Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find a singular value decomposition of the matrix A.
1)
A =–9 0
0 5
1)
A)
A = 1 0
0 1 –9 0
0 51 0
0 1
B)
A =–1 0
0 1 9 0
0 5 1 0
0 1
C)
A =–9 0
0 59 0
0 5 1 0
0 1
D)
A =–9 0
0 59 0
0 5 –1 0
0 1
Find a unit vector at which the quadratic form xTAx is maximized, subject to the constraint xTx = 1.
2)
A = 8 –6 4
–6 9 –2
4 –2 4
2)
A)
2
–2
1
B)
2/3
–2/3
1/3
C)
1/3
2/3
2/3
D)
2/ 5
–2/ 5
1/ 5
Determine whether the matrix is symmetric.
3)
1 2
5 0
3)
A)
No
B)
Yes
1
Find the maximum value of Q(x) subject to the constraint xTx = 1.
4)
Q(x) = 14x2
1+ 14x2
2+ 18x2
3+ 26x1x2+ 18x1x3+ 18x2x3
4)
A)
1
B)
9
C)
36
D)
25
Find the singular values of the matrix.
5)
–9 0
0 4
5)
A)
81, 16
B)
–81, 16
C)
–9,4
D)
9, 4
Find the matrix of the quadratic form.
6)
3x 2
1+5x 2
2+2x 2
3–12x1x2+x2x3
6)
A)
3 0 0
0 5 0
0 0 2
B)
3–12 0
–12 5 1
0 1 2
C)
3–6 1/2
–6 5 0
1/2 0 2
D)
3–6 0
–6 5 1/2
0 1/2 2
Make a change of variable, x = Py, that transforms the given quadratic form into a quadratic form with no cross–product
term. Give P and the new quadratic form.
7)
Q(x) =4 x 2
1+7x 2
2+4x1x2
7)
A)
P =1–2
2 1 ; 8y1+3y2
B)
P =1/ 5–2/ 5
2/ 5 1/ 5; 8 y 2
1+3y 2
2
C)
P =–1/ 5–2/ 5
2/ 5 1/ 5; –8y 2
1–3y 2
2
D)
P =1–2
2 1 ; 8 y 2
1+3 y 2
2
2
Use the given covariance matrix to compute the percentage of the total variance that is contained in the first principal
component. Round to one decimal place.
8)
S = 60 –55
–55 280
8)
A)
13.8%
B)
82.4%
C)
89.7%
D)
86.2%
Find the maximum value of Q(x) subject to the constraint xTx = 1.
9)
Q(x) =2 x 2
1+7x 2
2+5x 2
3
9)
A)
2
B)
14
C)
7
D)
5
Find a singular value decomposition of the matrix A.
10)
A =4 0 –4
4 4 4
10)
A)
A =0–1
1 0 4 3 0
0 4 2
1 1 1
–1 0 1
1 –2 1
B)
A =0–1
1 0 48 0 0
032 0
1/ 3 1/ 3 1/ 3
–1/ 2 0 1/ 2
1/ 6–2/ 6 1/ 6
C)
A =0–1
1 0 4 3 0 0
0 4 2 0
1/ 3–1/ 2 1/ 6
1/ 3 0 –2/ 6
1/ 3 1/ 2 1/ 6
D)
A =0–1
1 0 4 3 0 0
0 4 2 0
1/ 3 1/ 3 1/ 3
–1/ 2 0 1/ 2
1/ 6–2/ 6 1/ 6
3
Compute the quadratic form xTAx for the given matrix A and vector x.
11)
A =9 3 0
3 1 5
0 5 –2, x=x1
x2
x3
11)
A)
9x 2
1+x2
2– 2x2
3+3x1+8x2+5x3
B)
9x 2
1+x2
2– 2x2
3
C)
9x 2
1+x2
2– 2x2
3+6x1x2+10x2x3
D)
9x 2
1–x2
2+ 2x2
3–6x1x2–10x2x3
Find a unit vector at which the quadratic form xTAx is maximized, subject to the constraint xTx = 1.
12)
A =4 2
2 7
12)
A)
–1/ 5
2/ 5
B)
1
2
C)
–2/ 5
1/ 5
D)
1/ 5
2/ 5
Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D.
13)
13 10 4
10 12 6
4 6 5
13)
A)
P =2/3 –2/3 1/3
2/3 1/3 –2/3
1/3 2/3 2/3 , D =1 0 0
0 4 0
0 0 25
B)
P =2/3 –2/3 1/3
2/3 1/3 –2/3
1/3 2/3 2/3 , D =25 0 0
0 4 0
0 0 1
C)
P =2–2 1
2 1 –2
1 2 2 , D =25 0 0
0 4 0
0 0 1
D)
P =2–2 1
2 1 –2
1 2 2 , D =1 0 0
0 4 0
0 0 25
4
Find the maximum value of Q(x) subject to the constraints xTx = 1 and xTu = 0, where u is a unit eigenvector
corresponding to the greatest eigenvalue of the matrix of the quadratic form.
14)
Q(x) =2 x 2
1+7x 2
2+4x 2
3
14)
A)
7
B)
4
C)
0
D)
2
Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D.
