diagonal matrix

Diagonal matrices are of different types, such as identity matrix, scalar matrix, and null or zero matrix. In identity and scalar matrices, the diagonal elements are non-zero, while all other elements are zero. 

Example:

\( \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix} \)

3. Identity matrix

The identity matrix is a special type of matrix where all the diagonal elements are 1, and the rest of the elements are 0. 

Example:

\( \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \)
 

4. Addition of diagonal matrices

When two diagonal matrices of the same order are added, the result is also a diagonal matrix. The corresponding diagonal elements are added together. 

Example:

\( \text{Add } A = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 0 \\ 0 & 5 \end{bmatrix} \)

  A × B = \( \begin{bmatrix} 5 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 3 \end{bmatrix} \)

5. Multiplication of diagonal matricesThe result will also be a diagonal matrix if we multiply two diagonal matrices of the same size. To multiply the diagonal matrices, multiply the corresponding diagonal elements.

Example:

 \( \text{Add } A = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 0 \\ 0 & 5 \end{bmatrix} \)

  A × B = \( \begin{bmatrix} 5 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 3 \end{bmatrix} \)

6. Transpose of a diagonal matrix

When we transpose (flip) a diagonal matrix, the result is the same diagonal matrix. This is because all the non-diagonal elements are zero. 

Example:

\( A = \begin{bmatrix} 3 & 0 \\ 0 & 5 \end{bmatrix} \)

A^T =\( \begin{bmatrix} 3 & 0 \\ 0 & 5 \end{bmatrix} \)
 

7. Commutative property

Diagonal matrices satisfy the commutative property for both addition and multiplication. This means the order in which we add or multiply the diagonal matrices does not affect the result.   

A + B = B + A
A × B = B × A

8. Diagonal matrices are symmetric

A matrix is said to be symmetric when it looks the same after flipping it across the diagonal. Since all non-diagonal elements in a diagonal matrix are zero, transposing it does not affect the matrix. 
 

An anti-diagonal matrix has non-zero elements only on the anti-diagonal, which is from the top-right to the bottom-left. It is a representation of a diagonal matrix where non-zero elements are on the anti-diagonal.

In an anti-diagonal matrix, all the elements are 0 except the diagonal elements from the upper-right edge to the lower-left edge. An anti-diagonal  diagonal. An example of an anti-diagonal matrix is,

\( \begin{bmatrix} 0 & 0 & 2 \\ 0 & 6 & 0 \\ 7 & 0 & 0 \end{bmatrix} \)

The determinant of a diagonal matrix is found by multiplying its main diagonal elements. The determinant of a diagonal matrix is non-zero only when all its diagonal elements are non-zero. 

Consider a 3 × 3 matrix, A =

\( \begin{bmatrix} 2 & 0 & 0 \\ 0 & -4 & 0 \\ 0 & 0 & 6 \end{bmatrix} \)
 

The determinant of the given matrix |A| = 2 × (-4) × 6 = -48.
 

The inverse of a diagonal matrix is also a diagonal matrix. To find the inverse of a diagonal matrix, just take the reciprocal of each number on the diagonal. All the other numbers stay zero, but the diagonal matrix cannot be zero.

If we have a diagonal matrix of 3 × 3, A =

\( \begin{bmatrix} a_{11} & 0 & 0 \\ 0 & a_{22} & 0 \\ 0 & 0 & a_{33} \end{bmatrix} \)
 

Step 1: Find the determinant.
The determinant of a diagonal matrix is the product of all the numbers on the diagonals.

|A| = a11 × a22 × a33
 

Step 2: Find the adjoint
The adjoint of a diagonal matrix is also a diagonal matrix.
 

Step 3: Use the formula

The inverse formula for a diagonal matrix is A-1 =

\( \begin{bmatrix} \dfrac{1}{a_{11}} & 0 & 0 \\ 0 & \dfrac{1}{a_{22}} & 0 \\ 0 & 0 & \dfrac{1}{a_{33}} \end{bmatrix} \)
 

Example:

Find the inverse of a given diagonal matrix, A = 

\( \begin{bmatrix} 2 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 5 \end{bmatrix} \)

A^-1 =
\( \begin{bmatrix} \dfrac{1}{2} & 0 & 0 \\ 0 & \dfrac{1}{4} & 0 \\ 0 & 0 & \dfrac{1}{5} \end{bmatrix} \)

A block matrix is a square matrix that is split into smaller sections or blocks. It is made by dividing the matrix into smaller square blocks along the diagonals. The blocks that are not on the diagonals are filled with zeros. The simple rules of a block diagonal matrix are:
The blocks on the diagonal are square matrices like 2 × 2, 3 × 3, etc.

All the other blocks are zero matrices.

If we say [Aij] = 0, for i ≠ j, it means all the blocks not on the diagonal are zero.
For example,

A = 

\( \begin{bmatrix} B_{1} & 0 & 0 \\ 0 & B_{2} & 0 \\ 0 & 0 & B_{3} \end{bmatrix} \)
 

Here, B1, B2, and B3 are square matrices, and all the other parts are zero matrices.

Block matrices are used to break big problems into smaller ones. Each part on the diagonal works separately, which saves time and effort in big calculations like matrix multiplication or finding inverses.

Diagonal matrices are useful because they make complex problems easier, especially when working with computations, physics, computer graphics, and many other fields. Below are some real-life examples of how diagonal matrices are used in different areas.

• Physics: Diagonal matrices are used to simplify the moment of inertia of an object when it is rotating. In physics, diagonal matrices help in calculating how an object resists rotation around an axis.
   
• Computer Graphics: Diagonal matrices are used to scale the images in computer graphics. It is also used to resize, stretch, or shrink images or shapes easily.
   
• Machine Learning and AI: In some machine learning algorithms, diagonal matrices help in simplifying covariance matrices. It makes it easier to analyze and reduce large datasets. 
   
• Economics: In economic models, diagonal matrices represent systems where each sector, like farming, manufacturing, etc., affects only itself and not others. It shows how much one sector produces or consumes by itself.