What is principal minors?

Principal minors are the determinants of submatrices obtained by deleting rows and columns from a square matrix. Specifically, the kth principal minor of an n × n matrix A is the determinant of the k × k submatrix obtained by retaining the first k rows and columns of A. The first principal minor is simply the determinant of the matrix itself, while the nth principal minor is the determinant of the entire submatrix (i.e., the determinant of A).

Principal minors play an important role in several areas of mathematics, including linear algebra, optimization, and differential equations. They provide a way to test for various properties of a matrix, such as positive-definiteness, stability, and invertibility.

In particular, a matrix A is positive definite if and only if all of its principal minors are positive. A matrix is stable if all of its principal minors have the same sign, while a matrix is invertible if and only if its determinant (i.e., the nth principal minor) is nonzero. Finally, the maximum and minimum eigenvalues of a symmetric matrix can be found by considering its principal minors.

Overall, principal minors are a powerful tool for analyzing the properties of matrices and are instrumental in many applications.