Diagonal Matrix: Types, Properties & Determinants

Diagonal Matrix forms a significant part of linear algebra that expresses its properties and operations. Matrix (plural matrices) in general, is a mathematical concept represented as a rectangular array of numbers, symbols, or expressions. These are arranged in the form of rows and columns. Matrices find wide application in the field of engineering, physics, economics, statistics, and computer graphics. For example, consider a matrix with m number of rows and n number of columns, it is thus a “m × n” matrix.

When the number of rows and columns, is the same (m=n), the matrix is called a square matrix. A diagonal Matrix is a type of square matrix.

Key Takeaways: Square Matrix, Diagonal Matrix, Upper Triangular, Lower Triangular, Block diagonal, Inverse, Anti Diagonal, Determinant

---

What is a Diagonal Matrix?

[Click Here for Sample Questions]

The square matrix in which all elements other than diagonal are zero is called a diagonal matrix. In other words, we can say that a diagonal matrix is a square matrix where except for the diagonal it's all elements are zero.

Consider a matrix D = [d_ij]_{n × n}.

This will be a diagonal matrix when

d_ij = 0, when i ≠ j.

Hence, the following conditions are to be met for a matrix to be a diagonal matrix:

1. The matrix should be square
2. All the elements of the matrix other than the diagonal should be zero

Example of a Diagonal Matrix

The video below explains this:

Matrices Detailed Video Explanation:

---

Types of Diagonal Matrix

[Click Here for Sample Questions]

Diagonal matrix are of two types:

1. Lower Triangular: A square matrix with elements above the principal diagonal as zeroes. For example

2. Upper Triangular: A square matrix with elements below the principal diagonal as zeroes.

3. Identity Matrix: The diagonal matrix where all the diagonal elements of the matrix are 1 and all other elements are zero.

4. Zero Matrix: The diagonal matrix where all the elements of the square matrix are zero.

Also Read:

---

How to Find a Diagonal Matrix?

[Click Here for Sample Questions]

Following are the steps to identify the diagonal matrix:

1. The first step is to identify is the matrix is square or not. To identify a square count the number of rows and columns so if the number of rows and columns are equal it is a diagonal matrix.
2. The next step is to identify if all the elements other than the diagonal elements are zero.

Example 

1. \(\begin{bmatrix} 4 & 0\\ 0 & 4 \end{bmatrix}\)
2. \(\begin{bmatrix} 4 & 0 & 0 & 5 \\ 0 & 5 & 0 & 4 \end{bmatrix}\)
3. \(\begin{bmatrix} 1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{bmatrix}\)

Explanation 

The matrix A is square as the number of rows and columns are equal and all the elements other than the diagonal elements (4, 4) are zero. Matrix A is a diagonal matrix.

The matrix B is not a square matrix as the number of rows and columns are not equal and thus matrix B is also not a diagonal matrix.

The matrix C is a square matrix as the number of rows and columns are equal and all the elements other than the diagonal elements are zero.

According to the definition of the diagonal matrix should be. The square matrix and the elements other than the principal diagonal should be zero. So, the definition doesn't mention that the elements of diagonal cannot be zero. So, matrix C is a diagonal matrix by the definition of a diagonal matrix.

---

Properties of a Diagonal Matrix

[Click Here for Sample Questions]

1. When 2 diagonal matrices are multiplied or added the resultant matrix is another matrix of the same order.

For Example,

Matrix A = \(\begin{bmatrix} 3 & 0 \\ 0 & 4 \end{bmatrix}\)

Matrix B = \(\begin{bmatrix} 1 & 0 \\ 0 & 5 \end{bmatrix}\)

Matrix A+B = \(\begin{bmatrix} 3+1 & 0+0 \\ 0+0 & 4+5 \end{bmatrix}\)

Result = \(\begin{bmatrix} 4 & 0 \\ 0 & 9 \end{bmatrix}\)

The resultant matrix is a diagonal matrix and is of the same order. Hence the property is proved

Similarly,

Matrix A×B = \(\begin{bmatrix} 3 \times 1 & 0 \times 0 \\ 0 \times 0 & 4 \times 5 \end{bmatrix}\)

Result = \(\begin{bmatrix} 3 & 0 \\ 0 & 20 \end{bmatrix}\)

The resultant matrix is a diagonal matrix and is of the same order. 

