Question 1

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D.&#10;

Accepted Answer

The correct orthogonal matrix P and diagonal matrix D are given in option C. The orthogonal matrix P has columns that are normalized eigenvectors of the original matrix, and the diagonal matrix D contains the corresponding eigenvalues. The eigenvalues of the given matrix are 25, 4, and 1, and the corresponding eigenvectors, when normalized, match the columns of P in option C.

Question 2

Find a singular value decomposition of the matrix A.&#10;

Accepted Answer

D

Question 3

Make a change of variable, x = Py, that transforms the given quadratic form into a quadratic form with no cross-product&#10;term. Give P and the new quadratic form.&#10;

Accepted Answer

B

Question 4

Find the singular values of the matrix.&#10;

Accepted Answer

The singular values of a matrix are the square roots of the eigenvalues of $A^TA$, where $A$ is the given matrix. For the given matrix, $A^TA$ is a diagonal matrix with diagonal entries 64 and 36, so the singular values are the square roots of these, which are 8 and 6, respectively. Singular values are always non-negative.

Question 5

Find the matrix of the quadratic form.&#10;

Accepted Answer

The matrix of a quadratic form involving terms $x_i x_j$ has elements that are half the coefficients of these terms when $i \neq j$, and the diagonal elements are the coefficients of $x_i^2$ terms. Since there are no $x_i^2$ terms in the given quadratic form, all diagonal elements are 0. For the off-diagonal elements, the coefficients should be halved, resulting in the matrix $\left[\begin{array}{rrr} 0 & 3 & -9 \ 3 & 0 & 9 \ -9 & 9 & 0 \end{array}\right]$, which corresponds to option C.

Question 6

Find the singular values of the matrix.&#10;

Accepted Answer

The singular values of a matrix are the square roots of the eigenvalues of $A^TA$, where $A$ is the given matrix. For the given matrix, $A^TA$ is a 3x3 matrix. The characteristic polynomial of $A^TA$ will generally have three roots, corresponding to three eigenvalues. Since the given matrix is 2x3, it is rank-deficient, implying at least one singular value is zero. The other two singular values are the positive square roots of the non-zero eigenvalues of $A^TA$, which are $6$ and $\sqrt{66}$, making the correct answer $0, 6, \sqrt{66}$.

Question 7

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D.&#10;

Accepted Answer

The correct orthogonal matrix P must have orthonormal columns, and option C provides this with vectors normalized to have a length of 1. The corresponding diagonal matrix D correctly lists the eigenvalues 16 and 3.

Question 8

Find the matrix of the quadratic form.&#10;

Accepted Answer

The matrix of the quadratic form is:$\left[ \begin{array} { c r c } 4 & - 2 & 1 / 2 \ - 2 & 5 & 0 \ 1 / 2 & 0 & 3 \end{array} \right]$The quadratic form is:$$4 x _ { 1 } ^ { 2 } + 5 x _ { 2 } ^ { 2 } + 3 x _ { 3 } ^ { 2 } - 4 x _ { 1 } x _ { 2 } + x _ { 2 } x _ { 3 }$$The matrix of the quadratic form is:$\left[ \begin{array} { c r c } a _ { 11 } & a _ { 12 } & a _ { 13 } \\ a _ { 21 } & a _ { 22 } & a _ { 23 } \\ a _ { 31 } & a _ { 32 } & a _ { 33 } \end{array} \right]$where:$a _ { 11 } = 4$$a _ { 12 } = - 2$$a _ { 13 } = 1 / 2$$a _ { 21 } = - 2$$a _ { 22 } = 5$$a _ { 23 } = 0$$a _ { 31 } = 1 / 2$$a _ { 32 } = 0$$a _ { 33 } = 3$

Question 9

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D.&#10;

Accepted Answer

To orthogonally diagonalize a matrix, we find its eigenvalues and eigenvectors. For the given matrix, the eigenvalues are 7 and 2. The corresponding normalized eigenvectors form the columns of P, and the eigenvalues are the entries of D. Choice D correctly represents the orthogonal matrix P with normalized eigenvectors and the diagonal matrix D with the eigenvalues 7 and 2.

Question 10

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D.&#10;

Accepted Answer

The correct orthogonal matrix P and diagonal matrix D are obtained by finding the eigenvectors and eigenvalues of the given matrix. The eigenvalues are 25 and 4, with 25 having a multiplicity of 1 and 4 having a multiplicity of 2. The corresponding eigenvectors are orthogonalized and normalized to form the columns of P, and the eigenvalues form the diagonal entries of D. Choice D correctly represents the orthogonal matrix P and the diagonal matrix D with the eigenvalues 25 and 4.

Question 11

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Accepted Answer

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Question 12

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Accepted Answer

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Question 13

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Accepted Answer

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Question 14

Make a change of variable, x = Py, that transforms the given quadratic form into a quadratic form with no cross-product&#10;term. Give P and the new quadratic form.&#10;

Accepted Answer

The answer of Make a change of variable, x =...

Question 15

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Accepted Answer

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Question 16

Determine whether the matrix is symmetric.&#10;

Accepted Answer

The answer of Determine whether the matrix is symmetric.&#10; ...

Question 17

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Accepted Answer

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Question 18

Find a singular value decomposition of the matrix A.&#10;

Accepted Answer

The answer of Find a singular value decomposition of the...

Question 19

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Accepted Answer

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Question 20

Determine whether the matrix is symmetric.&#10;

Accepted Answer

The answer of Determine whether the matrix is symmetric.&#10; ...

Question 21

Use the given covariance matrix to compute the percentage of the total variance that is contained in the first principal&#10;component. Round to one decimal place.&#10;

Accepted Answer

The answer of Use the given covariance matrix to compute...

Question 22

Use the given covariance matrix to compute the percentage of the total variance that is contained in the first principal&#10;component. Round to one decimal place.&#10;

Accepted Answer

The answer of Use the given covariance matrix to compute...

Question 23

Find a singular value decomposition of the matrix A.&#10;

Accepted Answer

The answer of Find a singular value decomposition of the...

Question 24

Convert the matrix of observations to mean-deviation form, and construct the sample covariance matrix.&#10;

Accepted Answer

The answer of Convert the matrix of observations to mean-deviation...

Question 25

Convert the matrix of observations to mean-deviation form, and construct the sample covariance matrix.&#10;

Accepted Answer

The answer of Convert the matrix of observations to mean-deviation...

Deck 7: Symmetric Matrices and Quadratic Forms