Test matrix for negative semi definiteness
This function returns TRUE if the argument, a square symmetric real matrix x, is negative semi-negative.
is.negative.semi.definite(x, tol=1e-8)
x |
a matrix |
tol |
a numeric tolerance level |
For a negative semi-definite matrix, the eigenvalues should be non-positive.
The R function eigen is used to compute the eigenvalues.
If any of the eigenvalues in absolute value is less than the given tolerance, that eigenvalue
is replaced with zero. Then, if any of the eigenvalues is greater than zero, the matrix
is not negative semi-definite. Otherwise, the matrix is declared to be negative semi-definite.
TRUE or FALSE.
Frederick Novomestky fnovomes@poly.edu
Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.
### ### identity matrix is always positive definite I <- diag( 1, 3 ) is.negative.semi.definite( I ) ### ### positive definite matrix ### eigenvalues are 3.4142136 2.0000000 0.585786 ### A <- matrix( c( 2, -1, 0, -1, 2, -1, 0, -1, 2 ), nrow=3, byrow=TRUE ) is.negative.semi.definite( A ) ### ### positive semi-defnite matrix ### eigenvalues are 4.732051 1.267949 8.881784e-16 ### B <- matrix( c( 2, -1, 2, -1, 2, -1, 2, -1, 2 ), nrow=3, byrow=TRUE ) is.negative.semi.definite( B ) ### ### negative definite matrix ### eigenvalues are -0.5857864 -2.0000000 -3.4142136 ### C <- matrix( c( -2, 1, 0, 1, -2, 1, 0, 1, -2 ), nrow=3, byrow=TRUE ) is.negative.semi.definite( C ) ### ### negative semi-definite matrix ### eigenvalues are 1.894210e-16 -1.267949 -4.732051 ### D <- matrix( c( -2, 1, -2, 1, -2, 1, -2, 1, -2 ), nrow=3, byrow=TRUE ) is.negative.semi.definite( D ) ### ### indefinite matrix ### eigenvalues are 3.828427 1.000000 -1.828427 ### E <- matrix( c( 1, 2, 0, 2, 1, 2, 0, 2, 1 ), nrow=3, byrow=TRUE ) is.negative.semi.definite( E )
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