a) Show that V₁-A = 0 0-201 3is an eigenvector of A and find the correspondingeigenvalue.(b) Find the characteristic polynomial of A and factorise it. Hint: the answer to (a)may be useful.(c) Determine all eigenvalues of A and find bases for the corresponding eigenspaces.(d) Find an invertible matrix P and a diagonal matrix D such that P-¹AP = D.

Answered: Image /qna-images/answer/f51cee61-d832-49d8-a0f1-1f9d23f3738a.jpg

Answered: a) Show that V₁ - A = 0 0 -2 01 3 is an…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. To save time, the eigenvalues are 11 and 2. 6 4-2 4 6 -2 -2 -2 3 Enter the matrices P and D below. (Use a comma to separate matrices as needed. Type exact answers, using radicals as needed. Do not label the matrices.)
Suppose the characteristic equation for a matrix A is given by X3 + 3X2 – 9A – 27 = 0. Find the eigenvalues of the matrix A and give the algebraic multiplicity of each eigenvalue. - Eigenvalue: A = 3 Eigenvalue: A = Algebraic multiplicity: Ex: 5 Algebraic multiplicity:
Consider the matrix A = - 13 24 8 15 (a) Calculate the det(A - xI), where x is a variable. We call the result the characteristic polynomial of A. (b) Give the two zeros 1 and 2 of this polynomial. λ1 and 22 are called the eigenvalues of the matrix A. (c) For each eigenvalue 21, find an eigenvector vi = ( ) + (8) xi # such 7 that Avi = hivi. (d) Construct the matrix D = (2). are eigenvectors v1 and v2. Calculate QDQ λ1 0 22 0 and the matrix Q whose columns -1 ; what do you notice?
A 2x2 upper triangular matrix A has eigenvalues a and - 5 (with a ≠ -5). An eigenvector corresponding to the eigenvalue a is (1,0). Which of the following could be the matrix A? A. (-5 a/2 0) B. (a 0/2 -5) C. (-5 2/0 a) D. (a -5/0 2) E. (a 2/0 -5)
Consider the matrix A =−6 4−2 0 A) Determine the eigenvalues of the matrix A B) For each eigenvalue from part (A), determine the corresponding eigenvectors of the matrix A. Use row operation notation to indicate steps
Consider the matrix 0 1 0 0 0010 0001 1000 (a) Find the eigenvalues by finding the characteristic polynomial and looking at its roots. (b) Find the eigenvalues by setting up a system of equations Ar = λ and solving the easy system of equations in the variables x1, x2, x3, x4 for possible X's. (c) For (each/the) eigenvalue, find the corresponding eigenvectors and eigenspaces.
Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D. To save time, the eigenvalues are 10 and -9. A = 1 -3 9 -3 93 9 3 1 Enter the matrices P and D below. (Use a comma to separate answers as needed. Type exact answers, using radicals as needed. Do not label the matrices.)
Find all eigenvalues and a basis for each eigenspace of A. Then use them to find values of lambda and phi such that A can be expressed as the product of a scaling matrix with factor lambda and a rotation matrix with angle phi. A = 3 -4 4 3
Let A = (a) By calculating and factorizing C₁(2), the characteristic polynomial of A, show that -1 is the only eigenvalue of A, and then find a corre- sponding eigenvector u. -9 2 (b) Find a vector v R2 such that (A + I)v = u, and show by direct calculation that P-¹AP = B where (c) Using the fact that P = [u v] and B = Bn PA 0 = (-1)" [ 7] for every integer n, find An explicitly in terms of n. Hint: Begin by using the equality A = PBP-¹ to express A" in terms of P and B.
(a) By finding the characteristic polynomial for A, show that the eigenvalues of A are 1 and 2. (b) Find a basis for each eigenspace of A (c) Find, if possible, an invertible matrix P and a diagonal matrix D such that P^−1 AP = D. If it is not possible, explainwhy.

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