Let p and q be two linearly independent vectors in Rn such t

Let p and q be two linearly independent vectors in R^n such that _2 = _2 = 1. Let A=pq^T + qp^T. Show that A is a symmetric matrix. Show that p + q and p - q are eigenvectors of A. Determine the corresponding eigenvalues. Determine the kernel and rank of A. Find an eigenvalue decomposition of A in terms of p and q.

Solution

Ans-

I
Suppose now that B ½ R is compact. By the Heine-Borel theorem, B is then closed and
bounded. Since B is bounded, its closure must contain sup B by above. Since B is closed,
however, it is equal to its own closure. This means that B must contain its supremum.
41.
Given any point y 2 A, we have x 6 = y. Since X is Hausdor®, we may thus ¯nd disjoint
open sets U(y) and V (y) containing y and x, respectively. Since the sets U(y) form an
open cover of A, ¯nitely many of them do. Say A ½ U(y
U = U(y
1
) [ ¢ ¢ ¢ [ U(y
n
); V = V (y
11
1
1
) [ ¢ ¢ ¢ [ U(y
) \\ ¢ ¢ ¢ \\ V (y
n
n
) and let
):

U = U(x
1
) \\ ¢ ¢ ¢ \\ U(x
n
); V = V (x
1
) [ ¢ ¢ ¢ [ V (x
):
Then U and V are open sets containing A and B, respectively. Moreover, we have
z 2 U =) z 2 U(x
) for each i
=) z =2 V (x
i
) for each i
=) z =2 V:
i
n
In particular, U and V are also disjoint, as needed.