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BlackOps February 17th, 2010 10:27 PM

Symmetric matrix and eugenvalues..
 
I just dont get the following problem:

Find a real 2 x 2 symmetric matrix A with eugenvalues:

l = 1 and l = 4 and eigenvector u = (1,1) belonging to l = 1;


well i looked to definition which says that if A is real symmetric matrix with eugenvectors u and v belonging to l1 and l2, then u and v are orthogonal...

from my problem i know u... but i do not know eigenvector v... and even if id knew that...how to derive A then? kinda messed up a little... any ideas?

thank u!

Erebos February 17th, 2010 11:42 PM

Re: Symmetric matrix and eugenvalues..
 
Ans: A = [5/2 -3/2]
[-3/2 5/2].

Proof: Let a be a 2x2 real symmetric matrix. Then A = [a b]. Since A is symmetric, then A = A-Transpose = [a c]. This implies that b = c, so we can now
[c d] [b d]

now write A = [a b]. Now, since we know that [1,1]-transpose is the eigenvector for l = 1, then we also know that [a-1 b] *u[1 1]-transpose = [0 0]-transpose.
[b d] [b d-1]

So, the above system tells us that (a-1)*u + bu = 0 and bu + (d-1)*u = 0. Multiplying the second equation by -1, we obtain -bu + (1-d)*u = 0.

Adding this equation to the first equation in our series gives us (a-1)*u + (1-d)*u = 0, dividing by u, we obtain a-1+1-d = a-d = 0, which gives us a = d, which means we can now rewrite A = [a b].
[b a]

Now, we are also given that l1 = 1 and l2 = 2, which means that the characteristic polynomial of the matrix must be (l-1)(l-4) = l^2 - 5l + 4.

So the det(A - lI) = l^2 - 5l +4, and since we know the form of A, we know that the det(A-lI) is also equal to (a-l)^2 - b^2 =

l^2 - 2al +a^2 - b^2. So l^2 - 2al +a^2 - b^2 = l^2 - 5l +4. Subtracting l^2 off of each side, we now have -2al + a^2 - b^2 = -5l + 4. Then from here, we see that
-2al = -5l and a^2 - b^2 = 4. The first equation gives us a = 5/2. Then, this gives us in turn that (5/2)^2 - b^2 = 4, which implies that b^2 = +/-(3/2).

With further validation (which is not shown here), you can actually rule out 3/2, because if you check the eigenvector of l=1 with b = c = 3/2, you actually get u = [1, -1], which contradicts our given assumption. Therefore, A = [5/2 -3/2]
[-3/2 5/2].



-QED-

BlackOps February 18th, 2010 08:01 AM

Re: Symmetric matrix and eugenvalues..
 
based on ur proof i have figured it all out and understood concepts. thank u very much!


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