Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
expand_more
expand_more
format_list_bulleted
Question
Chapter 28.3, Problem 1E
Program Plan Intro
To prove that every diagonal element of a symmetrical positive defined matrix is positive.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
Find the product of the product of the elements of a matrix and the other elements of a matrix with random integer elements of 4 x 2 each
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
Need Help? Read it
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.
#UsePython
Knowledge Booster
Similar questions
- Find a basis for the nullspace of the matrix. (If there is no basis, enter NONE in any single cell.) 1 2 4 A = 0 1 0arrow_forwardFind the matrix product of integer random elements and the size of each (4 x 2) of them.arrow_forwardFind the product of the elements of a matrix by other elements of a matrix with integer random elements and the size of each (4 x 2) of them.arrow_forward
- List any two properties of eigenvalues of a square matrix. Explain it in your own way and provide it with examples.arrow_forward. The determinant of an n X n matrix can be used in solving systems of linear equations, as well as for other purposes. The determinant of A can be defined in terms of minors and cofactors. The minor of element aj is the determinant of the (n – 1) X (n – 1) matrix obtained from A by crossing out the elements in row i and column j; denote this minor by Mj. The cofactor of element aj, denoted by Cj. is defined by Cy = (-1y**Mg The determinant of A is computed by multiplying all the elements in some fixed row of A by their respective cofactors and summing the results. For example, if the first row is used, then the determi- nant of A is given by Σ (α(CI) k=1 Write a program that, when given n and the entries in an n Xn array A as input, computes the deter- minant of A. Use a recursive algorithm.arrow_forwardFor the matrix A, find (if possible) a nonsingular matrix P such that p-lAP is diagonal. (If not possible, enter IMPOSSIBLE.) 6 -3 A = -2 P = Verify that P1AP is a diagonal matrix with the eigenvalues on the main diagonal. p-1AP =arrow_forward
- Find the Inverse of Matrix ( 5 x 5 ) and Prove that the theory of ( A x A - ¹ ) = eye is true by using random matrix ?arrow_forwardDetermine the value(s) of x so that the matrix below is the augmented matrixof an inconsistent linear system. x 1 5 5 -3 2arrow_forwardLet A be an m × n matrix with m > n. (a) What is the maximum number of nonzero singular values that A can have? (b) If rank(A) = k, how many nonzero singular values does A have?arrow_forward
arrow_back_ios
arrow_forward_ios
Recommended textbooks for you
Operations Research : Applications and AlgorithmsComputer ScienceISBN:9780534380588Author:Wayne L. WinstonPublisher:Brooks Cole
Operations Research : Applications and Algorithms
Computer Science
ISBN:9780534380588
Author:Wayne L. Winston
Publisher:Brooks Cole