26. Prove that A = 0 is an eigenvalue of A if and only if A issingular.||27. In this problem we show that the eigenvalues of a Hermitianmatrix A are real. Let x be an eigenvector corresponding to theeigenvalue A.a. Show that (Ax, x) = (x, Ax). Hint: See Problem 21c.b. Show that X(x, x) = X(x, x). Hint: Recall that Ax = AX.c. Show that l = X; that is, the eigenvalue A is real.

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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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11. Let (b) (c) (d) 1 1 3 3 -3 -5 -3 3 3 1 Find the eigenvalues of A. (Hint: X = -2 is an eigenvalue with multiplicity 2.) A = Find the eigenvectors of A. Find matrices P and such that A = PDP-¹, Use the information above to find A4.
4. A real, symmetric 3-by-3 matrix M has eigenvalues X₁ = 4, X₂ 1 vector corresponding to the eigenvalue X₁ = 4 is v₁ -1 0 [4] -2 to the eigenvalue X₂ = 6 is v₂ = 6 and X3 = -3. An eigen- and an eigenvector corresponding What is M?
Two numbers are greater than 10 but less than 40. Their GCF is 17. What are the two numbers?
6. Prove that if A' is an all-zero matrix for some k > 1, then 0 is the only eigenvalue of A.
46. Let A and B be n× n matrices, each with n distinct eigenvalues. Prove that A and B have the same eigenvectors if and only if AB = BA.
Provide a coherent answer for items C and D.
7. Let A = -2 -2 2 2 -1 4 -2 4 (a). Find the eigenvalues and corresponding eigenvectors for A. (b). Find a matrix P that orthogonally diagonalizes A.
Let λ1, λ2, ..., λn the eigenvalues of matrix A. Show that det(A) = λ1λ2...λn
A King in ancient times agreed to reward the inventor of chess with one grain of wheat on the first of the 64 squares of a chessboard. On the second square, the King would place two grains of wheat, on the third square, four grains of wheat, and on the fourth square eight grains of wheat. If the amount of wheat is doubled in this way on each of the remaining squares, how many grains of wheat should be placed on a square 16? Also, find the total number of grains of wheat on the board at this time and their total weight in pounds. (Assume that each grain of wheat weighs 1/7000 pounds.)
2 Consider the matrix A = 3 4. Show that p(A)v = p(c)v holds in general, if v is an eigenvector of A for the eigenvalue c. 5. Assume now that v1, v2, ..., Un are eigenvectors to different eigenvalues c1, C2, ..., Cn. We want to show that these eigenvectors are linearly indepdendent. To do so, write down a linear combination of the form v = bv1 + bzvzt... +bn Un = 0 and show that all coefficient by are zero. To show, for example, that bị is zero, you can consider the polynomial P1 (z) = (z – c2)(r – c3)... (z - Cn) and evalutate pi (A)v. %3D %3D
Mark as TRUE or FALSE. You will need to give an explanation. (a) If v1, v2 ..., Vn are linearly independent, then v1, V2 are linearly independent. (b) If v and w are two eigenvectors for the same eigenvalue c, then the sum v + w is also an eigenvector for the eigenvalue c. (c) Any 2 x 2 matrix has a real eigenvalue. (d) If v is an eigenvector for a matrix A and a matrix B then v is an eigenvector for A+ B.
9. Determine which of the following vectors 1 -1 is an eigenvector of the matrix 1 1 1 1 -1 1 -1 W = -1 -1 1 -1 -1 1 and find the corresponding eigenvalue.

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