Definite Matrix¶

Definition¶

Let $\boldsymbol{A}$ be a real symmetric matrix. The definiteness of $\boldsymbol{A}$ classifies the sign of the quadratic form $\boldsymbol{x}^{\top} \boldsymbol{A} \boldsymbol{x}$ :

Positive definite: $\boldsymbol{x}^{\top}\boldsymbol{A}\boldsymbol{x}>0$ for every nonzero vector $\boldsymbol{x}$ .
Positive semidefinite: $\boldsymbol{x}^{\top}\boldsymbol{A}\boldsymbol{x}\ge 0$ for every vector $\boldsymbol{x}$ .
Negative definite: $\boldsymbol{x}^{\top}\boldsymbol{A}\boldsymbol{x}<0$ for every nonzero vector $\boldsymbol{x}$ .
Negative semidefinite: $\boldsymbol{x}^{\top}\boldsymbol{A}\boldsymbol{x}\le 0$ for every vector $\boldsymbol{x}$ .
Indefinite: $\boldsymbol{x}^{\top}\boldsymbol{A}\boldsymbol{x}$ takes both positive and negative values.

Interpretation¶

Definiteness tells whether the quadratic form determined by a symmetric matrix is always positive, always nonnegative, always negative, always nonpositive, or changes sign. It is the matrix version of the sign of $a$ in $ax^2$ .

Four panels of level curves of quadratic forms: ellipses for a positive definite and a negative definite matrix, parallel lines for a semidefinite one, and hyperbolas for an indefinite one.

The four panels show the curves on which $\boldsymbol{x}^{\top}\boldsymbol{A}\boldsymbol{x}$ is constant, its level curves, for $\operatorname{diag}(1,2)$ , $\operatorname{diag}(-1,-2)$ , $\operatorname{diag}(0,1)$ and $\operatorname{diag}(1,-1)$ . For a diagonal matrix the form is $d_1x_1^2+d_2x_2^2$ $d_{2} x_{2}^{2}$ , so its signs can be read off the diagonal. The first two give ellipses, the first form positive and the second negative in every direction. The third form, $x_2^2$ , gives horizontal lines and is zero along the whole $x_1$ -axis: nonnegative but not positive, so positive semidefinite. The fourth, $x_1^2-x_2^2$ $x_{2}^{2}$ , gives hyperbolas and takes both signs.

Positive definite means $\boldsymbol{x}^{\top}\boldsymbol{A}\boldsymbol{x}>0$ for every $\boldsymbol{x}\neq\boldsymbol{0}$ $0$ : the curve on the right stays above $0$ . Try the examples, and look for directions where the form vanishes.

A positive definite $\boldsymbol{A}$ gives a bowl, a negative definite one a cap, and an indefinite one a saddle; the eigenvalues are shown below.

The definition quantifies over all nonzero $\boldsymbol{x}$ , not only the axes. Testing the standard basis vectors gives $\boldsymbol{e}_i^{\top}\boldsymbol{A}\boldsymbol{e}_i=a_{ii}$ $a_{ii}$ , so a positive definite matrix has positive diagonal entries. The converse fails: $\begin{pmatrix}1&2\\2&1\end{pmatrix}$ has a positive diagonal, but the value $1-4+1=-2$ at $\boldsymbol{x}=(1,-1)$ $(1, - 1)$ , so it is indefinite. The categories overlap: every positive definite matrix is also positive semidefinite, and the zero matrix is both positive and negative semidefinite.

In one variable, the second-order condition reads the sign of $f''(x_0)$ at a stationary point. For functions of several variables, a symmetric matrix of second derivatives takes the place of $f''(x_0)$ , and its definiteness separates local minima, maxima and saddle points.

Key Results¶

A symmetric matrix is positive definite if and only if all its eigenvalues are positive.
A symmetric matrix is positive semidefinite if and only if all its eigenvalues are nonnegative.
A symmetric matrix is negative definite if and only if all its eigenvalues are negative.
A symmetric matrix is negative semidefinite if and only if all its eigenvalues are nonpositive.
A symmetric matrix is indefinite if and only if it has both positive and negative eigenvalues.

Examples¶

$\boldsymbol{I}_n$ is positive definite, while $-\boldsymbol{I}_n$ is negative definite.
$\begin{pmatrix}1&0\\0&0\end{pmatrix}$ is positive semidefinite, and $\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ is indefinite.

Prerequisites¶

Sources¶

Math II: Linear Algebra · Week 6

Aliases: nonnegative definite Keywords: positive definite matrix, positive semidefinite matrix, negative definite matrix, negative semidefinite matrix, indefinite matrix, definiteness

Last updated: 20 October 2026 03:02