is.indefinite {matrixcalc}

Test matrix for positive indefiniteness

Description

This function returns TRUE if the argument, a square symmetric real matrix x, is indefinite. That is, the matrix has both positive and negative eigenvalues.

Usage

is.indefinite(x, tol=1e-8)

Arguments

x

a matrix

tol

a numeric tolerance level

Details

For an indefinite matrix, the matrix should positive and negative eigenvalues. The R function eigen is used to compute the eigenvalues. If any of the eigenvalues is absolute value is less than the given tolerance, that eigenvalue is replaced with zero. If the matrix has both positive and negative eigenvalues, it is declared to be indefinite.

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.negative.semi.definite

Examples

###
### identity matrix is always positive definite
###
I <- diag( 1, 3 )
is.indefinite( 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.indefinite( 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.indefinite( 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.indefinite( 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.indefinite( 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.indefinite( E )