Principal Minors and Traces of Principal Minors 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
   
   \(A=\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\a_{n1} & a_{n2} & ... & a_{nn}\end{bmatrix}\)
   
   a \(K^{th}\) Principal Minor of a \(N\times N\) Square Matrix \(A\) (where \(1 \leq K < N \) ) is the Determinant Value of the \(K\times K\) Sub-Matrix Obtained by Removing \(N-K\) Number of Rows (and Corresponding Columns) from the Matrix.
2. For any \(N\times N\) Square Matrix \(A\), the Total Number of \(K^{th}\) Principal Minors is \(C(N,K)\) (or \(C(N,N-K)\) which gives the Number of Combinations Without Repeatition that can be formed out of \(N\) Items taking \(K\) Items a a time).
3. The Sum of All \(K^{th}\) Principal Minors of a Square Matrix is called the Trace of \(K^{th}\) Principal Minors of the Square Matrix.
4. When \(K =1 \), the \(K^{th}\) Principal Minors are the Main Diagonal Elements and Trace of \(K^{th}\) Principal Minors of the Square Matrix is Same as Trace of the Matrix.
5. The Absolute Values of Traces of \(K^{th}\) Principal Minors of a Square Matrix give the Absolute Values Co-efficients of the Charateristic Polynomial/Polynomial Equation of the Square Matrix .
6. Following example calculates the \(K^{th}\) Principal Minors and Traces of \(K^{th}\) Principal Minors of a \(2\times 2\) Matrix \(A\)
   
   \(A=\begin{bmatrix} -5 & 4 \\ 3 & 7\end{bmatrix}\)
   
   Total Number of \(1^{st}\) Principal Minors = \(C(2,1)=2\)
   
   \(1^{st}\) Principal Minor on removing the \(1^{st}\) Row and Column=\(\begin{vmatrix}7\end{vmatrix}=7\)
   
   \(1^{st}\) Principal Minor on removing the \(2^{nd}\) Row and Column=\(\begin{vmatrix}-5\end{vmatrix}=-5\)
   
   Trace of the \(1^{st}\) Principal Minors \(T_1 =7-5=2\)
7. Following example calculates the \(K^{th}\) Principal Minors and Traces of \(K^{th}\) Principal Minors of a \(3\times 3\) Matrix \(A\)
   
   \(A=\begin{bmatrix} 3 & -4 & 1\\7 & 2 & 6 \\-2 & 8 & 9\end{bmatrix}\)
   
   Total Number of \(1^{st}\) Principal Minors = \(C(3,1)=3\)
   
   \(1^{st}\) Principal Minor on removing the \(1^{st}\) and \(2^{nd}\) Rows and Columns=\(\begin{vmatrix}9\end{vmatrix}=9\)
   
   \(1^{st}\) Principal Minor on removing the \(1^{st}\) and \(3^{rd}\) Rows and Columns=\(\begin{vmatrix}2\end{vmatrix}=2\)
   
   \(1^{st}\) Principal Minor on removing the \(2^{nd}\) and \(3^{rd}\) Rows and Columns=\(\begin{vmatrix}3\end{vmatrix}=3\)
   
   Trace of the \(1^{st}\) Principal Minors \(T_1 =9+2+3=14\)
   
   
   Total Number of \(2^{nd}\) Principal Minors = \(C(3,2)=3\)
   
   \(2^{nd}\) Principal Minor on removing the \(1^{st}\) Row and Column=\(\begin{vmatrix}2 & 6 \\ 8 & 9\end{vmatrix}=-30\)
   
   \(2^{nd}\) Principal Minor on removing the \(2^{nd}\) Row and Column=\(\begin{vmatrix}3 & 1 \\ -2 & 9\end{vmatrix}=29\)
   
   \(2^{nd}\) Principal Minor on removing the \(3^{rd}\) Row and Column=\(\begin{vmatrix}3 & -4 \\ 7 & 2\end{vmatrix}=34\)
   
   Trace of the \(2^{nd}\) Principal Minors \(T_2 =-30+29+34=33\)
8. Following example calculates the \(K^{th}\) Principal Minors and Traces of \(K^{th}\) Principal Minors of a \(4\times 4\) Matrix \(A\)
   
