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Sylvester's criterion verifies positive definiteness for a concrete symmetric $3 \times 3$ matrix

Example

The symmetric matrix

$A = (\begin{array}{ccc} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2 \end{array})$

is positive definite.

Facts & Assumptions

Given: The displayed real symmetric matrix $A$ .

[L1]

A real symmetric matrix is positive definite exactly when all of its leading principal minors are positive (Sylvester's criterion: a real symmetric $n \times n$ matrix with $n \geq 1$ is positive definite if and only if all leading principal minors are positive).

Verification

technique · determinant computation with an independent square completion

1.1

The leading principal minors are $Δ_{1} = 2$ , $Δ_{2} = \det (\begin{array}{cc} 2 & 1 \\ 1 & 2 \end{array}) = 3$ , and $Δ_{3} = 2 (4 - 1) - (2) = 4$ . All three are positive.

algebra

1.2

Independently, for $(x, y, z) \neq 0$ , one has $(\begin{array}{ccc} x & y & z \end{array}) A (\begin{array}{c} x \\ y \\ z \end{array}) = 2 {(x + \frac{y}{2})}^{2} + \frac{3}{2} {(y + \frac{2 z}{3})}^{2} + \frac{4}{3} z^{2} > 0$ , because the three squared linear forms vanish simultaneously only at $(0, 0, 0)$ .

algebra

2.1

Hence [L1] proves that $A$ is positive definite.

step 1.1L1

3.1

Both calculations agree, and each boundary minor $k = 1, 2, 3$ has been evaluated explicitly.

step 1.1step 2.1step 1.2∎

Depends on

Sylvester's criterion: a real symmetric $n\times n$ matrix with $n\geq1$ is positive definite if and only if all leading principal minors are positive

Used by

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Sylvester's criterion verifies positive definiteness for a concrete symmetric 3×33×3 matrix