Negative Definite Matrix Checker
Check whether a symmetric matrix is negative definite — the mirror image of positive definite.
Check whether a symmetric matrix is negative definite — the mirror image of positive definite.
How Negative Definite Matrix Checker Works
A symmetric matrix is negative definite exactly when its negation is positive definite, which translates to the leading principal minors alternating in sign, starting negative (the 1×1 minor negative, the 2×2 minor positive, the 3×3 minor negative, and so on). The calculator computes each leading minor and checks this pattern.
Who Uses Negative Definite Matrix Checker and Why
- Confirming a Hessian matrix at a critical point represents a true local maximum during an optimization problem.
- Checking the mirror-image condition of positive definiteness when analyzing a symmetric matrix that's expected to be "all negative" in its quadratic form.
- Verifying a matrix in a stability or control-theory context where negative definiteness signals a system settling rather than diverging.
Mistakes to Avoid
- Expecting a simple "all entries negative" check — negative definiteness has nothing to do with the sign of individual entries; it's about the alternating-sign pattern of the leading principal minors.
- Forgetting the alternating pattern requirement — the 1×1 leading minor must be negative, the 2×2 positive, the 3×3 negative, and so on; checking for uniformly negative minors will give the wrong answer.
- Testing a non-symmetric matrix, since this test (like positive definiteness) is only defined for symmetric matrices.
Tips for Best Results
- The fastest way to reason about this: a matrix A is negative definite exactly when −A is positive definite, so if you already understand positive definiteness, you understand this too.
- A negative definite Hessian at a critical point is the standard confirmation of a local maximum, the direct mirror of how positive definite confirms a minimum.
Fixing Common Problems
I expected a negative result but got a semi-definite or indefinite classification instead. — Check whether any leading principal minor came out exactly zero rather than strictly negative/positive — a zero anywhere in the sequence means the matrix is at most semi-definite, not strictly definite.
Terms Explained
Negative definite: A symmetric matrix whose leading principal minors alternate in sign starting negative (1×1 negative, 2×2 positive, 3×3 negative, and so on).
Hessian matrix: A matrix of second derivatives used in optimization; its definiteness at a critical point determines whether that point is a minimum, maximum, or saddle point.