Positive definite vs. positive semidefinite
A symmetric matrix $A$ is positive semidefinite if $x^{\top} A x \geq 0$ for every vector $x$, and positive definite if $x^{\top} A x > 0$ for every nonzero $x$. Equivalently, the eigenvalues of $A$ are nonnegative or positive.
positive definite ⊂ positive semidefinite
Definitions
Level:
Examples
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The $3 \times 3$ identity matrix
The $3 \times 3$ identity matrix is positive definite, because $x^{\top} I x = \lVert x \rVert^2 > 0$ for every nonzero $x$. So it is also positive semidefinite.
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The matrix \(\begin{bmatrix} 4 & 2 \\ 2 & 3 \end{bmatrix}\)
The matrix \(A = \begin{bmatrix} 4 & 2 \\ 2 & 3 \end{bmatrix}\) is positive definite, because it is symmetric and \(x^{\top} A x = 4 x_1^2 + 4 x_1 x_2 + 3 x_2^2 = (2 x_1 + x_2)^2 + 2 x_2^2,\) which is positive unless $x_2 = 0$ and $2 x_1 + x_2 = 0$, that is, unless $x = 0$.
Properties
- A symmetric matrix is positive definite if and only if all of its eigenvalues are positive.
- A symmetric matrix is positive definite if and only if all of its leading principal minors (the determinants of its upper-left $k \times k$ submatrices) are positive (Sylvester’s criterion).
- A symmetric matrix is positive definite if and only if it has a Cholesky decomposition $A = L L^{\top}$ with $L$ invertible.
- A positive definite matrix $A$ defines an inner product $\langle x, y \rangle = x^{\top} A y$ on $\reals^n$.
Examples
-
The $3 \times 3$ identity matrix
The $3 \times 3$ identity matrix is positive definite, because $x^{\top} I x = \lVert x \rVert^2 > 0$ for every nonzero $x$. So it is also positive semidefinite.
-
The matrix \(\begin{bmatrix} 4 & 2 \\ 2 & 3 \end{bmatrix}\)
The matrix \(\begin{bmatrix} 4 & 2 \\ 2 & 3 \end{bmatrix}\) is positive definite, because its leading principal minors $4$ and $4 \cdot 3 - 2 \cdot 2 = 8$ are positive. So it is also positive semidefinite.
Properties
- A symmetric matrix is positive semidefinite if and only if all of its eigenvalues are nonnegative.
- Every matrix of the form $B^{\top} B$ is positive semidefinite, because $x^{\top} B^{\top} B x = \lVert B x \rVert^2 \geq 0$. Conversely, every positive semidefinite matrix can be written in this form.
- A positive semidefinite matrix is positive definite if and only if it is invertible.
- Sylvester’s criterion with $\geq$ does not test positive semidefiniteness: all leading principal minors of \(\begin{bmatrix} 0 & 0 \\ 0 & -1 \end{bmatrix}\) are $\geq 0$, but $x = (0, 1)$ gives $x^{\top} A x = -1$. All principal minors (not only the leading ones) must be nonnegative.
Examples
| Example | positive definite | positive semidefinite | Why |
|---|---|---|---|
| The $3 \times 3$ identity matrix | ✓ | ✓ | The $3 \times 3$ identity matrix is positive definite, because $x^{\top} I x = \lVert x \rVert^2 > 0$ for every nonzero $x$. So it is also positive semidefinite. |
| The matrix \(\begin{bmatrix} 4 & 2 \\ 2 & 3 \end{bmatrix}\) | ✓ | ✓ | The matrix \(\begin{bmatrix} 4 & 2 \\ 2 & 3 \end{bmatrix}\) is positive definite, because its leading principal minors $4$ and $4 \cdot 3 - 2 \cdot 2 = 8$ are positive. So it is also positive semidefinite. |
Missing example: begin_bmatrix_1_2_2_4_end_bmatrix | |||
Missing example: begin_bmatrix_1_2_2_3_end_bmatrix | |||
Missing example: matrix_1_1_minus_1_1 | |||
Notes
- Negative definite and negative semidefinite matrices are defined the same way, with the inequalities reversed: $A$ is negative definite exactly when $-A$ is positive definite.
- A matrix that is neither positive nor negative semidefinite is indefinite.
- A positive definite function, as used for Lyapunov functions, is a different notion: a function with $V(0) = 0$ and $V(x) > 0$ for $x \neq 0$. The quadratic form $V(x) = x^{\top} A x$ is a positive definite function exactly when $A$ is a positive definite matrix.