Question 1

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

Accepted Answer

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

Question 2

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

Accepted Answer

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

Question 3

Find the 
  matrix A of the quadratic form 
  
.

Accepted Answer

The answer of Find the 
  matrix A of...

Question 4

Find the 
  matrix A of the quadratic form 
  
.

Accepted Answer

The answer of Find the 
  matrix A of...

Question 5

Find the 
  matrix A of the quadratic form 
  
.

Accepted Answer

The answer of Find the 
  matrix A of...

Question 6

Find an 
  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 an 
  orthogonal matrix P...

Question 7

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 8

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 9

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 10

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 11

If B is any 
  matrix, then the formula 
  defines a positive semidefinite quadratic form on 
  
.

Accepted Answer

The formula defines a quadratic form using the matrix B that is positive semidefinite, meaning that the associated quadratic form evaluates to a non-negative value for all inputs.

Question 12

If the quadratic form 
  is positive definite, then A is invertible.

Accepted Answer

If the quadratic form is positive definite, then all of the eigenvalues of A are positive, which implies that A is invertible.

Question 13

If 
  
, and 
  
, then the graph of the equation 
  is an ellipse.

Accepted Answer

The answer of If 
  
, and 
 ...

Question 14

If  is a quadratic form, then  for every scalar

Accepted Answer

Every quadratic form is homogeneous of degree 2, which means that scaling the variables by a constant scales the entire quadratic form by the square of that constant. Therefore, 11edaba5_20b5_7683_a975_73c80244aac2_TBMC1039_00 is simply a scaled version of 11edaba5_20b5_7682_a975_7178aaa19a3b_TBMC1039_00 and is still a quadratic form.

Question 15

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

Accepted Answer

The answer of If 
  is a positive definite...

Question 16

Determine if the matrix A is positive definite.

Accepted Answer

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

Question 17

Determine if the matrix A is positive definite.

Accepted Answer

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

Question 18

Determine if the matrix A is positive definite.

Accepted Answer

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

Question 19

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

The answer of Show that the given symmetric matrix A...

Question 20

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...

Quiz 11: Additional Topics and Applications

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

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

Find the matrix A of the quadratic form .

Find the matrix A of the quadratic form .

Find the matrix A of the quadratic form .

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

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

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

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

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

If B is any matrix, then the formula defines a positive semidefinite quadratic form on .

If the quadratic form is positive definite, then A is invertible.

If , and , then the graph of the equation is an ellipse.

If is a quadratic form, then for every scalar

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

Determine if the matrix A is positive definite.

Determine if the matrix A is positive definite.

Determine if the matrix A is positive definite.

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.

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