Question

4 7 5 0 2 2 Problem 7 Let A= -1 2 9 -4 1 5 -1 3 7 3 1 -4 2 0 1 1 0 10 2 a) (4 pts] Using the [V, D] command in MATLAB with rational format, find a diagonal matrix D and a matrix V of maximal rank satisfying the matrix equation A * V = V * D. Is A real-diagonalizable? b) [4 pts) Write down the eigenvalues of A. For each eigenvalue, find a basis for the corresponding eigenspace of A. c) [4 pts] Define a pairing on R5 by < V, w >= vT * A *w. Show that this pairing is a bilinear symmetric pairing on R5. Is the pairing an inner product? (you should justify your answer either way)

Answer 1

(a) Matlab Code: A = [4 5 -1 3 7;5 0 2 1 2;-1 2 9 -4 1;3 1 -4 2 0;7 2 1 0 10]
[V1,D1] = eig(A);
V = rats(V1)
D = rats(D1)

Output:

A =

     4     5    -1     3     7
     5     0     2     1     2
    -1     2     9    -4     1
     3     1    -4     2     0
     7     2     1     0    10

V =

5×70 char array

    '   -351/527       400/1491     -193/531       -88/677       150/259   '
    '    233/341       199/487       -47/89         19/230       116/409   '
    '    -29/218      -209/623       -43/155       186/209        11/392   '
    '     63/514      -941/1210     -207/475      -178/429        65/483   '
    '     46/193       -23/110        21/37         45/409       103/137   '

D =

5×70 char array

    '   -443/101          0             0             0             0      '
    '       0        -1507/1172         0             0             0      '
    '       0             0          567/179          0             0      '
    '       0             0             0         2015/178          0      '
    '       0             0             0             0         6943/429   '

A is real diagonalizable.

(b) The eigenvalues are the diagonal entries of the matrix D say lambda_i. Let v_i denote the ith column of the matrix V, then the eigenspace corresponding to the eigenvalue, lambda_i, is spanned by v_i, since the eigenvalues are distinct.

(c) Since A is symmetric, the bilinear pairing is also symmetric. The pairing is not an inner product since A is not a positive definite matrix.