Question 1

Evaluate 
   for 
   
, where the inner product is defined by 
   
.

Accepted Answer

The answer of Evaluate 
   for 
 ...

Question 2

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 
 ...

Question 3

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

Accepted Answer

Maximum: 4

Question 4

Use the technique of constrained optimization (not calculus) to find the absolute maximum and minimum values of the function 
   
, for 
   
.

Accepted Answer

Maximum:

Question 5

Show that the given symmetric matrix A is positive definite by finding the determinants of its leading principal submatrices. Then find the LDU-factorization of A.

Accepted Answer

Determinan

Question 6

If A is an 
   matrix with complex entries, then the matrix 
   is Hermitian.

Accepted Answer

A) True 
 B)False

Question 7

If 
   is a positive definite quadratic form and 
   is a negative definite quadratic form, then 
   is indefinite.

Accepted Answer

A) True 
 B)False

Question 8

Determine if the given matrix A is Hermitian.

Accepted Answer

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

Question 9

If A is positive definite, then 
   exists, and 
   is positive definite.

Accepted Answer

A) True 
 B)False

Question 10

Find 
   for the matrix 
   
.

Accepted Answer

The answer of Find 
   for the matrix...

Question 11

Let 
   
, where A is a symmetric 
   matrix with eigenvalues 
   and associated orthonormal eigenvectors 
   
. Then subject to the constraints 
   and 
   
, the minimum value of 
   is 
   
, attained at 
   
.

Accepted Answer

A) True 
 B)False

Question 12

Use the technique of constrained optimization (not calculus) to find the absolute maximum and minimum values of the function 
   
, for 
   
.

Accepted Answer

Maximum:

Question 13

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

Accepted Answer

Maximum: 6

Question 14

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 
 ...

Question 15

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

Accepted Answer

Maximum: 3

Evaluate for , where the inner product is defined by .

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

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

Use the technique of constrained optimization (not calculus) to find the absolute maximum and minimum values of the function , for .

Show that the given symmetric matrix A is positive definite by finding the determinants of its leading principal submatrices. Then find the LDU-factorization of A.

If A is an matrix with complex entries, then the matrix is Hermitian.

If is a positive definite quadratic form and is a negative definite quadratic form, then is indefinite.

Determine if the given matrix A is Hermitian.

If A is positive definite, then exists, and is positive definite.

Find for the matrix .

Let , where A is a symmetric matrix with eigenvalues and associated orthonormal eigenvectors . Then subject to the constraints and , the minimum value of is , attained at .

Use the technique of constrained optimization (not calculus) to find the absolute maximum and minimum values of the function , for .

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

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

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

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

Evaluate for , where the inner product is defined by .

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

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

Use the technique of constrained optimization (not calculus) to find the absolute maximum and minimum values of the function , for .

Show that the given symmetric matrix A is positive definite by finding the determinants of its leading principal submatrices. Then find the LDU-factorization of A.

If A is an matrix with complex entries, then the matrix is Hermitian.

If is a positive definite quadratic form and is a negative definite quadratic form, then is indefinite.

Determine if the given matrix A is Hermitian.

If A is positive definite, then exists, and is positive definite.

Find for the matrix .

Let , where A is a symmetric matrix with eigenvalues and associated orthonormal eigenvectors . Then subject to the constraints and , the minimum value of is , attained at .

Use the technique of constrained optimization (not calculus) to find the absolute maximum and minimum values of the function , for .

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

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

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

Systems of Linear Equations

Euclidean Space

Matrices

Subspaces

Determinants

Eigenvalues and Eigenvectors

Vector Spaces

Orthogonality

Linear Transformations

Inner Product Spaces

Filters