Types of matrices

Matrices are fundamental concepts in mathematics, physics, computer science, and engineering. Let us explore more various forms of matrices and their unique characteristics. Here are some different types of matrices with examples.

1. Periodic Matrix: A periodic matrix is a square matrix. A that meets the requirements.

Example:

Thus A^2=I and the period of A is k=2.

2. Real Matrix: A real matrix has entries (elements) that are all real numbers. A real matrix is a rectangular array whose elements are all real integers (R).

Example:

Here, A is a 3×3 real matrix because all its entries are real numbers.

3.Imaginary Matrix: An imaginary matrix is a matrix in which all the elements are purely imaginary numbers. A purely imaginary number is of the form bi, where $b\in \mathbb{R}$ (a real number) and $i$ is the imaginary unit . (i=)

Example:

Here, All the entries are purely imaginary numbers like 0i, -3i, 5i etc.

4. Unitary matrix: A unitary matrix is a complex square matrix  U that meets the following condition:

Where,

• is the conjugate transpose of U ( transpose of the matrix after obtaining the complex conjugate of each member).
• $I$ is the identity matrix of the same dimension.

Example:

Compute the conjugate transpose.

Verify the Unitary property

Thus,U is a unitary matrix.

5. Normal Matrix: A normal matrix is a square matrix $A$ that commutes with its conjugate transpose, meaning:

Where,

• is the conjugate transpose of A ( transpose of A with the  complex conjugate each element  )
• $A$ is defined over the field of complex numbers but can also be real.

Example:

Conjugate transpose:

Verify

Thus, A is a normal matrix.

6. Hermitian Matrix: A Hermitian matrix is a square matrix that equals its own conjugate transpose. In other words, a matrix A is Hermitian if:

where A^H is the conjugate transpose of $A$. This means that the element in the $i$-th row and j-th column is the complex conjugate of the element in the j-th row and i-th column. Mathematically:

Example:

Here,

• The diagonal elements 4, 3, 5 are real.
• A12=2+i ,A21=2-i, showing the required conjugate symmetry.
• A13=6, A31=6 as 6 is its own conjugate.
• A23=-i,A32=i

Since , the matrix A is Hermitian.

7.Skew Hermitian: A skew-Hermitian matrix is a square matrix 𝐴 that meets the following condition:

Here, A^H is the conjugate transpose of $A$. This means the element in the $i$-th row and $j$-th column is the negative complex conjugate of the element in the $j$-th row and $i$-th column. Mathematically:

Example:

Here,

• The diagonal elements (0, 0, 0) are purely imaginary or zero.
• A12= 2+i, A21=-2-i
• A13=-i, A31=i

Since, the matrix A is skew hermitian.

8. Sub Matrix: A submatrix is a smaller matrix that is formed by selecting certain rows and/or columns from a larger matrix while maintaining their order.

How to create sub matrix

• Start with a given matrix
• Select particular rows and columns to add to the submatrix.
• Remove the unwanted rows and/or columns.

Example 1 : Submatrix from a 3×3 Matrix

Submatrix by Removing the 1st Row and 2nd Column:

• Remove the 1st row: {1, 2, 3}
• Remove the 2nd column: {2, 5, 8}

The submatrix is:

Example 2: Submatrix from a 4×4 Matrix

Submatrix by Keeping the 1st, 3rd Rows and 2nd, 4th Columns:

• Keep rows {1,3}: {1,2,3,4} and {9,10,11,12}
• Keep columns {2,4}: {2,4} and {10,12}

The submatrix is:

9. Sparse Matrix: A Sparse Matrix  has a majority of its elements as zero.

Example:

Each Types of matrices serves a unique purpose, from simplifying computations to modeling real-world problems. We have to know all Different Types of matrices for tackling challenges in mathematics, science, and technology.