Quadratic Forms and Positive Semidefinite Matrices

Practice Questions

Linear Algebra › Quadratic Forms and Positive Semidefinite Matrices

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$\begin{align*}&\text{Use eigenvalues to determine if the following matrix is positive semi-definite:}\begin{bmatrix}12&-5\-5&9\end{bmatrix}\end{align*}$

$\begin{align*}&\text{See with eigenvalues if the following matrix is positive semi-definite:}\begin{bmatrix}-9&-15&8\-15&12&6\8&6&6\end{bmatrix}\end{align*}$

$D = \begin{bmatrix} 4 & \log_{3} 4 & \log_{3}4 & \log_{3}4 \ \log_{5}6 & 4 & \log_{5}6& \log_{5} 6 \ \log_{7} 8 & \log_{7}8 & 4 & \log_{7}8 \ \log_{9} 10 & \log_{9}10 & \log_{9}10 & 4 \end{bmatrix}$

True or false: is an example of a diagonally dominant matrix.

$\begin{align*}&\text{Decide whether or not the matrix }\begin{bmatrix}15&-11&8\-11&20&-1\8&-1&10\end{bmatrix}\&\text{is positive semi-definite using eigenvalues.}\end{align*}$

$\begin{align*}&\text{Decide whether or not the matrix }\begin{bmatrix}17&-5&9\-5&18&-5\9&-5&18\end{bmatrix}\&\text{is positive semi-definite using eigenvalues.}\end{align*}$

$\begin{align*}&\text{Use eigenvalues to determine if the following matrix is positive semi-definite:}\begin{bmatrix}6&5&12\5&19&3\12&3&7\end{bmatrix}\end{align*}$

$\begin{align*}&\text{See with eigenvalues if the following matrix is positive semi-definite:}\begin{bmatrix}-14&2\2&-15\end{bmatrix}\end{align*}$

$\begin{align*}&\text{Decide whether or not the matrix }\begin{bmatrix}15&-15\-15&20\end{bmatrix}\&\text{is positive semi-definite using eigenvalues.}\end{align*}$

$A = \begin{bmatrix} 20 & 0.1 & 0.2 & -0.3 & 0 \ 0.2 & -15 & 0.4 & 0.1 & -0.3 \ 0.3 & -0.2 & 18 & -0.2 & 0.2 \ 0.2 & - 0.2 & 0.5 & 16 & 0.2 \ 0.4 & 0 & 0.1 & 0.8 & -40 \end{bmatrix}$

True or false: is an example of a diagonally dominant matrix.

Consider the diagonally dominant matrix

$A = \begin{bmatrix} 20 & 0.1 & 0.2 & -0.3 & 0 \ 0.2 & -15 & 0.4 & 0.1 & -0.3 \ 0.3 & -0.2 & 18 & -0.2 & 0.2 \ 0.2 & - 0.2 & 0.5 & 16 & 0.2 \ 0.4 & 0 & 0.1 & 0.8 & -40 \end{bmatrix}$

Of the five Gershgorin discs of , give the center of the one with the greatest radius.

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