R: Test matrix for positive indefiniteness

is.indefinite {matrixcalc} | R Documentation

## 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 fnovomes@poly.edu

### 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 )

[Package matrixcalc version 1.0-3 Index]