svrep (version 0.7.0)

get_nearest_psd_matrix: Approximates a symmetric, real matrix by the nearest positive semidefinite matrix.

Description

Approximates a symmetric, real matrix by the nearest positive semidefinite matrix in the Frobenius norm, using the method of Higham (1988). For a real, symmetric matrix, this is equivalent to "zeroing out" negative eigenvalues. See the "Details" section for more information.

Usage

get_nearest_psd_matrix(X)

Value

The nearest positive semidefinite matrix of the same dimension as X.

Arguments

X

A symmetric, real matrix with no missing values.

Details

Let \(A\) denote a symmetric, real matrix which is not positive semidefinite. Then we can form the spectral decomposition \(A=\Gamma \Lambda \Gamma^{\prime}\), where \(\Lambda\) is the diagonal matrix whose entries are eigenvalues of \(A\). The method of Higham (1988) is to approximate \(A\) with \(\tilde{A} = \Gamma \Lambda_{+} \Gamma^{\prime}\), where the \(ii\)-th entry of \(\Lambda_{+}\) is \(\max(\Lambda_{ii}, 0)\).

References

- Higham, N. J. (1988). "Computing a nearest symmetric positive semidefinite matrix." Linear Algebra and Its Applications, 103, 103–118.

Examples

X <- matrix(
  c(2, 5, 5,
    5, 2, 5,
    5, 5, 2),
  nrow = 3, byrow = TRUE
)
get_nearest_psd_matrix(X)