Positive definite matrix
Examples
Examples
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The $3 \times 3$ identity matrix
High school
The $3 \times 3$ identity matrix is positive definite, because $x^{\top} I x = \norm{x}^2 > 0$ for every nonzero $x$.
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The matrix \(\begin{bmatrix} 4 & 2 \\ 2 & 3 \end{bmatrix}\)
Undergraduate
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 $\ip{x}{y} = x^{\top} A y$ on $\reals^n$.
Notes
Some authors do not require $A$ to be symmetric. Then $x^{\top} A x > 0$ depends only on the symmetric part $\frac{1}{2}(A + A^{\top})$, so, for example, a rotation by less than $90°$ counts as positive definite.
For complex matrices, $A$ is required to be Hermitian, and $x^{\top}$ is replaced by the conjugate transpose $x^{\ast}$.
Distinctions involving this term
- Positive definite vs. positive semidefinite: Positive definiteness and semidefiniteness describe the sign of the quadratic form $x^{\top} A x$ of a symmetric matrix, and they appear in optimization, stability, and statistics. 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.