Eigenvectors, Eigenvalues and Idempotent

In summary, the conversation discusses a question about idempotent matrices and eigenvalues. The question asks for help in showing that if λ is an eigenvalue of an independent matrix, it must be either 0 or 1. The solution involves using the statement A*v=lamda*v and rearranging terms to solve for lambda.
  • #1
mpm
82
0
I have a question that deals with all three of the terms in the title. I'm not really even sure where to begin on this. I was hoping someone could help.

Question:

An n x n matrix A is said to be idempotent if A^2 = A. Show that if λ is an eigenvalue of an independent matrix, then λ must either be 0 or 1.

If I could get some help on this I would really appreciate it.

Thanks,

mpm
 
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  • #2
Well you know that there must exist a nonzero vector v such that A*v=lamda*v. Now play with this statement by applying A again, and rearanging terms so that you end up with only expressions involving lambda. Then solve for lambda.
 

1. What are eigenvectors and eigenvalues?

Eigenvectors and eigenvalues are mathematical concepts that are used to represent and analyze linear transformations on vector spaces. Eigenvectors are vectors that do not change direction when a linear transformation is applied, only their magnitude may change. Eigenvalues are scalars that represent how much the eigenvectors are stretched or compressed by the linear transformation.

2. How are eigenvectors and eigenvalues calculated?

Eigenvectors and eigenvalues can be calculated using a mathematical process called diagonalization. This involves finding the characteristic equation of a matrix, solving for the eigenvalues, and then finding the corresponding eigenvectors.

3. What is an idempotent matrix?

An idempotent matrix is a square matrix that, when multiplied by itself, results in the same matrix. In other words, the matrix is its own inverse. This means that the matrix does not change the result of a linear transformation when applied multiple times.

4. How are idempotent matrices used in science?

Idempotent matrices are used in various fields of science, such as physics, engineering, and economics. In physics, they are used to represent physical systems with energy conservation properties. In engineering, they are used in control systems and signal processing. In economics, they are used to model production processes and market equilibrium.

5. Can an idempotent matrix have more than one eigenvalue?

No, an idempotent matrix can only have eigenvalues of 0 and 1. This is because the characteristic equation of an idempotent matrix is x(x-1) = 0, which only has solutions of 0 and 1. Additionally, the only eigenvectors of an idempotent matrix are the zero vector and the ones corresponding to the eigenvalue of 1.

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