Problem 1 (a) problem Given a symmetric matrix A = Rn and a vector b = R^ consider the minimization Use the diagonalization 1 min ±xªAx+b²x. XER" 2 A = QAQT, where A = diag(~1,..., λn) ER×n and Q = Rnˇn is orthogonal, to transform 1 into (1) 1 min ZERn 2 ZT AZ + c² z. (2) (b) How are x and z related? How are b and c related? Solution? Under what conditions on 21,..., λ does 2 have a unique solution? What is the (Hint: If g; R→ R, j = 1,...,n, are given functions, then the minimizer z = (Z1,...,Zn) E 18(z) is obtained by minimizing g¡(z;), j = 1,...,n, indi- R" of the function g(z) def vidually.) (c) Let = 2/5 6 -2/√5 (-2 3²) - (-2/5/5 1/√3) (01) (2/√5 1/5 5 -Q =A =QT =A and b: = () Compute the solution z of 2 and the solution x of 1.
Problem 1 (a) problem Given a symmetric matrix A = Rn and a vector b = R^ consider the minimization Use the diagonalization 1 min ±xªAx+b²x. XER" 2 A = QAQT, where A = diag(~1,..., λn) ER×n and Q = Rnˇn is orthogonal, to transform 1 into (1) 1 min ZERn 2 ZT AZ + c² z. (2) (b) How are x and z related? How are b and c related? Solution? Under what conditions on 21,..., λ does 2 have a unique solution? What is the (Hint: If g; R→ R, j = 1,...,n, are given functions, then the minimizer z = (Z1,...,Zn) E 18(z) is obtained by minimizing g¡(z;), j = 1,...,n, indi- R" of the function g(z) def vidually.) (c) Let = 2/5 6 -2/√5 (-2 3²) - (-2/5/5 1/√3) (01) (2/√5 1/5 5 -Q =A =QT =A and b: = () Compute the solution z of 2 and the solution x of 1.
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter2: Matrices
Section2.3: The Inverse Of A Matrix
Problem 80E
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Transcribed Image Text:Problem 1
(a)
problem
Given a symmetric matrix A = Rn and a vector b = R^ consider the minimization
Use the diagonalization
1
min ±xªAx+b²x.
XER" 2
A = QAQT,
where A = diag(~1,..., λn) ER×n and Q = Rnˇn is orthogonal, to transform 1 into
(1)
1
min
ZERn 2
ZT AZ + c² z.
(2)
(b)
How are x and z related? How are b and c related?
Solution?
Under what conditions on 21,..., λ does 2 have a unique solution? What is the
(Hint: If g; R→ R, j = 1,...,n, are given functions, then the minimizer z = (Z1,...,Zn) E
18(z) is obtained by minimizing g¡(z;), j = 1,...,n, indi-
R" of the function g(z)
def
vidually.)
(c)
Let
=
2/5 6
-2/√5
(-2 3²) - (-2/5/5 1/√3) (01) (2/√5 1/5
5
-Q
=A
=QT
=A
and
b:
=
()
Compute the solution z of 2 and the solution x of 1.
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