Exam 7: Symmetric Matrices and Quadratic Forms

Use the given covariance matrix to compute the percentage of the total variance that is contained in the first principal component. Round to one decimal place. - $\left[ \begin{array} { l l l l l l } 26 & 31 & 26 & 31 & 11 & 31 \\ 10 & 40 & 55 & 25 & 50 & 30 \end{array} \right]$

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. - $\left[ \begin{array} { l l } 3 & 2 \\ 2 & 6 \end{array} \right]$

$\text { Find the maximum value of } Q ( x ) \text { subject to the constraint } _x T _ { x = 1 } \text {. }$ - $Q ( x ) = 14 x _ { 1 } ^ { 2 } + 14 x _ { 2 } ^ { 2 } + 18x \frac { 2 } { 3 } + 26 x _ { 1 } x _ { 2 } + 18 x _ { 1 } x _ { 3 } + 18 x _ { 2 } x _ { 3 }$

Use the given covariance matrix to compute the percentage of the total variance that is contained in the first principal component. Round to one decimal place. - $\left[ \begin{array} { r r r r } 10 & 13 & 1 & 4 \\6 & 3 & 9 & 6 \\17 & 5 & 5 & 5\end{array} \right]$

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. - $\left[ \begin{array} { r r r } 11 & 7 & 7 \\7 & 11 & 7 \\7 & 7 & 11\end{array} \right]$