-57 26] The matrix has eigenvalues \1 = -5 and X2 = 8. Find eigenvectors -130 60 corresponding to these eigenvalues. v1 = [8] and v2 = help (matrices) Find the solution to the linear system of differential equations x' = -57x+26y satisfying the initial conditions x (0) y' = -130x + 60y = 1 and y(0) = 1. x(t) = = help (formulas) y(t) = help (formulas) Book: Section 3.4 of Notes on Diffy Qs

When I multiplied I got -28+7i+i^2, where i is the eigenvalue

For the matrix find the eigenvalue corresponding to the eigenvector 3 ^=(², ²) A -3 2 (9).

a) Find the eigenvalues of A. b) Find three linearly independent solutions of x' = Ax. c) Find the exponential e^(tA) Let A = 2 -1 ا دیا 0 0 3 0 -30

- find the eigenvectors of the diagonal matrix (A) J where. 2 5 0-12 2 4 10 A = and P=[^}; 2 1 5 4 ♡ 1

Suppose a system of differential equations yields eigenvalue λ=1+3i with 2+i eigenvector = Find any real-valued solution.

7.'., Given A = Find the eigenvalues and eigenvectors of matrix A

The matrix v₁ = 1 x₁ (t) = -13 12] -8 7 x₂ (t) = has eigenvalues A₁ = -1 and ₂ = -5. Find eigenvectors corresponding to these eigenvalues. satisfying the initial conditions 1 2 Find the solution to the linear system of differential equations and 7₂ x₁ (0)] _x₂ (0) = [3] help (formulas) help (formulas) = x 3 x = - 13 -8 12 7 X1 X help (matrices)

For the following matrix A = ^-[20] 1 C (a) Find all real values c for which the matrix A has real eigenvalues (b) Find the eigenvectors which correspond to the situation where all eigenvalues are real (CR). Hint: You should confirm the eigenvectors are reasonable for all c being considered.

(6) Write the formula for the condition number of A, ||A||||A-¹||, in terms of the eigenvalues of A when A is s.p.d. Given a small (2 x 2) matrix 1 1 A ^ = [ {₁+c]; its eigenvalues and its inverse, compute its condition number using one of the norms. and investigate what happens with the condition number when € → 0.