Diagonal Matrix
2026-02-28 13:49 Diff
https://brightchamps.com/en-us/math/algebra/diagonal-matrix

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:

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

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

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.