Solving Matrix Basis Problem with Orthogonal Matrix B | Need Help Urgently

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The discussion revolves around solving the Matrix Basis Problem involving an orthogonal matrix B. The user seeks assistance in proving that the eigenvalues (lambdas) are equal when given two bases. A key insight provided is that the eigenvalues will only be the same if the two bases are identical, which implies that the identity matrix M must be used. Additionally, it is noted that vector norms and inner products are preserved by orthogonal matrices, suggesting that these properties are crucial to the solution.

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Lindsayyyy
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Hi everyone

Homework Statement



File at attachment. Given are two basis and the orthogonal matrix B. When r=...(see attachment) I shall proof that the lambdas are equal.

Homework Equations



-

The Attempt at a Solution



I have much trouble with this exercise and it is quite urgent. I tried to express v1' via v1 and v2, but this doesn't bring me to the solution, for example I have: v1' = av1 + bv2 etc.

Can anyone help me with this?

Thanks for the help in advance
 

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Hi Lindsayyyy! :smile:

I don't really get your problem statement.

Any vector r can be represented uniquely with respect to a basis.
With respect to a different basis the representation is again unique, but will always be different.
So as I understand your problem, you can only proof that the lambdas are different.

The lambdas will only be the same iff the 2 basis are the same (that is, if M is the identity matrix).So I suspect you're not supposed to proof the lambdas are the same.
Especially seeing the last equation saying something about what appears to be the vector norm of an inner product of the lambdas.
Still not quite sure what it says though. Can you clarify?

I can say that vector norm and inner product are preserved by an orthogonal matrix, so you probably need to do something with that.
 
Last edited:

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