Problem 3. Suppose A has eigenvalues 0, 3, 5 with corresponding independent eigenvectors u, v,w. (a)...
Suppose A is a symmetric 3 by 3 matrix with eigenvalues 0, 1, 2 (a) What properties 4. can be guaranteed for the corresponding unit eigenvectors u, v, w? In terms of u, v, w describe the nullspace, left nullspace, (b) row space, and column space of A (c) Find a vector x that satisfies Ax v +w. Is x unique? Under what conditions on b does Ax = b have a solution? (d) (e) If u, v, w are...
11. (adapted from 1.6 8) Prove the characteristic polynomial of matrix A - is p(x) = 12 - (a+d)X + ad-bc = 0. Show that p(A) = A - (a + d) A+ (ad - bc)1 = 0. 12. (adapted from 1.6 14) Suppose A has eigenvalues 0,0,3 with independent eigenvectors u, v,w. (a) Give the vectors span the nullspace and the column space. (b) Find a particular solution to Ax=w. Find all solutions. (c) Does w + u in...
Suppose the eigenvalues of a 3x 3 matrix A are λ1-5 λ2-4 andh"4 with corresponding eigenvectors v1 = 0 v' 1 and V' -4 Let Cur Atte x,-1-1 | Find the solution or the equation xk + 1 . AXk for the spected xo, and describe what happens as k→00 Find the solution of the equation xx-1 = A4 choose the correct answer below. @a. ½=(5)kl o lli| | | |+2 This c Suppose the eigenvalues of a 3x 3...
Slove 2nd problem plz (1) Find the eigenvalues and corresponding eigenvectors of [o1 0 0 0 1 2 1 -2 HINT: Note that 13 + 2/2 - 1 - 2 can be regrouped as 1(12 - 1)+2(12-1). Then factor out the common (12 - 1). (2) Solve the equation Y" + 2y' - - 2y = 0) using the method of converting to a linear system of first-order ODE's. Show that the coefficient matrix is the 3 x 3 matrix...
Let A be a 2x2 matrix with eigenvalues 4 and and corresponding eigenvectors V, = and v2 Let} be a solution of the difference equation X: 1 -AX. Xo' - a Computex, = Ax (Hint: You do not need to know itselt b. Find a formula for x, involving k and the eigenvectors V, and v2 a x Ax=(Type an integer or simplified fraction for each matrix element) b. xxv.v2 (Type expressions using k as the variable.)
please answer both a and b Problem 2 (Eigenvalues and Eigenvectors). (a) If R2-R2 be defined by f(x,y) = (y,z), then find all the eigenvalues and eigenvectors of f Hint: Use the matrix representation. (b) Let U be a vector subspace (U o, V) of a finite dimensional vector space V. Show that there exists a linear transformation V V such that U is not an invariant subspace of f. Hence, or otherwise, show that: a vector subspace U-o or...
Problem 2 (Eigenvalues and Eigenvectors). (a) If R2 4 R2 be defined by f(x,y) (y,x), then find all the eigenvalues and eigenvectors of f Hint: Use the matrix representation. (b) Let U be a vector subspace (U o, V) of a finite dimensional vector space V. Show that there exists a linear transformation V V such that U is not an invariant subspace of f Hence, or otherwise, show that: a vector subspace U-0 or U = V, if and...
Problem 2. Find the eigenvalues Xi and the corresponding eigenvectors v; of the matrix -4 6 -12 A-3 -16, (3 3 8 and also find an invertible matrix P and a diagonal matrix D such that D=P-AP or A = PDP-
Please show complete and neat steps for all the problems 8. The eigenvalues and corresponding eigenvectors for this matrix are given below. 1 -3 1 b+3c a) Verify that these are indeed the correct and valid eigenvector/eigenvalue combinations for this matrix. x(t) y(t) z(t) Give the complete solution to the differential equation X'- AX, where X b) Please give your answers for x(t), y(t), and z(t) explicitly. solvé if you dont 8. The eigenvalues and corresponding eigenvectors for this matrix...
Please solve in details. Show that, u, v, w are orthonormal eigenvectors of matrix M, corresponding to eigenvalues 1, 12, 13 respectively, and L is a square matrix whose columns are u, v, w , then (D = Ll' (M L ) is a diagonal matrix.