Eigenvalues and Eigenvectors

In summary, if B is the inverse of A, then if |psi> is an eigenvector of A with eigenvalue a not equal to 0, then |psi> is also an eigenvector of B with eigenvalue 1/a. This can be proven by simplifying A as a 2x2 matrix, finding the inverse of A, and using the fact that [A,B]=0 to show that the eigenvalue of B must be 1/a.
  • #1
azone
7
0
Suppose that B is the inverse of A. Show that if |psi> is an eigenvector of A with eigenvalue a not equal to 0, then |psi> is an eigenvector of B with eigenvalue 1/a.


So I know that A|psi> = a|psi>, and I'm trying to prove that A^(-1)|psi> = 1/a|psi>. I tried simplifying A as a 2x2 matrix and then doing the inverse of that. And then I assumed that the inverse of A has an eigenvalue b. So then I did the determinant of A^(-1)-b = 0 in the hopes to find b and see that it's equal to 1/a. But that became really messy.

Any suggestions on how to solve this problem? Thank you so much!
 
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  • #2
Start with A|psi> = a|psi> and solve for (isolate) |psi> in terms of inv(A). You should be able to take it from there.
 
  • #3
well it is not too difficult:

you supposed A invertible and B=A^(-1).

since AB=BA=1---->[A,B]=0.

(1)|psi>=BA|psi>=Ba|psi>=AB|psi>=ab|psi>.


but ab=1 so the eigenvalue b must be b=1/a
 

1. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are mathematical concepts used in linear algebra to understand and analyze the behavior of linear transformations. Eigenvalues are scalar values that represent the amount by which an eigenvector is scaled when it is transformed by a linear transformation. Eigenvectors are non-zero vectors that remain in the same direction after a linear transformation is applied.

2. How are eigenvalues and eigenvectors calculated?

Eigenvalues and eigenvectors can be calculated by solving the characteristic equation of a square matrix. The characteristic equation is obtained by subtracting the identity matrix multiplied by a scalar, lambda, from the original matrix and setting the determinant of the resulting matrix equal to zero. The solutions to this equation are the eigenvalues, and the corresponding eigenvectors can be found by plugging in the eigenvalues into the original matrix and solving for the corresponding eigenvectors.

3. What is the significance of eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are important in many areas of mathematics, including linear algebra, differential equations, and physics. They are used to solve systems of linear equations, diagonalize matrices, and understand the behavior of systems that evolve over time. In data analysis, they are used in techniques such as principal component analysis to reduce the dimensionality of data and identify patterns in data.

4. Can a matrix have multiple eigenvalues and eigenvectors?

Yes, a matrix can have multiple eigenvalues and eigenvectors. The number of eigenvalues and eigenvectors of a matrix is equal to its dimension. However, some eigenvalues may have multiplicity, meaning they have more than one corresponding eigenvector. This is often the case with symmetric matrices, where the eigenvectors form an orthogonal basis.

5. How are eigenvalues and eigenvectors used in real-world applications?

Eigenvalues and eigenvectors have numerous applications in various fields, including physics, engineering, and computer science. In physics, they are used to study quantum mechanics and the behavior of quantum systems. In engineering, they are used in structural analysis to find the natural frequencies and mode shapes of systems. In computer science, they are used in machine learning algorithms, image and signal processing techniques, and data compression methods.

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