is.negative.semi.definite {matrixcalc}

Test matrix for negative semi definiteness

Description

This function returns TRUE if the argument, a square symmetric real matrix x, is negative semi-negative.

Usage

is.negative.semi.definite(x, tol=1e-8)

Arguments

x

a matrix

tol

a numeric tolerance level

Details

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.

Value

TRUE or FALSE.

Author(s)

Frederick Novomestky [e-mail]

References

Bellman, R. (1987). Matrix Analysis, Second edition, Classics in Applied Mathematics, Society for Industrial and Applied Mathematics.

See Also

is.positive.definite, is.positive.semi.definite, is.negative.definite, is.indefinite

Examples

###
### 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 )