Symmetric Matrix Eigenvector Proof

In summary, the conversation discusses the relationship between eigenvalues and eigenvectors for a symmetric matrix. It is shown that if a non-zero vector x in the complex space C(n) satisfies the equation Ax = lambda x, where A is a symmetric matrix, then lambda is a real number and the real part of x is an eigenvector of A. The conversation also mentions using the complex inner product to further explore this relationship.
  • #1
chancellorpho
5
0
Eigenvalue and eigenvector for a symmetric matrix

Homework Statement



Let A be a n by n real matrix with the property that the transpose of A equals A. Show that if Ax = lambda x, for some non-zero vector x in C(n) then lambda is real, and the real part of x is an eigenvector of A.


Homework Equations





The Attempt at a Solution



Since transpose of A equals A, A must be a symmetric matrix. But beyond that, I don't know where to start. Any help would be appreciated!
 
Last edited:
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  • #2
Can anyone offer any insight?
 
  • #3
Start out with [itex](\boldsymbol{v},A \boldsymbol{v})[/itex]. In case this notation is unknown to you it's supposed to represent the complex inner product.
 

1. What is a transpose eigenvector?

A transpose eigenvector is a vector that, when multiplied by the transpose of a square matrix, produces a scalar multiple of itself. In other words, it is a vector that remains in the same direction after being transformed by the transpose operation.

2. How is the transpose eigenvector related to the regular eigenvector?

The transpose eigenvector is essentially the same as the regular eigenvector, but it is used in the context of transposed matrices. Both types of eigenvectors represent the direction in which a matrix transformation has the same effect as scalar multiplication.

3. What is the purpose of proving the transpose eigenvector?

Proving the transpose eigenvector is important because it allows us to understand the behavior of transposed matrices and their effect on vectors. It also helps us to solve systems of linear equations and find the eigenvalues of a matrix.

4. What are the steps involved in proving the transpose eigenvector?

The steps involved in proving the transpose eigenvector are as follows:

  1. Start with the definition of an eigenvector and an eigenvalue.
  2. Multiply the transpose of the matrix by the eigenvector.
  3. Use the properties of matrix multiplication to simplify the expression.
  4. Compare the resulting expression to the definition of an eigenvector and eigenvalue.
  5. Conclude that the original eigenvector is also a transpose eigenvector.

5. Can the transpose eigenvector be complex?

Yes, the transpose eigenvector can be complex. In fact, complex eigenvectors are often used in quantum mechanics and other areas of science. The only requirement is that the transpose of the matrix times the eigenvector must result in a scalar multiple of the eigenvector, regardless of whether it is complex or real.

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