6. A 2 x 2 matrix A is known to have the eigenvalues —3 and 4. What are the eigen- values of (a) 2A, (b) 5A, (c) A – 31, and (d) A + 4I? od hot are the
16. Assume that M is a 4 x 4 matrix with eigenvalues 11 = -1, 12 = 0, 13 = 5, y = 1. Choose the correct answer(s) (a) A basis of R* can be formed using eigenvectors of M (b) The matrix M is nonsingular c) The matrix M is diagonalizable (d) All of the above 17. Let S be a 3 x 3 symmetric matrix whose eigenvalues are 12 = 4, 13 = -1. Choose the correct answer(s)...
3. ( Find all eigenvalues and eigenvectors of the matrix A= [ 5 | 3 -1] and show the eigen- 1 vectors are linearly independent.
3. Suppose A is a real square n x n matrix with SVD given by A USVT Using MATLAB's eig and svd, investigate how the eigenvalues and eigen- vectors of the real symmetric matrix AT 0 depend on 2, U and V. Try a random matrix with n 2 to get started. Once you see the relationship, state it carefully, without proof. 4. (This is a continuation of the previous question.) Prove the property that you observed in the previous...
= ) 1 5 # 4 [10 points] Recall the real symmetric 2 by 2 matrix B of # 3. 5 1 # 4 a) From the matrix fact sheet, what are the eigenvalues of B and its characteristic polynomial ? # 4 b) Compute the trace of B, the sum of diagonal terms of the matrix B in three ways. 1- directly, 2- via eigenvalues, 3-via characteristic polynomial # 4 c) Compute the determinant of B, ad — bc,...
[18 Point Problem 5: Consider the following (2 x 2) matrix A: 1-4 -1] A= 13 2 a) Find the eigenvalues and the eigenvectors for the matrix. b) Compute the magnitude of the eigenvectors corresponding to both eigenvalues where a = 1. Observing your results, what conclusion can you draw. ('a' is the complex number replacing the free variables 11 or 12)
For a given system the system A-matrix is given by 4 3 2 5 367 1 A = 2 7 5 3 5 3 2 The matrix of left eigen vectors U and right eigen vectors Vare respectively -0.4633 -0.4633 0.4122 0.4343 -0.4711 0.6121 0.4538 0.6399 -0.5780 0.4538 0.5012 0.4569 0.4343 0.5012 -0.4338 0.6099 U = V = -0.4711 0.5780 0.6399 -0.4338 -0.3108 0.5529 -0.3108 0.5894 -0.4569 0.3328 0.4122 0.6099 0.5894 0.3328 0.6121 0.5529 Determine the eigen values of the...
-2 2 4 2-5 (10 pts) The matrix C 12 3 has two distinct eigenvalues. AI < has multiplicity 2 and has multiplicity
-2 2 4 2-5 (10 pts) The matrix C 12 3 has two distinct eigenvalues. AI
92 (a) The matrix A= 2 -5 -4 has an eigenvalue 2 -4 -5 Two of the entries of A are replaced by I, y so that it will not be convenient to find the eigenvalues by an application. 5 An eigenvector of A corresponding to the eigenvalue is 1 Find the value of and enter your answer in the box below. X= Number (b) Suppose that characteristic equation of a 8 x 8 matrix M is (1 - 2)4...
4.(5 pts)Give an example of a 3 x 3 matrix with eigenvalues of 2, 2, and -3 that is diagonalizable. Show that it is, in fact, diagonalizable, and find C and D such that C (you may make this as trivial as you wish!) AC = D 5.(5 pts) Give an example of a 3 x 3 matrix with eigenvalues of 2, 2, and -3 that is NOT diagonalizable. Show WHY it is not diagonalizable. 6. (5 pts) Let T:...