4. Prove the following:(a) If n x n matrix A is singular then it has 0 as an eigenvalue.(b) If 0 is an eigenvalue of a n x n matrix A then A is not invertible.(c) Combine statements from (a) and (b) into a single statement.

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Answered: 4. Prove the following: (a) If n x n…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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7. Determine all 2 × 2 matrices A with eigenvalues 1 and –1 and corresponding eigenvectors and respectively.
Let A be an n x n matrix and let λ be an eigenvalue of A. The collection of all eigenvectors corresponding to A, together with the zero vector, is called the eigenspace of A and is denoted by EA. qs)Why is it important that this definition includes the phrase "together with the zero vector"? (i.e., what would go wrong if this phrase was not included?)
Prove that if A is a positive semidefinite matrix, then the singular valuesof A are the same as the eigenvalues of A.
7. Recall that a nilpotent matrix is a nonzero square matrix A such that there exists a positive integer k where Ak = 0 (a) Prove that if A is an n x n nilpotent matrix, then A + I cannot have an eigenvalue of 0. (b) Find two square matrices of the same size, A and B such that A and B are both nilpotent, but A+ B and AB are not nilpotent.
Which of the following statements is(are) if
Consider that matrix 1 2 - A = 2 4 -2 36 -3 (a) Find the eigenvalues of A. Also find the corresponding eigenspaces and their dimensions. (b) Explain why A is diagonalizable. Calculate A2021. (c) Your friend claims "I think if P diagonalizes A, then P diagonalizes every integral power of A." Is his statement correct? If not, under what condition(s) the statement holds true? Can you prove it in general?
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.
1. For each of the following statements, check if it is True or False. (a) True or False: For each 2 x 2 mtrix A, there exists a matrix B such that det(A+B) + det(A) + det(B). (b) True or False: Let A be a 3 x 3 matrix with 3 distinct eigenvalues. Then there is an eigenbasis for A. (c) True or False: Let A be a n x n matrix with one eigenvalue A = 0. Then Ax infinitely many solutions for any b e R". b has %3| (d) True/False: An n x n matrix A is similar to a matrix B if and only if |A| = |B|.
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.
4. Consider the following matrix: 3. -2 A =-14 15 18 -18 -10 (a) Find the eigenvalues of A. (b) Find three linearly independent eigenvectors for A and state the algebraic and geometric multiplicity for each eigenvalue. Form a matrix C whose columns are the three linearly independent eigenvectors that you found in (b). Verify that C-1 AC is a diagonal matrix (which we call D). (d) Use your previous answer to find a matrix B with the property that B3 = A. (e) For any real matrix A, write down a sufficient condition for a real matrix E to exist such that E2 = A, and prove that this condition is sufficient. %3D
In quantum mechanics, when two states with energies ₁ and E2 mix, the resulting states have different energies E and E, which are the eigenvalues of the matrix H = V (EK). V Choosing to be the state with higher energy, derive expressions for E + E, and E - E in terms of E1, E2 and V.

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4. Prove the following: (a) If n x n matrix A is singular then it has 0 as an eigenvalue. (b) If 0 is an eigenvalue of a n x n matrix A then A is not invertible. (c) Combine statements from (a) and (b) into a single statement.