What is a principal minor of a matrix?

I was going through the book on operation research by Hamdy A.Taha. It referred to principal minor of a hessian matrix. Can someone explain what is meant by a principal minor? Is it different from 'minor of a matrix'?

Solution 1:

A minor of a matrix $A$ is the determinant of a submatrix formed from $A$ by deleting some (possibly empty,) set of rows and some (possibly empty) set of columns. In order for this determinant to exist, the number of remaining rows must be equal to the number of remaining columns (and there must be at least one remaining row and column). A principal minor of a square matrix is one where the indices of the deleted rows are the same as the indices of the deleted columns.

Thus for a $3 \times 3$ matrix $A$, you could delete nothing (resulting in the determinant of the matrix itself), delete one row and the corresponding column (resulting in one of three possible $2 \times 2$ determinants), or delete two rows and the corresponding two columns (resulting in one of the three diagonal elements).

Solution 2:

As pointed out by @RobertIsrael, the principal minor is a minor in which the indices of the omitted row and column match. for example for a $3*3$ matrix: a principal minor can be created by omitting '1st row and 1st column', or by omitting '1st row, 2nd row, 1 column, 2nd column' and so on.