Eigenvalue and invertible matrices

Thread starter stine23
Start date Dec 3, 2008
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Eigenvalue Matrices

In summary, an eigenvalue is a scalar associated with a square matrix that represents the amount of scaling a corresponding eigenvector undergoes. Eigenvectors are the directions in which a matrix operates as a scaling transformation and their associated eigenvalues determine the amount of scaling. Eigenvalues are important in linear algebra as they help understand matrix behavior, solve linear equations, and simplify calculations. A matrix can have multiple eigenvalues, with the number of distinct eigenvalues equal to the matrix's dimensions. An invertible matrix has a unique inverse solution and all of its eigenvalues must be non-zero.

Dec 3, 2008

stine23

How do I prove that if A is an invertible matrix and lambda does not equal zero then one dived by lambda is an eigenvalue of the inverse of A?

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Dec 3, 2008

Ben Niehoff

Science Advisor

Gold Member

First of all, if A is invertible, then none of the eigenvalues can be 0, by definition.

Second, consider

[tex]Av = \lambda v[/tex]

for some eigenvector v. What is

[tex]A^{-1}(\lambda v)[/tex]

?

1. What is an eigenvalue?

An eigenvalue is a scalar (single value) that is associated with a square matrix. It is calculated by solving the characteristic equation of the matrix and represents the factor by which the corresponding eigenvector is scaled.

2. How do eigenvalues and eigenvectors relate to each other?

Eigenvectors are the corresponding vectors to the eigenvalues of a matrix. They represent the directions in which the matrix operates as a scaling transformation. The eigenvalue determines the amount of scaling that occurs in the eigenvector's direction.

3. What is the importance of eigenvalues in linear algebra?

Eigenvalues are important in linear algebra because they help us understand the behavior of a matrix and its associated transformation. They can also be used to find solutions to systems of linear equations and to simplify complex calculations.

4. Can a matrix have more than one eigenvalue?

Yes, a matrix can have multiple eigenvalues. The number of distinct eigenvalues is equal to the number of rows (or columns) of the matrix. This means that a 3x3 matrix can have up to three distinct eigenvalues.

5. What is an invertible matrix and how does it relate to eigenvalues?

An invertible matrix is a square matrix that has a unique solution to its inverse. In other words, it is a matrix that can be multiplied by another matrix to get the identity matrix. For a matrix to be invertible, all of its eigenvalues must be non-zero.