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 
  ...

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

Determinan

Question 5

Find the Cholesky decomposition 
   of the positive definite matrix 
   
.

Accepted Answer

The answer of Find the Cholesky decomposition 
  ...

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

Maximum: 1

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

Maximum: 4

Question 12

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

Accepted Answer

Maximum:

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

Not positi

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...

Question 19

Find the Cholesky decomposition 
   of the positive definite matrix 
   
.

Accepted Answer

The answer of Find the Cholesky decomposition 
  ...

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 
 ...

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.

Systems of Linear Equations

Euclidean Space

Matrices

Subspaces

Determinants

Eigenvalues and Eigenvectors

Vector Spaces

Orthogonality

Linear Transformations

Inner Product Spaces

Filters

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.

Systems of Linear Equations

Euclidean Space

Matrices

Subspaces

Determinants

Eigenvalues and Eigenvectors

Vector Spaces

Orthogonality

Linear Transformations

Inner Product Spaces

Filters

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