Test whether a symmetric matrix is positive definite

Returns TRUE when every eigenvalue of the symmetric matrix M exceeds the tolerance tol. Used internally by the dispatcher to decide whether the Ribando hypergeometric series can be applied directly or whether the decomposition route is required.

Usage

is_positive_definite(M, tol = 1e-10)

Arguments

M

A symmetric numeric matrix.

tol

Numeric. Eigenvalue threshold below which a value is considered non-positive. Default 1e-10.

Value

A single logical: TRUE if every eigenvalue exceeds tol, FALSE otherwise.

Details

Eigenvalues are computed with eigen on the symmetric pathway, which is numerically stable and faster than the general non-symmetric routine. The threshold form (\(\lambda > tol\)) rather than \(\lambda > 0\) guards against eigenvalues that are numerically zero from finite-precision arithmetic on near-singular matrices.

References

Horn, R. A., & Johnson, C. R. (2013). Matrix Analysis, 2nd edition. Cambridge University Press. ISBN 978-0521548236. (Sylvester criterion and eigenvalue characterisation of positive definiteness.)

See also

compute_associated_matrix for the matrix on which this test is most often invoked; is_linearly_independent for the rank-based degeneracy test; is_tridiagonal for the tridiagonal-structure test; compute_solid_angle for the dispatcher that uses these predicates.

Examples

is_positive_definite(matrix(c(2, -0.5, -0.5, 2), nrow = 2))   # TRUE
#> [1] TRUE
is_positive_definite(matrix(c(1, -2,  -2,   1), nrow = 2))    # FALSE
#> [1] FALSE