Linear Algebra: UNIT IV: Matrix Decomposition
Orthogonal Transformation of a Symmetric Matrix to Diagonal Form: Exercise Problem Questions
Linear Algebra: UNIT IV: Matrix Decomposition: Orthogonal Transformation of a Symmetric Matrix to Diagonal Form: Exercise Problem Questions
Orthogonal Transformation of a Symmetric Matrix to Diagonal Form
EXERCISE
1.
Determine the l2 norm of
each of the following vectors.
[Ans: (a) 1 (b) √14 (c) √21 (d) √65 (e) √48]
2.
Determine the l1 norms of
the vectors in problem 1.
[Ans: (a) 1 (b) 6 (c) 7 (d) 9 (e) 12]
3.
Determine the l∞ norms of
the vectors in problem 1.
[Ans:
(a) 1 (b) 3 (c) 4 (d) 8 (e) 4]
4.
For each of the vectors in problem 1, determine the inner product generated
norm with respect to W = 
[Ans:
(a) √10 (b) √46 (c) √29 (d) √298 (e) √461]
5.
For the vectors given in problem 1, determine
(a)
||Z−Y||4
(b)
||Z−Y||2
(c)
||Z−Y||1
(d)
|| U−V||3
(e)
||U−V||1 and
(f)
|| U−V ||∞
[Ans: (a) 31/4
(b) √3 (c) 3 (d) (155)1/3 (e) 11 (f) 4]
6.
Determine the Euclidean norm of
[Ans: (a) 1 (b) √29 (c) 8 (d) √30 ]
7.
Determine the l1 norms of the vectors in problem 6.
[Ans: (a) 1 (b) 7 (c) 8 (d) 5+√5]
8.
Determine the l∞ norms of
the vectors in problem 6.
[Ans:
(a) 1 (b) 5 (c) 8 (d) 5]
9.
Determine the l3 norms of the vectors in problem 6.
[Ans: (a) 1 (b) (133)1/3 (c) 8
(d) 5.1448]
10.
Determine the Frobenius norms for the following matrices.
[Ans: (a) √30 (b) √79 (c) √66 (d) √22 (e) √75 ]
11.
Determine the L1 norms of the matrices in problem 10.
[Ans:
(a) 6 (b) 12 (c) 9 (d) 5 (e) 8.9443]
12.
Determine the L∞ norms of the matrices in problem 10.
[Ans:
(a) 7 (b) 8 (c) 9 (d) 5 (e) 9.4721]
13.
Determine the spectral norms of the matrices in problem 10.
[Ans:
(a) 5.4650 (b) 8.6099 (c) 8.1231 (d) 4.2992 (e) 7.1517]
14.
Determine the spectral radii of matrices.
[Ans: (a) 5.3723 (b) 6.7958 (c) 8.1231]
15.
Determine the spectral radius of A = 
[Ans: 4.8990]
16. For each matrix A,
find an orthogonal matrix P such that PTAP is diagonal.
17.
Determine the principal axes for the following quadratic forms.
Linear Algebra: UNIT IV: Matrix Decomposition : Tag: maths, mathematics : - Orthogonal Transformation of a Symmetric Matrix to Diagonal Form: Exercise Problem Questions
Linear Algebra: UNIT IV: Matrix Decomposition
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