Consider the real symmetric matrix A 0 1 1 2 1 3 0 0 0 a Fi

Consider the real symmetric matrix A = (0 -1 1 -2 -1 3 0 0 0) (a) Find the eigenvalues and eigenvectors of A. (b) Write/calculate the matrices for the diagonal factorization of A, A = X Lambda X^-1. Multiply out the matrix product to confirm that you get back A. (c) Find the eigenvectors of A^T. (A^T y = lambda y) (d) Show that x_i (the \"right-eigenvector\" for eigenvalue lambda_i) and y_j (the \"left-eigenvector\" for eigenvalue lambda_j) are orthogonal if lambda_i notequalto lambda_j. (Either work out all of the combinations for this specific A or adapt the dot-product-based proof that lambda = lambda for symmetric A from Lecture 31 to prove that this is a general property). (e) Calculate the dot products x_i middot y_i for i = 1, 2, 3. (f) Does the product X Lambda Y^T yield A? Show that if you scale either the x\'s or y\'s so that the products in (e) are all ones then X Lambda Y^T = A

Solution

to find the eigenvalues evaluate the determinant |A - K I |=0 where K is the eigen value

| 0-k -1 1

-2 -1-k 3

0 0 -k | =0 expanding the det ; -k[ k(k+1) -2] =0 => k=0 or k²+k-2=0 k=1 ,-2

the eigen values are k=0 ,1,- 2

to find the eigenvectors solve the matrix eqn AX= kX where X is the column matrix

X =( x₁,x₂,x₃)^T   

AX =KX => [ 0 -1 1

-2 -1 3

0 0 0 ] ( x₁   x₂   x₃)^T = ( kx₁ ,kx₂, kx₃]

the system when k=0 is : - x₂+x₃=0 => x₃=x₂

-2x₁-x₂+3x₃=0 when x₂=x₃we get x₁=x₂

   _{when k=0 the correseponing eigen vector is} X₁= ( 1,1,1)^T  -----I

when k=1 the system is : -x₂+x₃=x_{1 ,} -2x₁-x₂+3x₃=x₂  => x₃=0 and x₂= -x₁

when k=1 the eigenvector is X ₂= ( 1, -1 ,0) ^T ------II

when k= - 2 the system is : - x₂+x₃= -2x_{1 ,} -2x₁-x₂+3x₃= -2x₂ => x₃=0 x₂=2x₁

_{when k=- 2 the eigen vector is} X₃= ( 1,2,0)^T

   ^{the matrix formed by yhe eigen vectors} is X such that XAX^{-1 is a diogonal matrix} in which the main diogonal consists of the eigen values