Matrix with only real eigenvalues

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An nxn matrix with only real eigenvalues can be transformed into an upper triangular form using an orthogonal matrix Q, with eigenvalues positioned along the main diagonal. The proof can be approached through mathematical induction, where the induction step involves finding an orthogonal matrix P that brings the matrix closer to upper triangular form. The discussion emphasizes that gender stereotypes regarding math abilities are unfounded and should not deter anyone from seeking help. Overall, the focus is on the mathematical concept rather than personal feelings or biases. Understanding the process of diagonalization and the properties of orthogonal matrices is key to solving the problem.
Meistro
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Please help me I am hopelessly lost and don't even know where to start! I guess they're right when they said girls suck at math! It's not fair! :redface:

Let A be an nxn matrix with only real eigenvalues. Prove that there is an orthogonal matrix Q such that (Q^T)AQ is upper triangular with eigenvalues along the main diagonal.


Any of you boys out there help me solve this?
 
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Meistro said:
Please help me I am hopelessly lost and don't even know where to start! I guess they're right when they said girls suck at math! It's not fair! :redface:

No, "they" are wrong.

Meistro said:
Let A be an nxn matrix with only real eigenvalues. Prove that there is an orthogonal matrix Q such that (Q^T)AQ is upper triangular with eigenvalues along the main diagonal.

You can use induction on n. For a hint on the induction step, you can try to find an orthogonal matrix P where (P^T)AP is 1 step closer to being upper triangular- try to get the first column in the right shape.

Meistro said:
Any of you boys out there help me solve this?

Girls can excel at math just as well as boys, you shouldn't exclude a potential source of aid.
 

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