- The determinant of a matrix equals the product of its eigenvalues.
- A positive semidefinite matrix is a symmetric matrix with only nonnegative eigenvalues.
- A positive definite matrix is a symmetric matrix with only positive eigenvalues.
Combining (1) and (3) yields that a positive definite matrix is always nonsingular since its determinant never becomes zero.
Is is true that for a positive semidefinite matrix at least one of its eigenvalues equals zero and thus its determinant always equals zero => a positive semidefinite matrix is always singular?
You would say that specifically having a positive semidefinite matrix instead of a positive definite matrix implies that at least one of the eigenvalues equals zero.
Is this correct?