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Linear Algebra 2

This document contains 10 problems involving linear algebra concepts such as matrix diagonalization, eigenvalues and eigenvectors, Gauss elimination, quadratic forms, Cholesky decomposition, orthonormal bases of subspaces, and Moore-Penrose pseudo-inverses. The problems involve diagonalizing, solving systems of equations, analyzing properties of quadratic forms and matrices, finding bases and decompositions, and computing inverses.

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serena spaziani
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71 views2 pages

Linear Algebra 2

This document contains 10 problems involving linear algebra concepts such as matrix diagonalization, eigenvalues and eigenvectors, Gauss elimination, quadratic forms, Cholesky decomposition, orthonormal bases of subspaces, and Moore-Penrose pseudo-inverses. The problems involve diagonalizing, solving systems of equations, analyzing properties of quadratic forms and matrices, finding bases and decompositions, and computing inverses.

Uploaded by

serena spaziani
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Exercises

Problem 1. Diagonalize (if possible) the following matrices:


 
2 0 −1
(a) A =  0 1 0 ;
0 0 −1
 
2 0 −1
(b) A =  0 1 0 .
0 0 1
 
1 0 −1
Problem 2. Find the eigenvalues and eigenvectors of the matrix A =  0 0 0 . Then, determine
−1 0 1
an orthonormal matrix P such that PT AP is diagonal.
Problem 3. Using the Gauss elimination method, determine the solutions of the system

 x + y + 4z = 2
x + 2y + 3z = 3
2x + 4y − z = 1

Problem 4. Using the Gauss elimination method, determine the solutions of the system

 x+y−z =0
2x + y + 3z = 1
3x + 2y + 2z = 2

Problem 5. Using the Gauss elimination method, determine the solutions of the system

 x+y−z =0 perche non si fa la scala
2x + y + 3z = 1 anche in R4
3x + 2y + 2z = 1

Problem 6. Determine the sign of the quadratic form q(x) = xT Hx in the following cases:
 
2 1
(a) H = ;
1 4
 
−2 0 2
(b) H =  0 4 1 ;
2 1 1
 
1 0 3
(c) H =  0 2 2 .
3 2 20
Problem 7. Determine the Cholesky decomposition of the following matrix:
 
6 1
H= .
1 4
" √ #
6 q0
T
Solution. H = LL with L = . ■
√1 23
6 6

Problem 8. Let us consider the following vectors of R3 :


   
1 0
v1 =  0  , v2 =  1 
1 1
and denote by W the subspace of R3 generated by v1 , v2 . Determine an orthonormal basis of W .

1
Problem 9. Let us consider the following vectors of R4 :
     
2 0 1
 0   1   1 
v1 =   2  , v2 =  1  , v3 =  0
    ,

2 0 0

and denote by W the subspace of R4 generated by v1 , v2 , v3 . Determine an orthonormal basis of W .


 
0 1
Problem 10. Let A =  1 1 . Determine the Moore-Penrose pseudo-inverse of A.
0 1

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