Exam 7: Symmetric Matrices and Quadratic Forms

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. - $\left[ \begin{array} { c r } 12 & - 6 \\- 6 & 7\end{array} \right]$

$\text { Find the maximum value of } Q ( x ) \text { subject to the constraint } _x T _ { x = 1 } \text {. }$ - $Q ( x ) = 2 x _ { 1 } ^ { 2 } + 9 x _ { 2 } ^ { 2 } + 5 \times \frac { 2 } { 3 }$

Find the maximum value of $Q ( x )$ subject to the constraints $_x T _ { x } = 1$ and $_x { T } \mathbf { _u } = 0$ , where $u$ is a unit eigenvector corresponding to the greatest eigenvalue of the matrix of the quadratic form. - $Q ( x ) = 14 x _ { 1 } ^ { 2 } + 14 x _ { 2 } ^ { 2 } + 18 x _ { 3 } ^ { 2 } + 26 x _ { 1 } x _ { 2 } + 18 x _ { 1 } x _ { 3 } + 18 x _ { 2 } x _ { 3 }$

Find a singular value decomposition of the matrix A. - $A = \left[ \begin{array} { r r } - 9 & 0 \\0 & 6\end{array} \right]$

Make a change of variable, x = Py, that transforms the given quadratic form into a quadratic form with no cross-product term. Give P and the new quadratic form. - $Q ( x ) = 13 x _ { 1 } ^ { 2 } + 12 x _ { 2 } ^ { 2 } + 5 x _ { 3 } ^ { 2 } + 20 x _ { 1 } x _ { 2 } + 8 x _ { 1 } x _ { 3 } + 12 x _ { 2 } x _ { 3 }$

Convert the matrix of observations to mean-deviation form, and construct the sample covariance matrix. - $\mathrm { S } = \left[ \begin{array} { r c c } 30 & - 12 & 12 \\- 12 & 6 & 0 \\12 & 0 & 36\end{array} \right]$

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. - $\left[ \begin{array} { r r r } 13 & 10 & 4 \\10 & 12 & 6 \\4 & 6 & 5\end{array} \right]$

Find the matrix of the quadratic form. - $6 x _ { 1 } x _ { 2 } - 18 x _ { 1 } x _ { 3 } + 18 x _ { 2 } x _ { 3 }$

Find a singular value decomposition of the matrix A. - $A = \left[ \begin{array} { r r r } 4 & 0 & - 4 \\4 & 4 & 4\end{array} \right]$

$\text { Compute the quadratic form } x ^ { T } A x \text { for the given matrix } A \text { and vector } x \text {. }$ - $A = \left[ \begin{array} { r r } 2 & 0 \\0 & - 9\end{array} \right] , x = \left[ \begin{array} { l } x _ { 1 } \\x _ { 2 }\end{array} \right]$

Find the matrix of the quadratic form. - $4 x _ { 1 } ^ { 2 } + 5 x _ { 2 } ^ { 2 } + 3 x _ { 3 } ^ { 2 } - 4 x _ { 1 } x _ { 2 } + x _ { 2 } x _ { 3 }$

Find the maximum value of $Q ( x )$ subject to the constraints $_x T _ { x } = 1$ and $_x { T } \mathbf { _u } = 0$ , where $u$ is a unit eigenvector corresponding to the greatest eigenvalue of the matrix of the quadratic form. - $Q ( x ) = 3 x { } _ { 1 } ^ { 2 } + 8 x { } _ { 2 } ^ { 2 } + 4x \frac { 2 } { 3 }$

Find a singular value decomposition of the matrix A. - $A = \left[ \begin{array} { r r } 4 & - 1 \\- 1 & 4\end{array} \right]$

Convert the matrix of observations to mean-deviation form, and construct the sample covariance matrix. - $\mathrm { S } = \left[ \begin{array} { r c } 60 & - 55 \\- 55 & 280\end{array} \right]$

Determine whether the matrix is symmetric. - $\left[ \begin{array} { l l } 1 & 6 \\5 & 0\end{array} \right]$

$\text { Compute the quadratic form } x ^ { T } A x \text { for the given matrix } A \text { and vector } x \text {. }$ - $A = \left[ \begin{array} { r r r } 4 & 8 & 0 \\8 & 1 & 2 \\0 & 2 & - 2\end{array} \right] , x = \left[ \begin{array} { l } x _ { 1 } \\x _ { 2 } \\x _ { 3 }\end{array} \right]$

Determine whether the matrix is symmetric. - $\left[ \begin{array} { r r r } 3 & 9 & 4 \\9 & - 5 & - 2 \\4 & - 2 & 0\end{array} \right]$

Find the singular values of the matrix. - $\left[ \begin{array} { r r } - 8 & 0 \\0 & 6\end{array} \right]$

Find the singular values of the matrix. - $\left[ \begin{array} { r r r } 1 & 5 & 5 \\ 5 & 1 & - 5 \end{array} \right]$

Make a change of variable, x = Py, that transforms the given quadratic form into a quadratic form with no cross-product term. Give P and the new quadratic form. - $Q ( x ) = 5 x _ { 1 } ^ { 2 } + 8 x _ { 2 } ^ { 2 } + 4 x _ { 1 } x _ { 2 }$