a) Diagonalize the matrix in the form A = SAS-¹, with S a matrix containing the (normalized) eigenvectors and A a diagonal matrix containing the eigenvalues. b) Is the matrix S an orthogonal matrix? Why / why not? c) Using the eigenvalue decomposition computed in a), determine (including a short explanation!) a. the rank of the matrix A. b. the determinant of the matrix A.
a) Diagonalize the matrix in the form A = SAS-¹, with S a matrix containing the (normalized) eigenvectors and A a diagonal matrix containing the eigenvalues. b) Is the matrix S an orthogonal matrix? Why / why not? c) Using the eigenvalue decomposition computed in a), determine (including a short explanation!) a. the rank of the matrix A. b. the determinant of the matrix A.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.1: Introduction To Eigenvalues And Eigenvectors
Problem 35EQ
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Clear Handwriting and picture please !
Thanks
![Consider the matrix
A =
2
| 0
a.
1 2
-1 2
4 0
a) Diagonalize the matrix in the form A = SAS-1, with S a matrix containing the (normalized)
eigenvectors and A a diagonal matrix containing the eigenvalues.
b)
Is the matrix S an orthogonal matrix? Why / why not?
c) Using the eigenvalue decomposition computed in a), determine (including a short explanation!)
the rank of the matrix A.
b. the determinant of the matrix A.
C. the null space of the matrix A.
d) Determine if the matrix B = (A+A¹)´¹ is positive definite, negative definite or indefinite, without
computing its eigenvalue decomposition.
(Hint: use the elimination method and Hermite's theorem).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd6070c27-0824-4884-a918-195c8f609349%2F6514ec42-deb6-408b-9ec2-0a3cbc6c0b6e%2Fcw0ln69_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the matrix
A =
2
| 0
a.
1 2
-1 2
4 0
a) Diagonalize the matrix in the form A = SAS-1, with S a matrix containing the (normalized)
eigenvectors and A a diagonal matrix containing the eigenvalues.
b)
Is the matrix S an orthogonal matrix? Why / why not?
c) Using the eigenvalue decomposition computed in a), determine (including a short explanation!)
the rank of the matrix A.
b. the determinant of the matrix A.
C. the null space of the matrix A.
d) Determine if the matrix B = (A+A¹)´¹ is positive definite, negative definite or indefinite, without
computing its eigenvalue decomposition.
(Hint: use the elimination method and Hermite's theorem).
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