Background of Sylvester's Criterion

Sylvester's criterion offers a convenient way to determine whether a real symmetric matrix is positive definite. For a 3×3 matrix $A$ with entries $a_{ij}$, the criterion states that the determinants of the leading principal minors must all be positive. In practice we calculate the determinant of the upper-left 1×1, 2×2, and 3×3 submatrices. If any of these determinants is non-positive, the matrix fails to be positive definite.

Quadratic Form Interpretation

Positive definiteness arises naturally in the study of quadratic forms. Given a vector $x$, the value $x^{T}Ax$ represents a quadratic form. If this value is strictly positive for every nonzero $x$, then $A$ is positive definite. Sylvester's criterion provides an algebraic test that is equivalent to this geometric condition, making it a key tool in optimization, statistics, and physics.

Computation of Minors

Let the symmetric matrix be written as

$\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{12}& a_{22}& a_{23}\\ a_{13}& a_{23}& a_{33}\end{array}\right]$

The first leading principal minor is simply $a_{11}$. The second minor is the determinant of the top-left 2×2 block, given by $a_{11}{a}_{22}-a_{12}^2$. The third minor is the determinant of the full matrix, which involves a longer expression combining all the entries. The sign patterns of these minors determine definiteness.

Outcome Interpretation

If all three minors are positive, the matrix is positive definite. If the signs alternate beginning with negative, it is negative definite. Any mixture of positive and negative without a consistent pattern implies the matrix is indefinite. When one of the determinants is exactly zero, the matrix is semidefinite. These categories influence the behavior of quadratic functions, which might describe potential energy in physics or objective functions in optimization.

Significance in Applications

Checking positive definiteness is important when solving systems of linear equations, verifying convexity, or analyzing stability. In optimization, a positive definite Hessian indicates a local minimum. In statistics, covariance matrices must be positive semidefinite to make sense. Sylvester's criterion translates these abstract requirements into a manageable computation based solely on determinants.