Deck 7: Symmetric Matrices and Quadratic Forms

Question 1

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D.

Answer 1

C

Question 2

Find a singular value decomposition of the matrix A.

Answer 2

D

Question 3

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.

Answer 3

B

Question 4

Find the singular values of the matrix.

Question 5

Find the matrix of the quadratic form.

Question 6

Find the singular values of the matrix.

Question 7

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D.

Question 8

Find the matrix of the quadratic form.

Question 9

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D.

Question 10

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D.

Question 11

Question 12

Question 13

Question 14

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.

Question 15

Question 16

Determine whether the matrix is symmetric.

Question 17

Question 18

Find a singular value decomposition of the matrix A.

Question 19

Question 20

Determine whether the matrix is symmetric.

Question 21

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.

Question 22

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.

Question 23

Find a singular value decomposition of the matrix A.

Question 24

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

Question 25

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