Hence, the property is proved.

2. The transpose of a diagonal matrix is a diagonal matrix.

Transpose means to interchange the rows and columns of the matrix. 

For example,

Matrix A = \(\begin{bmatrix} 3 & 0 \\ 0 & 4 \end{bmatrix}\)

Interchanging the row and columns

Matrix B = \(\begin{bmatrix} 3 & 0 \\ 0 & 4 \end{bmatrix}\)

Hence, the property is proved.

3. Under multiplication, diagonal matrices are cumulative.

For example,

Matrix A = \(\begin{bmatrix} 3 & 0 \\ 0 & 4 \end{bmatrix}\)

Matrix B = \(\begin{bmatrix} 1 & 0 \\ 0 & 5 \end{bmatrix}\)

Multiplying (A×B),

\(\begin{bmatrix} (3 \times 1) + (0 \times 0) & (3 \times 0) + (0 \times 5) \\ (0 \times 1) + (4 \times 0) & (0 \times 0) + (4 \times 5) \end{bmatrix}\)

Result = \(\begin{bmatrix} 3 & 0 \\ 0 & 20 \end{bmatrix}\)

Multiplying (B×A),

\(\begin{bmatrix} (1 \times 3) + (0 \times 0) & (1 \times 0) + (0 \times 4) \\ (0 \times 3) + (5 \times 0) & (0 \times 0) + (5 \times 4) \end{bmatrix}\)

Result = \(\begin{bmatrix} 3 & 0 \\ 0 & 20 \end{bmatrix}\)

Therefore, A×B = B×A.

Hence, the property is proved.

Also Read:

---

Block Diagonal Matrix

[Click Here for Sample Questions]

A diagonal matrix split into different blocks is termed a Block Diagonal Matrix. The non-diagonal blocks form zero matrices and the main diagonal blocks the square matrices.

So, for a matrix D,

the non-diagonal blocks, D_ij = 0, when i ≠ j

Thus, D is called a block diagonal matrix.

n × n Order Block Diagonal Matrix

---

Inverse of Diagonal Matrix

[Click Here for Sample Questions]

When the elements on the main diagonal of a square matrix are the reciprocals of the corresponding elements of the original matrix, this forms the inverse of the diagonal matrix. 

For example,

n × n Diagonal Matrix and its Inverse

---

Anti Diagonal Matrix

[Click Here for Sample Questions]

When the elements of a square matrix are all zero except the ones on the diagonals (lower left to upper side corners are ≠ 0), it forms an anti diagonal matrix.

For example,

5×5 Anti Diagonal Matrix

---

Determinant of Diagonal Matrix

[Click Here for Sample Questions]

The product of the diagonal elements forms the determinant of a diagonal matrix. 

For example,

For A = \(\begin{bmatrix} 1 & 3 \\ 0 & 2 \end{bmatrix}\), the determinant is \(|A|=(1)(2)\)

For B = \(\begin{bmatrix} -2 & 4 & 0 \\ 0 & -1 & 4 \\ 0 & 0 & -4 \end{bmatrix}\), the determinant is \(|B|=(-2)(-1)(-4)\)

For C = \(\begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & -2 & 0 & 2 \\ 0 & 0 & 5 & 3 \\ 0 & 0 & 0 & -1 \end{bmatrix}\), the determinant is \(|C|=(1)(-2)(5)(-1)\)

Examples of Determinants of a Diagonal Matrix

Also Read:

---

Things to Remember

1. A square matrix with only the diagonal elements as non-zero is called a diagonal matrix. 
2. A square matrix with elements above the principal diagonal as zeroes is a lower triangular matrix.
3. A square matrix with elements below the principal diagonal as zeroes is an upper triangular matrix.
4. The diagonal matrix with all its diagonal elements as unity is called an identity matrix. 
5. The diagonal matrix where all the elements of the square matrix are zero is a zero matrix.
6. When 2 diagonal matrices are multiplied or added the resultant matrix is another matrix of the same order. 
7. The transpose of a diagonal matrix is a diagonal matrix.
8. Under multiplication, diagonal matrices are cumulative.
9. A diagonal matrix split into different blocks is termed a block diagonal matrix.
10. The product of the diagonal elements forms the determinant of a diagonal matrix. 

---