204PS5Solution2009

204PS5Solution2009 - Econ 204 Problem Set 5 Solutions...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Econ 204 Problem Set 5 Solutions Exercise 1 C is already diagonal. B can be diagonalized: it has eigenvalues 1 and 2 and a pair of eigenvectors (1 ; 0) and (1 ; 1) corresponding to these eigenvalues. Let V = f (1 ; 0) , (1 ; 1) g be the set with the two eigenvectors. It is also a basis for R 2 . Let W be the standard basis and consider the change of basis matrix Mtx V;W ( id ) = Mtx W;V ( id ) & 1 = & 1 1 1 ¡ & 1 . Thus, B can be written as B = Mtx W ( B ) = Mtx W;V ( id ) & Mtx V ( B ) & Mtx V;W ( id ) where Mtx V ( B ) is precisely the diagonal matrix & 1 2 ¡ (since (1 ; 0) and (1 ; 1) are both the eigenvectors of B and the basis vectors in V ). To check that this works, Mtx W;V ( id ) & Mtx V ( B ) & Mtx V;W ( id ) = & 1 1 1 ¡& 1 2 ¡& 1 1 1 ¡ & 1 = & 1 1 1 ¡& 1 2 ¡& 1 ¡ 1 1 ¡ = & 1 2 2 ¡& 1 ¡ 1 1 ¡ = & 1 1 2 ¡ = B: Eigenvalues for matrix A are 1 ; ¡ 1 ; and 2 with eigenvectors (1 ; ; ¡ 1) , (0 ; ; 1) , and (2 ; 1 ; ¡ 2) . Let M the matrix formed by these eigenvectors and K the di- agonal matrix with eigenvalues on the diagonal. You can proceed similarly for matrix A and verify that A = MKM & 1 ; M & 1 is thus again the change of basis matrix from the standard basis to the basis formed by the above eigenvectors. Exercise 2 If A is positive semide&nite and B is an n by m matrix, then B T AB is indeed positive semide&nite. Let x be a vector in R m . Then Bx is a vector in R n and since A is positive semide&nite, x T B T ABx = ( Bx ) T A ( Bx ) ¢ ; so that B T AB is positive semide&nite....
View Full Document

This note was uploaded on 10/04/2010 for the course ECON 204 taught by Professor Anderson during the Summer '08 term at Berkeley.

Page1 / 3

204PS5Solution2009 - Econ 204 Problem Set 5 Solutions...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online