Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter D.2, Problem 3E
Program Plan Intro
To prove that the permutation matrix, P is invertible and
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List any two properties of eigenvalues of a square matrix. Explain it in your own
way and provide it with examples.
Find the eigenvalues of the matrix and determine whether there is a sufficient number to guarantee that the matrix is diagonalizable. (Recall that the matrix may be diagonalizable even though it is not guaranteed to be diagonalizable by the theorem shown below.)
Sufficient Condition for Diagonalization
If an n xn matrix A has n distinct eigenvalues, then the corresponding eigenvectors are linearly independent and A is diagonalizable.
Find the eigenvalues. (Enter your answers as a comma-separated list.)
Is there a sufficient number to guarantee that the matrix is diagonalizable?
O Yes
O No
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If there is a non-singular matrix P such as P-1AP=D, matrix A is called a diagonalizable matrix. A, n x n square matrix is diagonalizable if and only if matrix A has n linearly independent eigenvectors. In this case, the diagonal elements of the diagonal matrix D are the eigenvalues of the matrix A.
A=({{1, -1, -1}, {1, 3, 1}, {-3, 1, -1}}) :
1
-1
-1
1
3
1
-3
1
-1
a)Write a program that calculates the eigenvalues and eigenvectors of matrix A using NumPy.
b)Write the program that determines whether the D matrix is diagonal by calculating the D matrix, using NumPy.
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