15.3.10  Computing inertia of a symmetric matrix

The inertia command computes the inertia of a real symmetric matrix, i.e. the number of positive, negative and zero eigenvalues, by performing LDL decomposition (see Section 15.3.9). It can also use the factorization for solving a system of linear equations if certain inertia-related conditions are satisfied.

• inertia takes one mandatory argument and one or two optional arguments:
  • A, a symmetric matrix of order n with real coefficients.
  • Optionally, B, a matrix of type m× n or a vector of length n.
  • Optionally, p₀, an integer such that 0≤ p₀≤ n.
  Note that inertia does not check whether A is symmetric.
• inertia(A) returns the list [p,n,z] where p, n and z are the numbers of positive, negative and zero eigenvalues of A, respectively.
• inertia(A,B ⟨,p₀ ⟩) returns the sequence containing two elements: the list [p,n,z] and the solution X to the system AX^T=B^T in case z=0. If p₀ is given, then X is not computed if p≠ p₀ and an empty list is returned instead.
• The inertia of A is computed from the LDL factorization of A. The matrix X can be computed at a small cost once L and D are known. The purpose of the argument p₀ is to prevent unnecessary computation if the matrix A has to be adjusted for the desired value of p before solving the system.

Examples

If we set p₀=1, then X is not computed.