Deck 11: Additional Topics and Applications

Question 1

Find a polynomial formula for the quadratic form
with the matrix

.

Answer 1

Question 2

Find a polynomial formula for the quadratic form
with the matrix

.

Answer 2

Question 3

Find the
matrix A of the quadratic form

.

Answer 3

Question 4

Find the
matrix A of the quadratic form

.

Question 5

Find the
matrix A of the quadratic form

.

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

.

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

.

Question 8

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

Question 9

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

Question 10

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

Question 11

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

.

Question 12

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

Question 13

If

, and

, then the graph of the equation
is an ellipse.

Question 14

If is a quadratic form, then for every scalar

Question 15

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

Question 16

Determine if the matrix A is positive definite.

Question 17

Determine if the matrix A is positive definite.

Question 18

Determine if the matrix A is positive definite.

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.

Question 20

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

Question 21

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

Question 22

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

Question 23

Find the Cholesky decomposition
of the positive definite matrix

.

Question 24

Find the Cholesky decomposition
of the positive definite matrix

.

Question 25

Find the Cholesky decomposition
of the positive definite matrix

.

Question 26

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

Question 27

If A is positive definite, every entry along the diagonal of A is positive.

Question 28

If A has an LDU-factorization, then A is positive definite.

Question 29

If A is positive definite with factorization
, then the diagonal entries of D are the eigenvalues of A.

Question 30

A symmetric matrix
is positive definite if and only if
is negative definite.

Question 31

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

.

Question 32

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

.

Question 33

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

.

Question 34

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

, for

.

Question 35

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

, for

.

Question 36

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

.

Question 37

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

.

Question 38

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

.

Question 39

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

.

Question 40

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

, where

.

Question 41

If m is the minimum value of a quadratic form
subject to the constraint

, then cm is the minimum value of
subject to the constraint

.

Question 42

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

.

Question 43

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

.

Question 44

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

.

Question 45

Suppose
and
are quadratic forms on

. If subject to the constraint
the maximum values of
and
are
and
respectively, then subject to the constraint
the maximum value of
is

.

Question 46

Evaluate

.

Question 47

Determine if
is in

.

Question 48

Evaluate

.

Question 49

Evaluate

.

Question 50

Normalize
using the complex dot product.

Question 51

Evaluate
for the matrices

.

Question 52

Evaluate
for the matrices

, where the inner product is defined by

.

Question 53

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

.

Question 54

Evaluate
for

, where the inner product is defined by

.

Question 55

In

, using the complex dot product, find a basis for

, where

.

Question 56

In a complex inner product space,
for all vectors u and

.

Question 57

The dimension of the complex vector space
is n.

Question 58

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

.

Question 59

If
is an orthonormal set in a complex inner product space V, then the set
is also an orthonormal set in V.

Question 60

If
and

, we have

, where the inner product is defined by

.

Question 61

Find
for the matrix

.

Question 62

Find
for the matrix

.

Question 63

Find
for the matrix

.

Question 64

Determine if the given matrix A is Hermitian.

Question 65

Determine if the given matrix A is Hermitian.

Question 66

Determine if the given matrix A is normal.

Question 67

Determine if the given matrix A is normal.

Question 68

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

.

​

Question 69

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

.

Question 70

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

.

Question 71

If A is a unitary matrix, and
is an eigenvalue of A, then

.

Question 72

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

Question 73

If A and B are
matrices that are both normal, and

, then both
and AB are normal.

Question 74

The formula
defines an inner product on the complex vector space
of all
complex matrices.

Question 75

An
matrix A is Hermitian if and only if it is normal and all of its eigenvalues are real.