   \(A=\begin{bmatrix} 2 & 1 & 5 & 3\\1 & -3 & -8 & 4 \\6 & -7 & 5 & 2 \\7 & 3 & 5 & -9\end{bmatrix}\)
   
   Total Number of \(1^{st}\) Principal Minors = \(C(4,1)=4\)
   
   \(1^{st}\) Principal Minor on removing the \(1^{st}\), \(2^{nd}\) and \(3^{rd}\) Rows and Columns=\(\begin{vmatrix}-9\end{vmatrix}=-9\)
   
   \(1^{st}\) Principal Minor on removing the \(1^{st}\), \(2^{nd}\) and \(4^{th}\) Rows and Columns=\(\begin{vmatrix}5\end{vmatrix}=5\)
   
   \(1^{st}\) Principal Minor on removing the \(1^{st}\), \(3^{rd}\) and \(4^{th}\) Rows and Columns=\(\begin{vmatrix}-3\end{vmatrix}=-3\)
   
   \(1^{st}\) Principal Minor on removing the \(2^{nd}\), \(3^{rd}\) and \(4^{th}\) Rows and Columns=\(\begin{vmatrix}2\end{vmatrix}=2\)
   
   Trace of the \(1^{st}\) Principal Minors \(T_1 =-9+5-3+2=-5\)
   
   
   Total Number of \(2^{nd}\) Principal Minors = \(C(4,2)=6\)
   
   \(2^{nd}\) Principal Minor on removing the \(1^{st}\) and \(2^{nd}\) Rows and Columns=\(\begin{vmatrix}5 & 2 \\ 5 & -9\end{vmatrix}=-55\)
   
   \(2^{nd}\) Principal Minor on removing the \(1^{st}\) and \(3^{rd}\) Rows and Columns=\(\begin{vmatrix}-3 & 4 \\ 3 & -9\end{vmatrix}=15\)
   
   \(2^{nd}\) Principal Minor on removing the \(1^{st}\) and \(4^{th}\) Rows and Columns=\(\begin{vmatrix}-3 & -8 \\ -7 & 5\end{vmatrix}=-71\)
   
   \(2^{nd}\) Principal Minor on removing the \(2^{nd}\) and \(3^{rd}\) Rows and Columns=\(\begin{vmatrix}2 & 3 \\ 7 & -9\end{vmatrix}=-39\)
   
   \(2^{nd}\) Principal Minor on removing the \(2^{nd}\) and \(4^{th}\) Rows and Columns=\(\begin{vmatrix}2 & 5 \\ 6 & 5\end{vmatrix}=-20\)
   
   \(2^{nd}\) Principal Minor on removing the \(3^{rd}\) and \(4^{th}\) Rows and Columns=\(\begin{vmatrix}2 & 1 \\ 1 & -3\end{vmatrix}=-7\)
   
   Trace of the \(2^{nd}\) Principal Minors \(T_2 =-55+15-71-39-20-7=-177\)
   
   
   Total Number of \(3^{rd}\) Principal Minors = \(C(4,3)=4\)
   
   \(3^{rd}\) Principal Minor on removing the \(1^{st}\) Row and Column=\(\begin{vmatrix}-3 & -8 & 4 \\ -7 & 5 & 2 \\ 3 & 5 & -9 \end{vmatrix}=421\)
   
   \(3^{rd}\) Principal Minor on removing the \(2^{nd}\) Row and Column=\(\begin{vmatrix}2 & 5 & 3 \\ 6 & 5 & 2 \\ 7 & 5 & -9 \end{vmatrix}=215\)
   
   \(3^{rd}\) Principal Minor on removing the \(3^{rd}\) Row and Column=\(\begin{vmatrix}2 & 1 & 3 \\ 1 & -3 & 4 \\ 7 & 3 & -9 \end{vmatrix}=139\)
   
   \(3^{rd}\) Principal Minor on removing the \(4^{th}\) Row and Column=\(\begin{vmatrix}2 & 1 & 5 \\ 1 & -3 & -8 \\ 6 & -7 & 5 \end{vmatrix}=-140\)
   
   Trace of the \(3^{rd}\) Principal Minors \(T_3 =421+215+139-140=635\)
9. You can use the Principal Minors Calculator to calculate Principal Minors of any Square Matrix.