Positive or Negative Semidefiniteness

Check whether a symmetric matrix is positive or negative semidefinite.

Usage

isSemidefinite( m, ... )

# Default S3 method
isSemidefinite( m, ... )

# S3 method for class 'matrix'
isSemidefinite( m, positive = TRUE,
   tol = 100 * .Machine$double.eps,
   method = ifelse( nrow( m ) < 13, "det", "eigen" ), ... )

# S3 method for class 'list'
isSemidefinite( m, ... )

semidefiniteness( m, ... )

Arguments

m

a symmetric quadratic matrix or a list containing symmetric quadratic matrices.

positive

logical. Check for positive semidefiniteness (if TRUE, default) or for negative semidefiniteness (if FALSE).

tol

tolerance level (values between -tol and tol are considered to be zero).

method

method to test for semidefiniteness, either checking the signs of the principal minors (if "det", default for matrices with up to 12 rows/columns) or checking the signs of the eigenvalues (if "eigen", default for matrices with 13 or more rows/columns).

...

further arguments of isSemidefinite.list are passed to isSemidefinite.matrix;. further arguments of semidefiniteness are passed to isSemidefinite; further arguments of other functions are currently ignored.

Details

Function semidefiniteness() passes all its arguments to isSemidefinite(). It is only kept for backward-compatibility and may be removed in the future.

If argument positive is set to FALSE, isSemidefinite() checks for negative semidefiniteness by checking for positive semidefiniteness of the negative of argument m, i.e. -m.

If method "det" is used (default for matrices with up to 12 rows/columns), isSemidefinite() checks whether all principal minors (not only the leading principal minors) of the matrix m (or of the matrix -m if argument positive is FALSE) are larger than -tol. Due to rounding errors, which are unavoidable on digital computers, the calculated determinants of singular (sub-)matrices (which should theoretically be zero) can considerably deviate from zero. In order to reduce the probability of incorrect results due to rounding errors, isSemidefinite() does not calculate the determinants of (sub-)matrices with reciprocal condition numbers smaller than argument tol but sets the corresponding principal minors to (exactly) zero. The number of principal minors of an \(N \times N\) matrix is \(\sum_{k=1}^N ( N\) choose \( k )\), which gets very large for large matrices. Therefore, it is not recommended to use method "det" for matrices with, say, more than 12 rows/columns.

If method "eigen" (default for matrices with 13 or more rows/columns) is used, isSemidefinite() checks whether all eigenvalues of the matrix m (or of the matrix -m if argument positive is FALSE) are larger than -tol. In case of a singular or nearly singular matrix, some eigenvalues that theoretically should be zero can considerably deviate from zero due to rounding errors, which are unavoidable on digital computers. isSemidefinite() uses the following procedure to reduce the probability of incorrectly returning FALSE due to rounding errors in the calculation of eigenvalues of singular or nearly singular matrices: if the reciprocal condition number of an \(N \times N\) matrix is smaller than argument tol and not all of the eigenvalues of this matrix are larger than -tol, isSemidefinite() checks whether all \(( N\) choose \( (N-k) )\) \((N-k) \times (N-k)\) submatrices are positive semidefinite, where \(k\) with \(0 < k < N\) is the number of eigenvalues in the interval -tol and tol. If necessary, this procedure is done recursively.

Please note that a matrix can be neither positive semidefinite nor negative semidefinite.

Value

isSemidefinite() and semidefiniteness() return a locigal value (if argument m is a matrix) or a logical vector (if argument m is a list) indicating whether the matrix (or each of the matrices) is positive/negative (depending on argument positive) semidefinite.

References

Chiang, A.C. (1984): Fundamental Methods of Mathematical Economics, 3rd ed., McGraw-Hill.

Gantmacher, F.R. (1959): The Theory of Matrices, Chelsea Publishing.

Author

Arne Henningsen

Examples

   # a positive semidefinite matrix
   isSemidefinite( matrix( 1, 3, 3 ))
#> [1] TRUE

   # a negative semidefinite matrix
   isSemidefinite( matrix(-1, 3, 3 ), positive = FALSE )
#> [1] TRUE

   # a matrix that is positive and negative semidefinite
   isSemidefinite( matrix( 0, 3, 3 ))
#> [1] TRUE
   isSemidefinite( matrix( 0, 3, 3 ), positive = FALSE )
#> [1] TRUE

   # a matrix that is neither positive nor negative semidefinite
   isSemidefinite( symMatrix( 1:6 ) )
#> [1] FALSE
   isSemidefinite( symMatrix( 1:6 ), positive = FALSE )
#> [1] FALSE
   
   # checking a list of matrices
   ml <- list( matrix( 1, 3, 3 ), matrix(-1, 3, 3 ), matrix( 0, 3, 3 ) )
   isSemidefinite( ml )
#> [1]  TRUE FALSE  TRUE
   isSemidefinite( ml, positive = FALSE )
#> [1] FALSE  TRUE  TRUE