Test a matrix for positive definiteness
This functionality does not run in MATLAB.
linalg::isPosDef(A) checks whether the matrix A is positive definite, so that for arbitrary vectors .
The component ring of A must be a field, i.e., a domain of category Cat::Field.
An error message is returned, if a result of an intermediate computation cannot be checked for being positive (which could happen, for example, if components of A are symbolic).
Here is an example of a positive definite matrix:
MatR := Dom::Matrix( Dom::Real ): A := MatR([[14, 6, 9], [6, 17, -4], [9, -4, 13]])
The following matrix is not positive definite:
B := MatR([[1, 2, 3], [2, 3, 4], [5, 6, 7]])
linalg::isPosDef in general does not work for matrices with symbolic entries. It may respond with an error message (because the system in general cannot decide whether a symbolic component is positive), such as for the following matrix:
delete a, b: C := matrix([[a, b], [b, a]])
Error: Cannot check whether the matrix component is positive. [linalg::factorCholesky]
However, properties of identifiers are taken into account, so that, for example, linalg::isPosDef is able to perform the test correctly for the following matrix:
assume(a > 1): C := matrix([[a, 1], [1, a]]):
Note that such computations depend on the power of the underlying property mechanism implemented in the property library.
A matrix of a domain of category Cat::Matrix