Hoofdminors

Hoofdminors, often called principal minors in English, are determinants of principal submatrices of a square matrix. For an n×n matrix A and a subset I of the index set {1,…,n}, the principal submatrix A[I,I] is formed by selecting the rows and columns whose indices lie in I. The determinant of this submatrix, det(A[I,I]), is the principal minor associated with I. The full determinant det(A) corresponds to the case I = {1,…,n}, and the 0×0 minor is conventionally taken as 1. In total a matrix has 2^n principal minors, counting all possible index subsets.

Leading principal minors are a common special case: they are the determinants of the top-left k×k submatrices A_k = A[{1,…,k},{1,…,k}] for k = 1,…,n. These leading minors play a central role in certain criteria for matrix definiteness, most notably Sylvester’s criterion, which states that a symmetric matrix is positive definite if and only if all its leading principal minors are positive.

Principal minors have several uses. They provide localized information about the determinant and the structure of a matrix, help in studying definiteness and rank properties, and are used in numerical linear algebra for testing matrix properties or in algorithms that rely on submatrix determinants. They also appear in theories involving inertia and factorization, where submatrix determinants relate to subspaces defined by subsets of indices.

Computationally, evaluating principal minors involves selecting an index set I, extracting the corresponding principal submatrix, and computing its determinant. Because the number of subsets grows exponentially with n, computing all principal minors is generally impractical for large matrices; in practice, one often analyzes specific families of minors, such as leading principal minors, or uses alternative criteria.