Matrix with only real eigenvalues

In summary: Plus, the conversation is about a mathematical proof, not a gender debate. Let's focus on the problem at hand.
  • #1
Meistro
5
0
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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  • #2
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.
 

FAQ: Matrix with only real eigenvalues

What is a matrix with only real eigenvalues?

A matrix with only real eigenvalues is a square matrix whose eigenvalues are all real numbers, as opposed to complex numbers. In other words, the solutions to the characteristic equation of the matrix are all real numbers.

How can you determine if a matrix has only real eigenvalues?

A matrix has only real eigenvalues if its characteristic polynomial has only real roots. This can be determined by finding the determinant of the matrix and solving the resulting characteristic equation for its roots.

What are the properties of a matrix with only real eigenvalues?

A matrix with only real eigenvalues has several important properties. First, its eigenvectors are all real and can form a basis for the vector space. Second, the matrix is always diagonalizable, meaning it can be written as a diagonal matrix with its eigenvalues on the main diagonal. Finally, the matrix is always symmetric if it is square.

Why are matrices with only real eigenvalues important?

Matrices with only real eigenvalues are important because they have many useful applications in mathematics and science. For example, they are commonly used in physics for modeling systems with real-world constraints. Additionally, they are used in various algorithms for solving systems of linear equations.

Can a matrix have both real and complex eigenvalues?

Yes, a matrix can have both real and complex eigenvalues. In fact, most matrices will have a mix of both. However, there are certain types of matrices, such as symmetric matrices, that will have only real eigenvalues. It is also possible for a matrix to have only complex eigenvalues.

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