Consider the real symmetric matrix A = (0 -1 1 -2 -1 3 0 0 0) (a) Find the eigen
ID: 3037117 • Letter: C
Question
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 = AExplanation / Answer
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=x1 , -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= -2x1 , -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