What is involutory matrix with example?

In mathematics, an involutory matrix is a square matrix that is its own inverse. That is, multiplication by the matrix A is an involution if and only if A2 = I, where I is the n × n identity matrix. Involutory matrices are all square roots of the identity matrix.

What is idempotent matrix with example?

Idempotent Matrix: Definition, Examples. An idempotent matrix is one which, when multiplied by itself, doesn’t change. If a matrix A is idempotent, A2 = A.

How do you find the involutory matrix?

Involutory Matrix: A matrix is said to be involutory matrix if matrix multiply by itself return the identity matrix. Involutory matrix is the matrix that is its own inverse. The matrix A is said to be involutory matrix if A * A = I.

When a matrix is equal to its inverse?

A matrix that is its own inverse is an involutory matrix: A² = I . A matrix is involutory if and only if: ( I – A ) ( I + A) = 0 . There is a one-to-one map from involutory matricies to idempotent matrices: “A is involutory“ <==> “ ½( A + I ) = B is idempotent” .

Which is an example of an involutory matrix?

An involutory matrix which is also symmetric is an orthogonal matrix, and thus represents an isometry (a linear transformation which preserves Euclidean distance ). Conversely every orthogonal involutory matrix is symmetric. As a special case of this, every reflection matrix is involutory.

Can a block diagonal matrix be an involutory matrix?

Any block-diagonal matrices constructed from involutory matrices will also be involutory, as a consequence of the linear independence of the blocks. An involutory matrix which is also symmetric is an orthogonal matrix, and thus represents an isometry (a linear transformation which preserves Euclidean distance ).

When is a reflection matrix an involutory matrix?

As a special case of this, every reflection matrix is an involutory. The determinant of an involutory matrix over any field is ±1. If A is an n × n matrix, then A is involutory if and only if ½ ( A + I) is idempotent. This relation gives a bijection between involutory matrices and idempotent matrices.

When is multiplication by a matrix an involution?

In mathematics, an involutory matrix is a square matrix that is its own inverse. That is, multiplication by the matrix A is an involution if and only if A2 = I, where I is the n × n identity matrix.