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

Source:`R/quadratic_forms.R`

`get_nearest_psd_matrix.Rd`

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.

## 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 \(\max(\Lambda_{ii}, 0)\).