Proving Real Eigenvalues for Symmetric Matrix Multiplication?

Thread starter tom08
Start date Dec 18, 2009
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Matrix Symmetric Symmetric matrix

In summary, a real symmetric matrix is a square matrix with real elements that is equal to its own transpose. It has many important properties and applications in mathematics and physics, such as solving systems of equations, optimization problems, and understanding eigenvalues and eigenvectors. A matrix can be determined to be real symmetric by comparing its elements row by row and column by column. Real symmetric matrices have real eigenvalues and are commonly used in physics to model various physical systems, particularly in quantum mechanics for representing operators and observables.

Dec 18, 2009

tom08

Homework Statement

Given a real diagonal matrix D, and a real symmetric matrix A,

Homework Equations

Let C=D*A.

The Attempt at a Solution

How to prove all the eigenvalues of matrix C are real numbers?

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Dec 18, 2009

Dick

Science Advisor

Homework Helper

Now why would you think that's true?

Dec 18, 2009

CompuChip

Science Advisor

Homework Helper

Also note this thread.

1. What is a real symmetric matrix?

A real symmetric matrix is a square matrix where the elements are real numbers and the matrix is equal to its own transpose. This means that the matrix is symmetric across its main diagonal.

2. What is the significance of a real symmetric matrix?

A real symmetric matrix has many important properties and applications in mathematics and physics. It is used in linear algebra for solving systems of equations, in optimization problems, and in spectral theory for understanding eigenvalues and eigenvectors.

3. How can you determine if a matrix is real symmetric?

A matrix is real symmetric if it is equal to its own transpose. This means that the elements on either side of the main diagonal are equal. One way to check this is to compare the elements row by row and column by column. If they are the same, the matrix is real symmetric.

4. Can a real symmetric matrix have complex eigenvalues?

No, a real symmetric matrix has real eigenvalues. This is because the eigenvectors of a real symmetric matrix are orthogonal, and the eigenvalues are the coefficients of the eigenvector expansion. Since the eigenvectors are real, the eigenvalues must also be real.

5. How is a real symmetric matrix used in physics?

In physics, real symmetric matrices are used to model many physical systems, including quantum mechanics, classical mechanics, and electromagnetism. They are particularly useful in quantum mechanics for representing operators and observables, such as the Hamiltonian and spin operators.