HesapHanem

Positive Definite Matrix Checker

Enter the entries a, b, c of the symmetric matrix.

How it is calculated

You give the three distinct entries of the symmetric matrix with rows (a, b) and (b, c). The tool computes det = a·c − b² and then applies these tests in order: positive definite if a > 0 and det > 0; negative definite if a < 0 and det > 0; positive semidefinite if a ≥ 0, c ≥ 0 and det ≥ 0; negative semidefinite if a ≤ 0, c ≤ 0 and det ≥ 0; anything else is indefinite. Because the matrix is built from a, b, c, it is symmetric by design. The determinant and the entry a are shown to 2 decimals. Entries are plain numbers and can be negative or zero.

Worked example

Take the matrix with a = 2, b = −1, c = 3. Determinant = 2 × 3 − (−1)² = 6 − 1 = 5.00. The leading entry a = 2 is above 0 and the determinant 5 is above 0, so the first test is satisfied and the matrix is Positive definite. As a check, both eigenvalues, (5 ± √5) / 2, are positive (about 1.38 and 3.62). Changing b to 2 would give det = 6 − 4 = 2, still positive definite, but b = 3 gives det = −3 and an indefinite matrix.

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Common mistakes

  • Checking only a and the determinant for semidefiniteness. With zeros allowed you must also check c; for a = 0, b = 0, c = −1 the tool correctly reports negative semidefinite.
  • Entering a non-symmetric matrix. The tool takes one value b for both off-diagonal positions, so for [[1, 2], [3, 1]] you would first have to symmetrise it, for example with b = (2 + 3) / 2.
  • Treating a positive determinant as proof of positive definiteness. If a is negative and det is positive, both eigenvalues are negative, so the matrix is negative definite.
  • Mixing up the roles of the entries. In [[a, b], [b, c]] the entries a and c are on the diagonal, and b is the repeated off-diagonal one.
  • Expecting a 3×3 or larger matrix to work. This tool classifies 2×2 matrices only.

About this calculator

For a 2×2 symmetric matrix written as [[a, b], [b, c]], its definiteness — whether the quadratic form it represents is always positive, always negative, or mixed — can be read directly from two numbers: the leading entry a and the determinant, det = ac − b². By Sylvester’s criterion applied to the 2×2 case, the matrix is positive definite if a is greater than 0 and the determinant is greater than 0; positive semidefinite if a, c and the determinant are all at least 0 (equality allowed — both diagonal entries must be checked, since leading minors alone do not settle semidefiniteness), with negative semidefinite as the mirrored case; negative definite if a is less than 0 and the determinant is greater than 0; and indefinite whenever the determinant is negative, meaning the matrix produces both positive and negative values depending on direction.

This classification matters across linear algebra and its applications: in optimization, the Hessian matrix of a function at a critical point must be positive definite for that point to be a local minimum (negative definite for a local maximum, indefinite for a saddle point); in statistics, a valid covariance matrix must be positive semidefinite; and in numerical methods, positive definiteness guarantees certain matrix decompositions (like the Cholesky decomposition) exist.

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Comments (1)

  • Editorial team

    For a Hessian-based test, the borderline cases deserve attention. The matrix with a = 1, b = 1, c = 1 comes back Positive semidefinite with a determinant of 0.00, which means the second-derivative test alone cannot call a minimum. A pair like a = 1, b = 2, c = 1 gives determinant -3.00 and Indefinite, a saddle. The tool handles 2 by 2 symmetric matrices only. What matrix sizes do you need?

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