1. Using the matrix Find 2. If Verify that (a) AA' and (AA')-¹. (b) The eigenvalues and eigenvectors of AA'. (c) The eigenvalues and eigenvectors of (AA)-¹ - (²3) A = e₁ = A = -- (31) (a) (A+B) + C = A + (B+C). (b) (AB)C = A(BC). (c) A(B+C) = AB + AC. 1/√3 1/√3 1/√3 3. Consider a random vector X'= (X₁, X2, X3) with mean vector ' = (1, 2, 3) and covariance matrix E. The eigenvalues of Σ are A₁ = 12, A26, A3 = 2 and the corresponding eigenvectors are (a) Find Σ and trace(). (b) Find E and 2-1. (c) Find E-1/2. B 3=(-2²4) c = (²1) C 2 e₂ 2/√6 -1/√6 -1/√6 " e3 = 0 1/√√2 -1/√2
1. Using the matrix Find 2. If Verify that (a) AA' and (AA')-¹. (b) The eigenvalues and eigenvectors of AA'. (c) The eigenvalues and eigenvectors of (AA)-¹ - (²3) A = e₁ = A = -- (31) (a) (A+B) + C = A + (B+C). (b) (AB)C = A(BC). (c) A(B+C) = AB + AC. 1/√3 1/√3 1/√3 3. Consider a random vector X'= (X₁, X2, X3) with mean vector ' = (1, 2, 3) and covariance matrix E. The eigenvalues of Σ are A₁ = 12, A26, A3 = 2 and the corresponding eigenvectors are (a) Find Σ and trace(). (b) Find E and 2-1. (c) Find E-1/2. B 3=(-2²4) c = (²1) C 2 e₂ 2/√6 -1/√6 -1/√6 " e3 = 0 1/√√2 -1/√2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I need help with number 3
Transcribed Image Text:1. Using the matrix
Find
2. If
Verify that
(a) AA' and (AA')-¹.
(b) The eigenvalues and eigenvectors of AA'.
(c) The eigenvalues and eigenvectors of (AA)-¹
A =
A = (²₁₁) B=(-²₁) Cc = (3²¹)
3
0
(131)
-1
(a) (A + B) + C = A + (B+C).
(b) (AB)C= A(BC).
(c) A(B+C) = AB + AC.
3. Consider a random vector X'= (X₁, X2, X3) with mean vector ' = (₁, 2, 3) and
covariance matrix Σ. The eigenvalues of E are di 12, A2 = 6, 32 and the
corresponding eigenvectors are
1/√3
-- ()--(-).
=
-1/√6
1/√3
1/√3
(a) Find Σ and trace().
(b) Find E and 2-1.
(c) Find E-1/2.
-1/√6
e3
0
1/√2
-1/√2
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