Question

Hey y'all, I would really appreciate some help with a kinda abstract question.

Answer 1

"Let A be a symmetric matrix and suppose that Au = \[\lambda u, u \neq \emptyset, and Av = \beta v, v \neq 0.\]\[ Assume \lambda \neq \beta.\]\Show that \[u^tv\] = 0.

Answer 2

Sorry for the sloppy formatting, but essentially I'm trying to prove (u^t)v equals zero given a symmetric matrix.

Answer 3

So you're essentially showing that u and v are orthogonal right since their dot product is 0. What have you tried so far in an attempt?

Answer 4

Well I set up (Av)^t u = u^t (Av) and used transposing properties and the given equations to convert \[(Av)^tu = v^tA^tu = v^tAu = v^t \lambda u \]
\[u^t(Av)= u^t(\beta v) = \beta(u^tv)\]
How's that looking so far? I was gonna put both sides onto one side and set equal to zero but since lambda and beta aren't equal, I'm not sure how to proceed. I'd appreciate any help.

Answer 5

So you are almost there. This is how I did it:

\[A \bar u = \lambda \bar u\]\[(A \bar u)^T = (\lambda \bar u)^T\]\[\bar u ^T A=\lambda \bar u ^T\]\[\bar u ^T A \bar v = \lambda \bar u ^T \bar v\]\[\beta \bar u ^T \bar v = \lambda \bar u ^T \bar v\]

Since beta != lambda then we know that the dot product must be 0!

Is that easy to follow? I sort of skipped a little bit.

Answer 6

Huh, that makes a lot more sense than what I was going for, thank you! Out of curiosity (and here is where I got fully stuck) why do we know the dot product is zero just based on their inequality?

Answer 7

Well if we know \[\lambda \ne \beta\] and we get an equation that looks like \[\lambda *x = \beta *x\] The only possible way to make this true is if x=0. A little algebra might make it easier to see depending on how your brain works, so I'll do it on the previous equation we got to.

\[\lambda * x - \beta * x =0\]

So I just subtracted beta*x from both sides, just algebra.
\[(\lambda - \beta ) x =0\]

factor out the x and you have a nice equation. Since we already know lambda isn't equal to beta, lambda-beta can't be zero. So to make this a true statement, x, our dot product, must be zero! =)

Answer 8

Oh, duh! I swear it's honestly the dumbest little bumps that just stonewall me with these problems. Thank you so much! I've been staring at this problem for much longer than I'm proud to admit, haha. I hope you have a nice day, friend

Answer 9

Haha no problem, I know how these matrices sort of feel overwhelming. I always would try to imagine what's going on "inside" the matrix as I was doing these and it would sort of side track me and distract me. There's a lot to think about, so don't feel too bad, everyone is a beginner.