Sample of quiz – Matrix decompositions

1. Singular value decomposition cloze

This problem is about the matrix Z given below. Using R, you will compute the singular value decomposition Z = UDVT and do numerical

computations before answering the following questionnaire.

Z =

2 52 54

−20 −70 −18

−70 −20 −18

−2 −52 −54

20 70 18

52 2 54

70 20 18

−52 −2 −54

A) Consider eigenvector (−1, 1, 1, 1, −1, −1, −1, 1)T /

√

8 of U. The

eigenvalue associated with this vector is

multi

• 176.36326

• 176.46326

• 176.86326

• 176.26326

B) The number 177.36326 is a (singular) eigenvalue of Z.

multi

• True

• False

C) Write the largest (singular) eigenvalue of Z:

numerical D) Consider the following matrix, which is derived by manipulation of results from the earlier decomposition:

1

−34 16 18

16 −34 18

−34 16 18

34 −16 −18

−16 34 −18

16 −34 18

34 −16 −18

−16 34 −18

This matrix was obtained as result of the following:

multi

• d1u1v

T

1 + d2u2v

T

2

• d1u1v

T

1

• d2u2v

T

2 + d3u3v

T

3

• d2u2v

T

2

• d3u3v

T

3

• d1u1v

T

1 + d3u3v

T

3

2. Singular value decomposition cloze

This problem is about a matrix Z given below. Using R, you will compute the singular value decomposition Z = UDVT and do numerical

computations before answering the following questionnaire.

Z =

−26 −8 10

1 −8 −17

−2 16 34

55 −44 37

1 −8 −17

1 −8 −17

1 −8 −17

−26 −8 10

−5 76 −23

A) Consider eigenvector (−2, 1, −2, 1, 1, 1, 1, −2, 1)T /

√

18 of U. The

eigenvalue associated with this vector is

multi

2

• 54

• 54.2

• 53.85

• 104.02305

B) Consider the eigenvector (1, 0, −1)T /

√

2 of V. The eigenvalue associated with this vector is

multi

• 54

• 53.9

• 54.2

• 53

C) Write the smallest (singular) eigenvalue of Z:

numerical D) Consider the following matrix, which is derived by manipulation of results from earlier decomposition:

−8 −8 −8

−8 −8 −8

16 16 16

46 −44 46

−8 −8 −8

−8 −8 −8

−8 −8 −8

−8 −8 −8

−14 76 −14

This matrix was obtained as result of the following:

multi

• d2u2v

T

2

• d2u2v

T

2 + d3u3v

T

3

• d1u1v

T

1 + d3u3v

T

3

• d1u1v

T

1

• d1u1v

T

1 + d2u2v

T

2

• d3u3v

T

3

3

3. Karhunen-Loeve decomposition cloze

This problem is about the matrix X given below. Using R, you will

compute the Karhunen-Loeve decomposition X = QΛQT and do numerical computations before answering the following questionnaire.

X =

84 −10 −26 8

−10 84 8 −26

−26 8 84 −10

8 −26 −10 84

A) Consider the eigenvalue of X with numerical value 60. The eigenvector associated with this value is

multi

• (1, −1, −1, 1)T /2

• (−1, 1, −1, 1)T /2

• (1, 1, 1, 1)T /2

• (1, 1, −1, −1)T /2

B) Write the smallest eigenvalue of X:

numerical C) Consider the following matrix, which is derived by manipulation of results from the earlier decomposition:

0.25 −0.25 −0.25 0.25

−0.25 0.25 0.25 −0.25

−0.25 0.25 0.25 −0.25

0.25 −0.25 −0.25 0.25

This matrix is the result of the following:

multi

• v3v

T

3

• v1v

T

1 + v2v

T

2

• v2v

T

2

• v1v

T

1 + v4v

T

4

• v2v

T

2 + v3v

T

3

• v3v

T

3 + v4v

T

4

4

• v1v

T

1

• v4v

T

4

D) Consider the following matrix, which is derived by manipulation of

the earlier results:

23 23 −23 −23

23 23 −23 −23

−23 −23 23 23

−23 −23 23 23

This matrix is the result of the following:

multi

• λ3v3v

T

3 + λ4v4v

T

4

• λ2v2v

T

2 + λ4v4v

T

4

• λ2v2v

T

2

• λ4v4v

T

4

• λ3v3v

T

3

• λ1v1v

T

1 + λ4v4v

T

4

• λ1v1v

T

1 + λ2v2v

T

2

• λ1v1v

T

1

4. Karhunen-Loeve decomposition cloze

This problem is about the matrix X given below. Using R, you will

compute the Karhunen-Loeve decomposition X = QΛQT and do numerical computations before answering the following questionnaire.

