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Minors of a Matrix – Explained with Examples

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The minors of a matrix are quantities that are obtained by deleting some rows and columns from a matrix and finding the determinant of the resulting submatrix. The minor  M_{ij} is the determinant of the submatrix formed by deleting the i^{th} row and the j^{th} column from the given matrix.

Example 1:

Consider the matrix A = \begin{bmatrix} 7 & -2 & 4\\ -8 & 6 & 3\\ 5 & 0 & -1 \end{bmatrix}.

We find all the minors of this matrix by calculating the determinants of the submatrices below.

M_{11} = \left|\begin{array}{cc}6 & 3\\0 & -1\end{array}\right| = (6)\cdot (-1) - (3)\cdot (0) = -6 M_{12} = \left|\begin{array}{cc}-8 & 3\\5 & -1\end{array}\right| = (-8)\cdot (-1) - (3)\cdot (5) = -7 M_{13} = \left|\begin{array}{cc}-8 & 6\\5 & 0\end{array}\right| = -30 M_{21} = \left|\begin{array}{cc}-2 & 4\\0 & -1\end{array}\right| = 2 M_{22} = \left|\begin{array}{cc}7 & 4\\5 & -1\end{array}\right| = -27 M_{23} = \left|\begin{array}{cc}7 & -2\\5 & 0\end{array}\right| = 10 M_{31} = \left|\begin{array}{cc}-2 & 4\\6 & 3\end{array}\right| = -30 M_{32} = \left|\begin{array}{cc}7 & 4\\-8 & 3\end{array}\right| = 53 M_{33} = \left|\begin{array}{cc}7 & -2\\-8 & 6\end{array}\right| = 26

Example 2:

We can similarly compute minors for a 4 \times 4 matrix. If we are given the matrix,

A = \begin{bmatrix} 4 & 9 & 5 & 3 \\ -2 & -3 & 8 & 6 \\ 0 & 0 & -4 & 8 \\ 8 & 1 & 0 & 2 \end{bmatrix}

we can similarly compute the minors M_{ij} by deleting the i^{th} row and the j^{th} column and computing the determinant of the resulting submatrix. For example, we can compute M_{23} as,

M_{23} = det \left|\begin{array}{ccc}4 & 9 & 3\\0 & 0 & 8\\8 & 1 & 2\end{array}\right| = 544

The other minors are computed in a similar fashion.

Principal Minors of a Matrix:

When we compute minors of a matrix we delete some rows and columns. We say that a minor is principal if whenever we delete the i^{th} row we also delete the i^{th} column and vice versa. The principal minors are denoted as M^{IJ} where I is the set of deleted row indices and J is the set of deleted column indices. Note that I=J for principal minors.

For example, consider the matrix A = \begin{bmatrix} 4 & 9 & 5 & 3 \\ -2 & -3 & 8 & 6 \\ 0 & 0 & -4 & 8 \\ 8 & 1 & 0 & 2 \end{bmatrix}.

Suppose we take I=J=\{2,4\} . This means that we delete the 2^{nd} and 4^{th} rows as well as the 2^{nd} and 4^{th} columns. The principal minor in this case is computed as,

M_{IJ} = \left|\begin{array}{cc}4 & 5\\0 & -4\end{array}\right| = (4)\cdot (-4) - (5)\cdot (0) = -16
Leading Principal Minors of a Matrix:

The minors corresponding to the upper left portion of a matrix are known as the leading principal minors of the matrix. Considering the same example as before,

A = \begin{bmatrix} 4 & 9 & 5 & 3 \\ -2 & -3 & 8 & 6 \\ 0 & 0 & -4 & 8 \\ 8 & 1 & 0 & 2 \end{bmatrix}

The leading principal minors are,

det \begin{bmatrix} 4 \end{bmatrix} = 4 det \begin{bmatrix} 4 & 9 \\ -2 & -3 \end{bmatrix} = -30 det \begin{bmatrix} 4 & 9 & 5 \\ -2 & -3 & 8 \\ 0 & 0 & -4 \end{bmatrix} = 120 det \begin{bmatrix} 4 & 9 & 5 & 3 \\ -2 & -3 & 8 & 6 \\ 0 & 0 & -4 & 8 \\ 8 & 1 & 0 & 2 \end {bmatrix} = -5112

If all the leading principal minors of a matrix are positive, the the matrix is a positive definite matrix.

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