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 $ax^2$.

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.

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¶

• Quadratic Form
• Symmetric Matrix

Related Concepts¶

• Variance and Standard Deviation
• Eigenvalue and Eigenvector

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