Positive Definite Symmetric Matrices

A symmetric matrix A is positive definite if any of the following equivalent conditions hold: all the eigenvalues of A are positive; the product xTAx > 0 for all nonzero vectors x; or there exists a nonsingular matrix W such that A = WTW.

A symmetric matrix A is positive semidefinite if all the eigenvalues of A are nonnegative, is negative definite if all the eigenvalues are negative, and is negative semidefinite if all the eigenvalues are nonpositive.

$\blacktriangleright$ Matrices + Definiteness Tests

$\left[\vphantom{
\begin{array}{cc}
2 & -1 \\
-1 & 2
\end{array}
}\right.$$\begin{array}{cc}
2 & -1 \\
-1 & 2
\end{array}$$\left.\vphantom{
\begin{array}{cc}
2 & -1 \\
-1 & 2
\end{array}
}\right]$, eigenvalues: 3, 1,    $\begin{array}{l}
\limfunc{negative}\limfunc{definite},\,\text{\emph{false}}\; ...
...\; \\
\limfunc{positive}\limfunc{definite},\,\text{\emph{true}}
\end{array}$

$\left[\vphantom{
\begin{array}{cc}
1 & -1 \\
-1 & 1
\end{array}
}\right.$$\begin{array}{cc}
1 & -1 \\
-1 & 1
\end{array}$$\left.\vphantom{
\begin{array}{cc}
1 & -1 \\
-1 & 1
\end{array}
}\right]$, eigenvalues: 0, 2,    $\begin{array}{l}
\limfunc{negative}\limfunc{definite},\emph{\,}\text{\emph{fal...
...; \\
\limfunc{positive}\limfunc{definite},\,\text{\emph{false}}
\end{array}$