In each of Problems 17 through 20, use a computer to find the eigenvalues and eigenvectors for the given matrix.
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- In this problem, if you give decimal answers then give at least three digits of accuracy beyond the decimal. The matrix has the following complex eigenvalues (give your answer as a comma separated list of complex numbers; use "i" for ✓-1 and feel free to use a computer to solve the relevant quadratic equation): λ = 1.65+1.548386257i, 1.65-1.548386257i Since A has non-real eigenvalues, it is not diagonalizable, but we can find a matrix C = section 5.5 of Lay): C = P = cos(6) A = sin(8) -sin(8) cos(8) [113] 1.8 Further, we may factor Cas C = XY, where X is a matrix that scales by the positive real number radians, with —π/2 ≤0 ≤ π/2) is and an invertible matrix P such that A = PCP-¹ (I want you to cook up C, P as in and Y is rotation matrix whose counter-clockwise rotation angle (in This means that if we let B be the basis for R² consisting of the columns of P, then the B-matrix of A is C. So if we're willing to change our basis for IR², then the linear transformation x Ax really is just…arrow_forwardLet A, B, and C be an n x n invertible matrices. Solve the following equation for X. Justify each step in your solution. A-1(A+ X)B = C.arrow_forwardSolve in 20 minutes pleasearrow_forward
- Problem 5. Let A be a diagonalizable matrix whose eigenvalues are all either 1 or -1. Show that A-¹ = A.arrow_forwardGiven matrix A with complex eigenvalues h1,2 = a +/- bi and complex eigenvectors v1,2 = c +/- di. Find a = _ , b = _ , c = < _ , _ >, and d = < _ , _ >.arrow_forwardProblem 1 Why does any Google matrix must have one as an eigenvalue? In other words, explain why a matrix whose sum of entries in all columns sum to k must have k as an eigenvalue.arrow_forward
- 3. Let 3 -7 5 A = -9 and B = 5 3 -4 Solve for Matrix M such that 2M + 5A = Barrow_forward4. For 3 A = -4 1 -1 find the matrix exponential exp(At). Combine similar terms and write your answer as one matrix or as a matrix multiplied by ect.arrow_forwardQuestion 7 Find all eigenvalues and eigenspaces for the following matrices. (a) A = (b) 1 -1 = -3 3 B = 0 -5 3 0 0 4arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning