Involutory Matrix

An involutory matrix is a special type of matrix in mathematics. For a matrix to be involutory, it needs to be an invertible matrix, i.e., a non-singular square matrix whose inverse exists. An involutory matrix is a square matrix whose product with itself is equal to the identity matrix of the same order. In other words, we can say that an involutory matrix is an inverse of itself. This implies if the square of a matrix is equal to the identity matrix, then it is an involutory matrix.

In this article, we will explore a special type of matrix, named involutory matrix, its definition, and formula. We will also see some examples of the involutory matrix for a better understanding of the concept.

An involutory matrix is a special kind of matrix as it satisfies the self-inverse function, i.e., an involutory matrix is its own inverse. In simple words, it can be said if the square of a square matrix A of order n is equal to the identity matrix of the same order, then A is an involutory matrix. All involutory matrices of order n are square roots of the identity matrix of order n. Some of the examples of involutory matrices are:

Let us go through the definition of an involutory matrix given below:

A square matrix A of order n × n is said to be an involutory matrix if and only if A² = I, where I is an identity matrix of order n × n. As we know that for two matrices A and B if AB = BA = I, then A and B are inverses of each other. Using the definition of inverse of a matrix, we can say that an involutory matrix is the inverse of itself. The formula for involutory matrix can also be written as, if A is an involutory matrix, then A = A^-1.

Matrix A = \(\left[\begin{array}{cc}
a & b \\
c & -a
\end{array}\right] \) is a general form of a 2 × 2 involutory matrix such that it satisfies a² + bc = 1, where a, b, c are real numbers. Using this we can make involutory matrices such as if we have a = 2, b = -1, and c = 3, then it satisfies the relation a² + bc = 1. So, the matrix \(\left[\begin{array}{cc}
2 & -1 \\
3 & -2
\end{array}\right] \) is an involutory matrix.

Now that we know the definition of the involutory matrix, let us go through some of its important properties that will help in its application and understanding the concept better:

• If A and B are involutory matrices such that AB = BA, then AB is also an involutory matrix.
• A block diagonal matrix A derived from an involutory matrix is also an involutory matrix.
• Eigenvalues of involutory matrices are always +1 and -1.
• The determinant of any involutory matrix is always ±1.
• Every symmetric involutory matrix is orthogonal and every orthogonal involutory matrix is symmetric.
• If a matrix A is involutory, then Aⁿ is also involutory for all integers n. We can say that Aⁿ = I if n is even and Aⁿ = A if n is odd.
• An involutory matrix A is an idempotent matrix if and only if A is an identity matrix.

Important Notes on Involutory Matrix

• A square matrix A is involutory if and only if A² = I or A = A^-1.
• All involutory matrices of order n are square roots of the identity matrix of order n.

Related Topics on Involutory Matrix

• Nilpotent Matrix
• Transformation Matrix
• Matrix Operations
• Equality of Matrices

What is Involutory Matrix in Maths?

An involutory matrix is a special kind of matrix as it satisfies the self-inverse function, i.e., an involutory matrix is its own inverse. An involutory matrix is a square matrix whose product with itself is equal to the identity matrix of the same order.

What is the Involutory Matrix Formula?

The formula for an involutory matrix A is A² = I or A = A^-1.

What is the Difference Between Idempotent Matrix and Involutory Matrix?

A matrix A is said to be an involutory matrix if A² = I and it is idempotent if A² = A.

How to Check if a Matrix is an Involutory Matrix?

To check if a matrix is involutory, we need to find its product with itself, i.e., A². If A² = I, where I is an identity matrix, then A is an involutory matrix.

How to Find Involutory Matrix?

Matrix A = \(\left[\begin{array}{cc}
a & b \\
c & -a
\end{array}\right] \) is a general form of a 2 × 2 involutory matrix such that it satisfies a² + bc = 1, where a, b, c are real numbers. We can find such values of a, b, c to find involutory matrix.