Exercise 1856Linear algebra › Quadratic formsL1Added 3 Nov 2026

Sylvester's criterion for positive definiteness

#Sylvester criterion#positive definite#minors

Statement

Show that the quadratic form
$Q(x,y,z) = 2x^2+2y^2+2z^2+2xy+2yz+2zx$
is positive definite using Sylvester's criterion.

Hint

The matrix of $Q$ is $A = \begin{pmatrix}2&1&1\\1&2&1\\1&1&2\end{pmatrix}$ . Compute the leading principal minors.

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Solution

Matrix.
$A = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix}.$

Leading principal minors.
- $\Delta_1 = 2 > 0$ .
- $\Delta_2 = \begin{vmatrix}2&1\\1&2\end{vmatrix}=3>0$ .
- $\Delta_3 = \det A = 2(4-1)-1(2-1)+1(1-2)=6-1-1=4>0$ .

Sylvester's criterion. All leading principal minors are positive, so $A$ is positive definite.

Conclusion. $Q$ is positive definite, i.e. $Q(x,y,z)>0$ for all $(x,y,z)\neq(0,0,0)$ .

Sylvester's criterion for positive definiteness

Statement

Solution

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