X =

75 3 −13 −41 35

3 75 35 31 −13

−13 35 75 −9 3

−41 31 −9 75 −57

35 −13 3 −57 75

A) Consider the eigenvalue of X with numerical value 27.6654. The

eigenvector associated with this value is

multi

• (0.6827, 0.4736, −0.4762, 0.2661, −0.1096)T

5

• (−0.3453, 0.4481, −0.365, −0.6822, −0.285)T

• (−0.143, −0.6353, −0.7231, 0.1232, −0.1945)T

• (−0.4218, 0.2953, −0.3293, 0.2733, 0.7429)T

• (−0.4651, 0.29, 0.0928, 0.6115, −0.5631)T

B) Write the smallest eigenvalue of X:

numerical C) Consider the following matrix, which is derived by manipulation of the earlier results:

0.0204 0.0908 0.1034 −0.0176 0.0278

0.0908 0.4036 0.4594 −0.0783 0.1236

0.1034 0.4594 0.5229 −0.0891 0.1406

−0.0176 −0.0783 −0.0891 0.0152 −0.024

0.0278 0.1236 0.1406 −0.024 0.0378

This matrix is the result of the following:

multi

• v2v

T

2

• v1v

T

1

• v1v

T

1 + v2v

T

2

• v1v

T

1 + v3v

T

3

• v3v

T

3 + v5v

T

5

• v4v

T

4

• v2v

T

2 + v4v

T

4

• v5v

T

5

• v3v

T

3

D) Consider the following matrix, which is derived by manipulation of

the earlier results:

37.2053 −23.1971 −7.4239 −48.9177 45.0462

−23.1971 14.4631 4.6287 30.4996 −28.0858

−7.4239 4.6287 1.4814 9.761 −8.9885

−48.9177 30.4996 9.761 64.3173 −59.227

45.0462 −28.0858 −8.9885 −59.227 54.5396

This matrix is the result of the following:

multi

6

• λ1v1v

T

1 + λ2v2v

T

2

• λ1v1v

T

1 + λ4v4v

T

4

• λ1v1v

T

1 + λ2v2v

T

2 + λ4v4v

T

4

• λ1v1v

T

1 + λ5v5v

T

5

• λ2v2v

T

2

• λ1v1v

T

1 + λ2v2v

T

2 + λ5v5v

T

5

• λ5v5v

T

5

• λ2v2v

T

2 + λ4v4v

T

4

• λ4v4v

T

4

• λ3v3v

T

3

• λ1v1v

T

1

5. Matrix decompositions multi

✄

✂

✁

Single

In what follows, the matrix X is of size n × p with n > p, stored in the

R object termed X. Select the correct statement(s) below.

(a) If we have computed the singular value decomposition svd(X),

then we can retrieve the results of Karhunen-Loeve decomposition

of XTX without needing to use the command eigen.

(b) If we have computed the Karhunen-Loeve decomposition eigen(X),

then we can retrieve the results of the singular value decomposition of XTX without needing the command svd.

(c) If we have computed the singular value decomposition svd(X),

then we can retrieve the results of Karhunen-Loeve decomposition

of XXT without needing to use the command eigen.

(d) If we have computed the Karhunen-Loeve decomposition eigen(t(X)%*%X),

then we can retrieve the results of the singular value decomposition of X without needing to run the command svd.

(e) If we have computed the Karhunen-Loeve decomposition eigen(X%*%t(X)),

then we can retrieve the results of the singular value decomposition of X without needing to run the command svd.