15)
11 7 7
7 11 7
7 7 11
15)
A)
P = 1/ 3 1/ 2 1/ 6
1/ 3–1/ 2 1/ 6
–1/ 3 0 2/ 6
, D =25 0 0
0 25 0
0 0 4
B)
P =1/ 3–1/ 2–1/ 6
1/ 3 1/ 2–1/ 6
1/ 3 0 2/ 6
, D =4 0 0
0 4 0
0 0 25
C)
P =1/ 3–1/ 2–1/ 6
1/ 3 1/ 2–1/ 6
1/ 3 0 2/ 6
, D =25 0 0
0 4 0
0 0 4
D)
P =1–1–1
1 1 –1
1 0 2 , D =25 0 0
0 4 0
0 0 4
5
Convert the matrix of observations to mean–deviation form, and construct the sample covariance matrix.
16)
22 27 22 27 727
13 43 58 28 53 33
16)
A)
S = 60 –55
–55 280
B)
S =–60 55
55 –280
C)
S =
50 –455
6
–455
6 2331
3
D)
S = 300 –275
–275 1400
Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D.
17)
12 – 6
– 6 7
17)
A)
P = 3/ 13 2/ 13
–2/ 13 3/ 13 , D =16 0
0 3
B)
P = 3/ 13 –2/ 13
–2/ 13 3/ 13 , D =3 0
016
C)
P = 3 –2
–2 3 , D =16 0
0 3
D)
P = 3 2
–2 3 , D =16 0
0 3
Find the maximum value of Q(x) subject to the constraints xTx = 1 and xTu = 0, where u is a unit eigenvector
corresponding to the greatest eigenvalue of the matrix of the quadratic form.
18)
Q(x) = 14x2
1+ 14x2
2+ 18x2
3+ 26x1x2+ 18x1x3+ 18x2x3
18)
A)
9
B)
36
C)
1
D)
16
6
Find the singular values of the matrix.
19)
1 3 3
3 1 –3
19)
A)
0, 16, 22
B)
0, 4, 22
C)
16, 22
D)
4, 22
Find the matrix of the quadratic form.
20)
8x1x2–16x1x3+6x2x3
20)
A)
1 4–8
4 1 3
–8 3 1
B)
4 0 0
0–8 0
0 0 3
C)
0 4–8
4 0 3
–8 3 0
D)
0 8–16
8 0 6
–16 6 0
Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D.
21)
2 2
2 5
21)
A)
P =1–2
2 1 , D =6 0
0 1
B)
P =1–2
2 1 , D =7 0
0 2
C)
P = 1/ 5–2/ 5
–2/ 5 1/ 5, D =1 0
0 6
D)
P =1/ 5–2/ 5
2/ 5 1/ 5, D =6 0
0 1
7
Make a change of variable, x = Py, that transforms the given quadratic form into a quadratic form with no cross–product
term. Give P and the new quadratic form.
22)
Q(x) = 13x2
1+ 12x2
2+ 5x2
3+ 20x1x2+ 8x1x3+ 12x2x3
22)
A)
P =2–2 1
2 1 –2
1 2 2 ; y2
1+ 4y2
2+ 25y2
3
B)
P =2/3 –2/3 1/3
2/3 1/3 –2/3
1/3 2/3 2/3 ; 25y2
1+ 4y2
2+y2
3
C)
P =2–2 1
2 1 –2
1 2 2 ; 25y2
1+ 4y2
2+y2
3
D)
P =–2/3 –2/3 1/3
–2/3 1/3 –2/3
1/3 –2/3 2/3 ; 5y2
1+ 2y2
2+y2
3
Determine whether the matrix is symmetric.
23)
5 2 4
2–5–2
4–2 0
23)
A)
Yes
B)
No
Find a singular value decomposition of the matrix A.
24)
A =4–1
–1 4
24)
A)
A =–1/ 2 1/ 2
1/ 2 1/ 29 0
0 9 –1/ 2 1/ 2
1/ 2 1/ 2
B)
A =–1/ 2 1/ 2
1/ 2 1/ 25 0
0 3 –1/ 2 1/ 2
1/ 2 1/ 2
C)
A =–1 1
1 1 5 0
0 3 –1 1
1 1
D)
A =–1 1
1 1 4–1
–1 4 –1 1
1 1
8
Compute the quadratic form xTAx for the given matrix A and vector x.
25)
A =–3 0
0–6, x =x1
x2
25)
A)
–3 x 2
1– 6x2
2
B)
–6 x 2
1– 3x2
2
C)
–6x1– 12x2
D)
–3x1– 6x2
Convert the matrix of observations to mean–deviation form, and construct the sample covariance matrix.
26)
912 0 3
7 4 10 7
13 1 1 1
26)
A)
S =
30 –3–15 –12
–3 18 –12 –3
–15 –12 18 9
–12 –3 9 2
B)
S = 30 –12 12
–12 6 0
12 0 36
C)
S =–30 12 –12
12 –6 0
–12 0 –36
D)
S = 90 –36 36
–36 18 0
36 0 108
Use the given covariance matrix to compute the percentage of the total variance that is contained in the first principal
component. Round to one decimal place.
27)
S = 30 –12 12
–12 6 0
12 0 36
27)
A)
79.1%
B)
41.7%
C)
65.0%
D)
34.4%
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Answer Key
Testname: C7
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