The matrix $\begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix}$

Terms

Term	This example	Why
symmetric	–
positive definite matrix	–	The matrix $A = \begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix}$ is not positive definite under the usual convention, because it is not symmetric, even though $x^{\top} A x = x_1^2 + x_2^2 > 0$ for every nonzero $x$. Authors who do not require symmetry call it positive definite.
positive semidefinite	✗

Positive definite vs. positive semidefinite: The matrix $\begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix}$ satisfies $x^{\top} A x = x_1^2 + x_2^2 > 0$ for every nonzero $x$, but it is not symmetric, so under the usual convention it is neither positive definite nor positive semidefinite. Authors who do not require symmetry call it positive definite.

Term	This example	Why
symmetric	–
positive definite matrix	–	The matrix \(A = \begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix}\) is not positive definite under the usual convention, because it is not symmetric, even though $x^{\top} A x = x_1^2 + x_2^2 > 0$ for every nonzero $x$. Authors who do not require symmetry call it positive definite.
positive semidefinite	✗