Let x, y be nonzero vectors in Rn, n ≥ 2, and letA = xyT . Show that λ = 0 is an eigenvalue of A with n − 1 linearly independent eigenvectors and consequently has multiplicity at least n − 1

Let M be a 2 × 2 matrix with eigenvalues ₁ = -0.8, 12 = 1 with corresponding eigenvectors Consider the difference equation with initial condition X = V₁ = V2 = Xk+1 = Mxk Write the initial condition as a linear combination of the eigenvectors of M. That is, write x0 = c1V1 + C2V2 = In general, X = Vi+ V2 k )* VI+ ) k V2 Specifically, X2 = For large k, xk →

(3) Let matrix A have eigenvectors . . associated with eigenvalues Xo = 4, A₁ = 2, A₂ = 0), respectively. Suppose some vector 7 is: 7 = №n +27₁ +30¹₂ Compute Ar. Give your result as a linear combination of eigenvectors ₁.

Let the eigenvalues of the matrix A be A1, A2, . . A,n and the linearly w m w w w w w ma ndependent eigenvectors corresponding to these eigenvalues V1,V2,.. Vn. .... ww m ww (S [V1V2 . .. Vn]nxn) || by placing the vectors V1,V2,., Vn in each column of A is the eigenvalue of V1,V2,.., Vn placed on the w w m w m w w the S matrix; If the matrix diagonal, prove that, diagonal matrix. Also show that eAt mw wm m m w ww diag[A1 A2 . .. A,]nxn) with in the A = SAS-1 (A SeAt s-1 . %3D ww w w w w m

If R is diagonalizable and Q is similar to R, then show that Q has n linearly independent eigenvectors. Also show that Q is invertible if R is invertible.

Show that if 0 is an eigenvector of A, then A is singular.