Recipe

Test definiteness using leading principal minors

Use when: you have a small symmetric matrix given as numbers, not eigenvalues, and want a definiteness verdict without computing a full eigendecomposition

Goal: a positive-definite, negative-definite, or 'neither' verdict, certified by a short list of ordinary determinants

Taught in Four tests for definiteness, and when each is the right one

Quick recipe

List the leading principal minors
Compute each minor in order, from $k=1$ up
Match the sign pattern to a verdict

1
For an $n\times n$ matrix, the $k$ -th leading principal minor $M_k$ is the determinant of the top-left $k\times k$ block, for every $k=1,\dots,n$ .

Example

For $B=\begin{bmatrix}4&1&1\\1&3&0\\1&0&2\end{bmatrix}$ , $M_1$ is the determinant of $[4]$ , $M_2$ is the determinant of $\begin{bmatrix}4&1\\1&3\end{bmatrix}$ , and $M_3$ is $\det(B)$ itself.

Why: Sylvester's criterion tests the sign pattern of exactly these $n$ determinants, and nothing else about $A$ matters to it.
2
Work through $M_1, M_2,\dots,M_n$ in increasing order, since a later minor is never needed once an earlier one breaks the pattern you're testing for.

Example

$M_1=4$ . $M_2=4(3)-1(1)=11$ . $M_3=\det(B)=19$ , expanding along the first row.

⚡ Shortcut

The moment $M_1$ already fails to be positive, stop there: $A$ can't be positive definite regardless of what $M_2,\dots,M_n$ turn out to be, so there's no need to compute them.

Why: each minor is itself the determinant of a genuine principal submatrix, and Sylvester's criterion only cares about the finished list of signs.
3
Compare the signs of $M_1,\dots,M_n$ against the two certified patterns below.

Example

$B$ 's minors are $4,\,11,\,19$ , all positive, matching the positive-definite pattern exactly.
- every $M_k>0$ — $A$ is positive definite.
- the signs strictly alternate starting negative ( $M_1<0$ , $M_2>0$ , $M_3<0,\dots$ ) — $A$ is negative definite.
- any other pattern occurs, including a minor equal to exactly $0$ — $A$ is neither; a zero minor needs the eigenvalue test to tell semidefinite from indefinite.
Why: these are the only two sign patterns Sylvester's criterion actually certifies, since flipping every sign of a negative definite matrix produces a positive definite one, alternating the signs of its minors along the way.

Common mistakes

Stopping after $M_1$ alone is positive and skipping the remaining minors that could still fail.
Testing only the leading principal minors when a semidefinite verdict is suspected; this test never certifies semidefiniteness, only strict definiteness.
Computing a non-leading principal minor (skipping rows and columns out of order) instead of always keeping the same top-left block.
Concluding "negative definite" from an alternating pattern that starts positive instead of negative.