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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.
Related Calculators
Principal Minors, Trace of Principal Minors, Determinant and Polynomial of Matrix Calculator
Related Topics and Calculators
Determinant, Minor, Cofactor and Adjoint of a Square Matrix,    Characteristic Polynomial / Polynomial Equation of a Square Matrix,    Combinations Without Repeatition,    Introduction to Matrix Algebra
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