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This document outlines Homework 19 for Math 21b: Linear Algebra, focusing on eigenspaces, due on March 25 and March 29, 2016. It includes problems related to finding eigenvalues and eigenvectors of given matrices, as well as theoretical concepts about eigenvalues, eigenspaces, and the relationships between similar matrices. The document emphasizes the definitions and properties of eigenvectors, algebraic and geometric multiplicities, and the implications of matrix similarity.

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1 views2 pages

hw19

This document outlines Homework 19 for Math 21b: Linear Algebra, focusing on eigenspaces, due on March 25 and March 29, 2016. It includes problems related to finding eigenvalues and eigenvectors of given matrices, as well as theoretical concepts about eigenvalues, eigenspaces, and the relationships between similar matrices. The document emphasizes the definitions and properties of eigenvectors, algebraic and geometric multiplicities, and the implications of matrix similarity.

Uploaded by

moien
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Math 21b: Linear Algebra Spring 2016

Homework 19: Eigenspaces


This homework is due on Friday, March 25, respectively on Tuesday, March 29, 2016.

1 Find all eigenvalues andeigenvectors for the following matrix A =


 2

−1 −1 0 0 
 −1 2 −1 0 0
 

 
 
 −1 −1 4 −1 −1 . What are the algebraic and geometric
 
 
 
 0 0 −1 1 0
 

 
 
0 0 −1 0 1
 

multiplicities? As a hint, we tell you that the eigenvalues 0, 1, 1, 3


appear. Use the trace trick to the 5th eigenvalue.
2 Assume that a 2×2 matrix has trace −5 and determinant 6. Find
its eigenvalues and find a non-diagonal matrix which realizes the
situation.
3 a) Verify that A and AT have the same eigenvalues.
b) Assume A is invertible. What is the relation between the eigen-
values of A and A−1?

4 The vector Anb gives pollution levels in the Silvaplana, Sils and
St Moritz lake n weeks  after an oil spill. The matrix is A =

 0.7 0 0   100 
  
 and b =  0  is the initial pollution level. Find
 0.1 0.6 0
   
  

   
0 0.2 0.8 0
   

a closed form solution for the pollution after n weeks.


5 1) Find all the eigenvalues and eigenvectors of the matrix
 
1 

0 0 0 
1 2 0 0 
 

A=  .


1 6 6 0 




1 14 36 24
 

b) Find the eigenvectors of A3, where A is the previous matrix.


c) Find the eigenvectors of (AT )−1, where A is the previous ma-
trix.

Eigenspaces

A nonzero vector ~v is called an eigenvector, if Av = λv for


some λ. The set of eigenvectors is called the eigenspace Eλ.
It is the kernel of A − λIn. The dimension of the eigenspace is
called the geometric multiplicity of λ. There is a general
result which tells that the geometric multiplicity of λ is always
smaller or equal to the algebraic multiplicity.
Recall that A is similar to B if there exists an invertible S such
that B = S −1AS. If A and B are similar, then they have the
same characteristic polynomial, the same eigenvalues and alge-
braic multiplicities as well as the same geometric multiplicities.
Similar matrices also have the same trace tr(A) = λ1 + ... + λn
as well as determinant det(A) = λ1 · · · λn. These formulas hold
in general if we allow the eigenvalues λi to be complex. More
on complex eigenvalues next week.

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