Question 1

If 2 is the maximum value of a quadratic form 
  subject to the constraint 
  
, then 1 is the maximum value of 
  subject to the constraint 
  
.

Accepted Answer

A) True 
 B)False  False

Question 2

Find the Cholesky decomposition 
  of the positive definite matrix 
  
.

Accepted Answer

The answer of Find the Cholesky decomposition 
  of...

Question 3

Determine if the given matrix A is Hermitian.

Accepted Answer

A is not Hermitian

Question 4

Show that the given matrix is positive definite, and then find the LDU-factorization.

Accepted Answer

The answer of Show that the given matrix is positive...

Question 5

Find the Cholesky decomposition 
  of the positive definite matrix 
  
.

Accepted Answer

The answer of Find the Cholesky decomposition 
  of...

Question 6

Verify that the given matrix A is Hermitian, where a, b denote real numbers, then diagonalize A by finding a unitary matrix P and diagonal matrix D such that 
  
.

Accepted Answer

The answer of Verify that the given matrix A is...

Question 7

Evaluate 
  for the functions 
  and 
  where the inner product is defined by 
  
.

Accepted Answer

The answer of Evaluate 
  for the functions 
...

Question 8

Suppose A is a positive definite matrix and subject to the constraint 
  the quadratic form 
  has maximum value M and minimum value m. Then subject to the constraint 
  the quadratic form 
  has maximum value 
  and minimum value 
  
.

Accepted Answer

A) True 
 B)False

Question 9

Find the maximum and minimum values of the quadratic form 
  subject to the constraint 
  
.

Accepted Answer

The answer of Find the maximum and minimum values of...

Question 10

Evaluate 
  
.

Accepted Answer

The answer of Evaluate 
  
....

Question 11

Find the maximum and minimum values of the quadratic form 
  subject to the constraint 
  
.

Accepted Answer

The answer of Find the maximum and minimum values of...

Question 12

Find the maximum and minimum values of the quadratic form 
  subject to the constraint 
  
.

Accepted Answer

The answer of Find the maximum and minimum values of...

Question 13

Verify that the given matrix A is unitary, then diagonalize A by finding a unitary matrix P and diagonal matrix D such that 
  
.
  
​

Accepted Answer

The answer of Verify that the given matrix A is...

Question 14

If 
  and 
  
, we have 
  
, where the inner product is defined by 
  
.

Accepted Answer

A) True 
 B)False

Question 15

Determine if the matrix A is positive definite.

Accepted Answer

The answer of Determine if the matrix A is positive...

Question 16

For vectors u, v in a complex inner product space, we have 
  if and only if 
  
.

Accepted Answer

A) True 
 B)False

Question 17

Find a polynomial formula for the quadratic form 
  with the matrix 
  
.

Accepted Answer

The answer of Find a polynomial formula for the quadratic...

Question 18

Find a 
  orthogonal matrix P and diagonal matrix D such that the change of variables 
  transforms the given quadratic form 
  into the form 
  
.

Accepted Answer

The answer of Find a 
  orthogonal matrix P...

Question 19

Find the Cholesky decomposition 
  of the positive definite matrix 
  
.

Accepted Answer

The answer of Find the Cholesky decomposition 
  of...

Question 20

Determine if the quadratic form 
  is positive definite, negative definite, indefinite, or none of these.

Accepted Answer

The answer of Determine if the quadratic form 
 ...

Exam 11: Additional Topics and Applications

If 2 is the maximum value of a quadratic form subject to the constraint , then 1 is the maximum value of subject to the constraint .

Find the Cholesky decomposition of the positive definite matrix .

Determine if the given matrix A is Hermitian.

Show that the given matrix is positive definite, and then find the LDU-factorization.

Find the Cholesky decomposition of the positive definite matrix .

Verify that the given matrix A is Hermitian, where a, b denote real numbers, then diagonalize A by finding a unitary matrix P and diagonal matrix D such that .

Evaluate for the functions and where the inner product is defined by .

Suppose A is a positive definite matrix and subject to the constraint the quadratic form has maximum value M and minimum value m. Then subject to the constraint the quadratic form has maximum value and minimum value .

Find the maximum and minimum values of the quadratic form subject to the constraint .

Evaluate .

Find the maximum and minimum values of the quadratic form subject to the constraint .

Find the maximum and minimum values of the quadratic form subject to the constraint .

Verify that the given matrix A is unitary, then diagonalize A by finding a unitary matrix P and diagonal matrix D such that . ​

If and , we have , where the inner product is defined by .

Determine if the matrix A is positive definite.

For vectors u, v in a complex inner product space, we have if and only if .

Find a polynomial formula for the quadratic form with the matrix .

Find a orthogonal matrix P and diagonal matrix D such that the change of variables transforms the given quadratic form into the form .

Find the Cholesky decomposition of the positive definite matrix .

Determine if the quadratic form is positive definite, negative definite, indefinite, or none of these.

Verify that the given matrix A is unitary, then diagonalize A by finding a unitary matrix P and diagonal matrix D such that .