Characteristic Polynomial / Polynomial Equation of a Square Matrix

1. Given a \(N \times N\) Square Matrix \(A\) having \(a_{ij}\) as elements of \(i^{th}\) Row and \(j^{th}\) Column as follows
   
   \(A=\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{n1} & a_{n2} & ... & a_{nn}\end{bmatrix}\)
   
   The Characteristic Polynomial of Matrix \(A\) is given by the following Determinant
   
   \(\begin{vmatrix} a_{11}-x & a_{12} & ... & a_{1n}\\ a_{21} & a_{22}-x & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{n1} & a_{n2} & ... & a_{nn}-x\end{vmatrix}\)
   
   The Characteristic Polynomial Equation of Matrix \(A\) is given by the following Determinant Equation
   
   \(\begin{vmatrix} a_{11}-x & a_{12} & ... & a_{1n}\\ a_{21} & a_{22}-x & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{n1} & a_{n2} & ... & a_{nn}-x\end{vmatrix}= \begin{vmatrix} x-a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & x-a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{n1} & a_{n2} & ... & x-a_{nn}\end{vmatrix}=0\)   ...(1)
2. On expanding the Determinant given in equation (1) we get the following Polynomial Equation
   
   \(x^N\hspace{.2cm}+\hspace{.2cm}(-1)^{1} T_1 x^{N-1}\hspace{.2cm}+\hspace{.2cm}(-1)^{2} T_2 x^{N-2}\hspace{.2cm}+\hspace{.2cm}(-1)^{3} T_3 x^{N-3}\hspace{.2cm}+\hspace{.2cm}\cdots\hspace{.2cm} +\hspace{.2cm}(-1)^{N-1} T_{N-1}x\hspace{.2cm}+\hspace{.2cm}(-1)^{N}D =0\)   ...(2)
   
   where \(T_1, T_2, T_3, ..., T_{N-1}\) are the Traces of K^th Principal Minors of Matrix \(A\) and \(D\) is the Determinant Value of Matrix \(A\).
3. The Roots of the Characteristic Polynomial Equation of a Matrix as given by equation (2) give the Eigen Values of the Matrix.
4. All Matrices Satisfy their own Characteristic Polynomial Equation. For example, for Matrix \(A\)
   
   \(A^N\hspace{.2cm}+\hspace{.2cm}(-1)^{1} T_1 A^{N-1}\hspace{.2cm}+\hspace{.2cm}(-1)^{2} T_2 A^{N-2}\hspace{.2cm}+\hspace{.2cm}(-1)^{3} T_3 A^{N-3}\hspace{.2cm}+\hspace{.2cm}\cdots\hspace{.2cm} +\hspace{.2cm}(-1)^{N-1} T_{N-1}A\hspace{.2cm}+\hspace{.2cm}(-1)^{N}DI =0\)   ...(3)
   
   where \(I\) is \(N \times N\) Identity Matrix.