is.posi_def: Check Positive-(Semi)definiteness of a matrix

In Keefe-Murphy/IMIFA: Infinite Mixtures of Infinite Factor Analysers and Related Models

Check Positive-(Semi)definiteness of a matrix

Description

Tests whether all eigenvalues of a symmetric matrix are positive (or strictly non-negative) to check for positive-definiteness and positive-semidefiniteness, respectively. If the supplied matrix doesn't satisfy the test, the nearest matrix which does can optionally be returned.

Usage

is.posi_def(x,
            tol = NULL,
            semi = FALSE,
            make = FALSE)

Arguments

x	A matrix, assumed to be real and symmetric.
tol	Tolerance for singular values and for absolute eigenvalues - only those with values larger than tol are considered non-zero. (default: tol = max(dim(x))*max(E)*.Machine$double.eps, where E is the vector of absolute eigenvalues).
semi	Logical switch to test for positive-semidefiniteness when TRUE or positive-definiteness when FALSE (the default).
make	Logical switch to return the nearest matrix which satisfies the test - if the test has been passed, this is of course just x itself, otherwise the nearest positive-(semi)definite matrix. Note that for reasons due to finite precision arithmetic, finding the nearest positive-definite and nearest positive-semidefinite matrices are effectively equivalent tasks.

Value

If isTRUE(make), a list with two components:

check	A logical value indicating whether the matrix satisfies the test.
X.new	The nearest matrix which satisfies the test (which may just be the input matrix itself.)

Otherwise, only the logical value indicating whether the matrix satisfies the test is returned.

Examples

x    <- cov(matrix(rnorm(100), nrow=10, ncol=10))
is.posi_def(x)                                           #FALSE
is.posi_def(x, semi=TRUE)                                #TRUE

Xnew <- is.posi_def(x, semi=FALSE, make=TRUE)$X.new
identical(x, Xnew)                                       #FALSE
identical(x, is.posi_def(x, semi=TRUE, make=TRUE)$X.new) #TRUE

Keefe-Murphy/IMIFA documentation built on Jan. 31, 2024, 2:15 p.m.