spd_test

spd_test, an Octave code which performs a few tests on a real matrix to determine whether it is symmetric positive definite (SPD).

A real or complex matrix A is positive definite if it is true that for every nonzero complex vector x, the product:

x^* * A * x > 0,

It turns out that if a matrix is positive definite in this sense, then it must also be Hermitian, or, if it is real, it must also be symmetric. The phrase "symmetric positive definite" arises so often that it is usually abbreviated as "SPD".

A real matrix A is restricted positive definite if it is true that for every nonzero real vector x:

x' * A * x > 0.

Here is a simple example of a restricted positive definite matrix which is not symmetric:

        1  1
       -1  1
      

Simple facts about a symmetric positive definite matrix A:

• A is nonsingular;
• No diagonal entry of A can be zero;
• The inverse of A is positive definite;
• the sum of two positive definite matrices is also positive definite;
• the product of two positive definite matrices is also positive definite;
• every eigenvalue of A is a strictly positive real number;
• Gauss elimination can be performed on A without pivoting;
• A has a Cholesky factorization A = L * L', where L is lower triangular.

Languages:

spd_test is available in a MATLAB version and an Octave version.

Related Data and Programs:

test_matrix, an Octave code which defines test matrices for which the condition number, determinant, eigenvalues, eigenvectors, inverse, null vectors, P*L*U factorization or linear system solution are known. Examples include the Fibonacci, Hilbert, Redheffer, Vandermonde, Wathen and Wilkinson matrices.

Source Code:

• spd_test.m, calls all the tests;
• spd_test.sh, runs all the tests;
• spd_test.txt, the output file;

• cheby_t_matrix.m, sets up the Chebyshev T matrix matrix;
• combin_matrix.m, sets up the combinatorial matrix;
• dif2_matrix.m, sets up the second difference matrix;
• r8mat_eigenvalues_positive.m, verifies that a matrix has only strictly positive real eigenvalues;
• r8mat_is_symmetric.m, reports how far away a matrix is from being symmetric.
• r8mat_xtax.m, computes x'Ax for given matrix A and vector x.