Involutory Matrix

The inverse of an involutory matrix is the matrix itself. Involutory matrices must be square and always have an inverse. A square matrix is involutory when it gives the identity matrix of the same order upon multiplication by itself. A square matrix A is involutory if it is equal to its inverse, i.e., A = A-1.
For example;
    
is an involutory matrix because it satisfies both conditions mentioned.
    

Involutory Matrix Formula

Consider a 2 × 2 matrix, 
 
The matrix is involutory if it satisfies the condition:
A2 = I
This means that when we multiply the matrix A by itself, the result is the identity matrix I.
  
Upon multiplication, we get:
  
Now, we can equate it to the identity matrix
  
Comparing the terms on each side, we get four conditions
a2 + bc = 1 
ab + bd = 0
ac + dc = 0
bc + d2 = 1
Let’s factor the second and third equations
ab + bd = b(a + d) = 0
ac + dc = c(a +d) = 0
This shows us that either b = 0 or a + d = 0, and either c = 0 or a + d = 0.
To satisfy both equations without forcing b or c to be zero, we choose a + d = 0, which means d = -a. 
So, the 2 × 2 matrix is involutory if:
a2 + bc = 1 and
d = -a
 

Involutory matrices have unique characteristics that set them apart from other matrices:

• Self-Inverse property: A square matrix A is involutory if multiplying it by itself gives the identity matrix, i.e., A2 = I, which also means A = A-1.
• Product of commuting involutory matrices: If A and B are both involutory and they commute (meaning AB = BA), then their product AB is also involutory.
• Diagonal or block diagonal forms: If you build a diagonal or block diagonal matrix using involutory matrices along the diagonal, the new matrix is also involutory.
• Eigenvalues: An involutory matrix can only have eigenvalues equal to 1 or –1.
• Determinant: The determinant of an involutory matrix is always either +1 or –1, not just 1.
• Symmetric and orthogonal relationship: All symmetric involutory matrices are orthogonal, and similarly, all orthogonal involutory matrices are symmetric.
• Powers of involutory matrix: For any integer n,
      An = {A, if n is oddI, if n is even
    So, An is also involutory for all integers n.
• Relation with idempotent matrix: If A is an involutory matrix, then B =12(A + I) is an idempotent matrix, i.e., B2 = B.
• Involutory and idempotent duality: A matrix can be both involutory and idempotent only if it is an identity matrix.
   

Involutory matrices have several real-life applications in various fields. Some of them are given below:

Image reflection in computer graphics

In computer graphics, involutory matrices are used to reflect objects or images across a line (in 2D) or a plane (in 3D). Example: A reflection matrix used to flip an image over the x-axis is involutory—because reflecting it again undoes the flip.

Pauli gates in quantum computing

In quantum computing, some gates (like the Pauli-X gate) act like matrix transformations. These are modeled using involutory matrices because applying the same gate twice returns the quantum state to its original form.

Encryption and decryption in cryptography

In certain encryption methods, involutory matrices are used so that the same matrix can be used for both encrypting and decrypting a message. Since an involutory matrix is its own inverse (A = A⁻¹), we don't have to calculate a separate inverse for decryption; we just have to apply the same matrix again.

Signal reversing in signal processing

Signal processing involves applying filters and transformations. Involutory matrices help the signal be reversed after processing using the same operation. This makes testing filters or making temporary modifications convenient.

Digital circuits in electrical engineering

Some circuits switch between ON and OFF states every time a signal is sent. An involutory matrix does the same thing, therefore helping model such circuits.