Matrix Factorization through Eigen-Value / Eigen-Vector Decomposition

1. Factorization of any Matrix through Eigen-Value/Eigen-Vector Decomposition is also called Matrix Diagonalization. This kind of factorization is only applicable for Square Matrices.
2. Following gives the Steps for Factorizing any \(N \times N\) Square Matrix \(A\) using Eigen-Value/Eigen-Vector Decomposition
   1. As given in the topic Eigen-Values and Eigen-Vectors of a Square Matrix, find the \(N\) Eigen-Values \(\lambda_1, \lambda_2, \cdots, \lambda_n\) and corresponding \(N\) Linearly Independent Eigen-Vectors \(V_1, V_2, ... , V_n\) for the Matrix \(A\).
      
      If the Matrix \(A\) does not have \(N\) Linearly Independent Eigen-Vectors, it is not possible to factorize Matrix \(A\) using Eigen-Value/Eigen-Vector Decomposition. In such case the Matrix \(A\) is called a Non Diagonalizable Matrix.
      
      If the Matrix \(A\) has \(N\) Linearly Independent Eigen-Vectors, it is possible to factorize Matrix \(A\) using Eigen-Value/Eigen-Vector Decomposition. In such case the Matrix \(A\) is called a Diagonalizable Matrix.
   2. The Eigen-Values and Eigen-Vectors of the Matrix \(A\) are related as follows
      
      \(A V_1=\lambda_1 V_1\)
      
      \(A V_2=\lambda_2 V_2\)
      
      \(\vdots\)
      
      \(A V_n=\lambda_n V_n\)   ...(1)
      
      The Eigen-Vectors Pre-Multiplied by Matrix \(A\) and the Eigen-Vectors Scaled by corresponding Eigen-Values can be arranged as Columns of \(N\times N\) Matrices. Thus the equation given by equation set (1) above can also be presented as
      
      \(\begin{bmatrix}AV_1 & AV_2 & \cdots & AV_n\end{bmatrix}=\begin{bmatrix}\lambda_1V_1 & \lambda_2V_2 & \cdots & \lambda_nV_n\end{bmatrix}\)
      
      \(\Rightarrow A\begin{bmatrix}V_1 & V_2 & \cdots & V_n\end{bmatrix}=\begin{bmatrix}V_1 & V_2 & \cdots & V_n\end{bmatrix}\begin{bmatrix}\lambda_1 & 0 & \cdots & 0 \\ 0 & \lambda_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda_n \end{bmatrix}\)   ...(2)
      
      Setting \(P=\begin{bmatrix}V_1 & V_2 & \cdots & V_n\end{bmatrix}\) and \(D=\begin{bmatrix}\lambda_1 & 0 & \cdots & 0 \\ 0 & \lambda_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda_n \end{bmatrix}\) we get
      
      \(AP=PD\)   ...(3)
      
      Post-Multiplying equation (3) above by \(P^{-1}\) on Both Sides we get
      
      \(APP^{-1}=PDP^{-1}\)
      
      \(\Rightarrow AI=PDP^{-1}\)
      
      \(\Rightarrow A=PDP^{-1}\)   ...(4)
      
      The equation (4) above gives the formula for Factorising Matrix \(A\) using Eigen-Value/Eigen-Vector Decomposition where \(P\) is the \(N \times N\) Eigen Vector Matrix corresponding to Matrix \(A\) (i.e. the Matrix containing the Eigen-Vectors of Matrix \(A\) as it's Columns) and \(D\) is the \(N \times N\) Diagonal Matrix having the Eigen-Values of Matrix \(A\) as it's Main Diagonal.
   3. The Diagonal Matrix \(D\) can also be obtained by Pre-Multiplying equation (3) above by \(P^{-1}\) on Both Sides as follows
      
      \(P^{-1}AP=P^{-1}PD\)
      
      \(\Rightarrow P^{-1}AP=ID\)
      
      \(\Rightarrow D=P^{-1}AP\)   ...(5)
      
      
3. Any of the Eigen-Vectors of the Eigen Vector Matrix \(P\) can be Scaled by any Arbitrary Numerical Constant without affecting it's calculations related to Decomposition/Factorization.
4. The Eigen-Vectors of the Eigen Vector Matrix \(P\) can be Arranged in Any Order (as long as the Eigen-Values are Arranged in Same Order in Matrix \(D\)) without affecting it's calculations related to Decomposition/Factorization.
5. The Eigen-Value/Eigen-Vector Decomposition can be used for Simplifying Calculation of Integer Powers of any Diagonalizable Square Matrix. For example, the value of \(A^M\) (where \(M\) is an Integer > 1) of any Square Matrix \(A\) can be calculated as follows
   
   \(A^M = AAAA\) ...\(M\) times = \(PDP^{-1}\hspace{1mm}PDP^{-1}\hspace{1mm}PDP^{-1}\) ...\(M\) times
   
   \(\Rightarrow A^M = PD^MP^{-1}\)
   
   \(\Rightarrow A^M = P{\begin{bmatrix}\lambda_1 & 0 & \cdots & 0 \\ 0 & \lambda_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda_n \end{bmatrix}}^MP^{-1}\)
   
   \(\Rightarrow A^M = P\begin{bmatrix}{\lambda_1}^M & 0 & \cdots & 0 \\ 0 & {\lambda_2}^M & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & {\lambda_n}^M \end{bmatrix}P^{-1}\)