Periodic, Indempotent, Involutary and Nilpotent Matrices

1. Periodic Matrix: If any Square Matrix when multiplied by itself \(K\) times gives back the Same Matrix then such Matrix is called a Periodic Matrix. The Positive Integer \(K+1\) is called the Period of the Matrix. That is, for any Periodic Matrix \(A\)
   
   \(A^{K+1}=A\)
   
   Also, for any Periodic Matrix \(A\) when \(K > 1\)
   
   \(A^K=I\)
   
   \(A^{K-1}=A^{-1}\)
   
   Periodic Matrices having a Period of 2 (i.e. when \(K = 1\) ) are called Indempotent Matrices
   
   Periodic Matrices having a Period of 3 (i.e. when \(K = 2\) ) are called Involutary Matrices
   
   Permutation Matrices, Reflection Matrices and Projection and Rejection Matrices are some examples of Periodic Matrices.
2. Indempotent Matrix: If any Square Matrix when multiplied by itself gives back the Same Matrix then such Matrix is called an Indempotent Matrix. Hence Indempotent Matrices are Periodic Matrices having a Period of 2. For any Indempotent Matrix \(A\)
   
   \(A^{N}=A\)   (where \(N\) is an Integer \(\geq 1\))
   
   Following are some properties of Indempotent Matrices
   1. All NULL Matrices and Identity Matrices are Indempotent
   2. All Non NULL and Non Identity Indempotent Matrices are Singular (i.e. their Determinant is 0)
   3. If \(M\) is an Indempotent Matrix and \(I\) is an Identity Matrix then \(I-M\) is also an Indempotent Matrix.
   Projection and Rejection Matrices are examples of Indempotent Matrices.
3. Involutary Matrix: If any Square Matrix when multiplied by itself gives back the Identity Matrix then such Matrix is called an Involutary Matrix. Hence, Any Involutary Matrix is an Inverse of itself. For any Involutary Matrix \(A\)
   
   \(AA=A^2=I\hspace{5mm}\Rightarrow A=A^{-1}\hspace{5mm}\Rightarrow AAA=A^{3}=AA^{2}=AI=A\)
   
   Thus, Involutary Matrices are Periodic Matrices having a Period of 3.
   
   Also, for any Involutary Matrix \(A\)
   
   \(A^N=I\)   (If \(N\) is Even)
   
   \(A^N=A\)   (If \(N\) is Odd)
   
   Reflection Matrices, Rotation By 180\(^\circ\) Matrices and Permutation Matrices containing only Disjoint Transpositions are some examples of Involutary Matrices.
4. Nilpotent Matrix: If any Square Matrix when multiplied by itself \(K\) times gives back a NULL Matrix then such Matrix is called a Nilpotent Matrix. The Positive Integer \(K +1\) is called the Degree of the Matrix. For any Nilpotent Matrix \(A\)
   
   \(A^{K+1}=\) NULL Matrix