Eigenvalues + Algebraic/Geometric Multiplicity

In summary, The conversation is about studying for a linear algebra final and coming across a problem involving a 3x3 matrix with lambda=4, algebraic multiplicity 3, and geometric multiplicity 1. The person is unsure about these concepts and asks for clarification. The expert explains that algebraic multiplicity is the multiplicity of the root in the characteristic polynomial and geometric multiplicity is the dimension of the eigenspace.
  • #1
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I'm studying for a linear algebra final, and I'm looking over an old final our prof gave us and I've come across something I don't remember ever hearing anything about... Here's the problem:

Write down a matrix A for the following condition:
A is a 3x3 matrix with lambda=4 with algebraic multiplicity 3 and with geometric multiplicity 1.

...I don't have a problem with eigenvalues or anything, but I don't believe he ever mentioned algebraic multiplicity or geometric multiplicity. Is this another concept in linear algebra?? Or is this something way simple that I'm looking way too far into.

...What does he mean by algebraic multiplicity and geometric multiplicity??


Thanks!
 
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  • #2
Algebraic multiplicity is easy. It's the multiplicity of the root in the characteristic polynomial. I checked wikipedia (always a good first stab) for geometric multiplicity and it says that it is the dimension of the eigenspace. In other words, there is only one linearly independent eigenvector with value 4.
 
  • #3
 
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What are eigenvalues and how are they related to algebraic and geometric multiplicity?

Eigenvalues are special numbers associated with a square matrix that represent the scaling factor of its corresponding eigenvectors. Algebraic multiplicity is the number of times an eigenvalue appears as a root of the characteristic polynomial of the matrix, while geometric multiplicity is the number of linearly independent eigenvectors corresponding to that eigenvalue.

How do I find the eigenvalues of a matrix?

To find the eigenvalues of a matrix, you must first compute the characteristic polynomial by taking the determinant of the matrix minus a scalar multiple of the identity matrix. Then, solve for the values of the scalar that make the characteristic polynomial equal to zero. These solutions will be the eigenvalues of the matrix.

What is the significance of algebraic and geometric multiplicity?

Algebraic multiplicity gives us information about the multiplicity of an eigenvalue, which affects the diagonalizability of a matrix. Geometric multiplicity tells us how many linearly independent eigenvectors a matrix has, which is important for finding the eigendecomposition of a matrix.

How can I determine the algebraic and geometric multiplicity of an eigenvalue?

The algebraic multiplicity can be determined by finding the power of the eigenvalue in the characteristic polynomial. The geometric multiplicity can be determined by finding the nullity of the matrix when the eigenvalue is used as a scalar in the equation (A - λI)x = 0.

Are algebraic and geometric multiplicity always equal?

Not necessarily. The geometric multiplicity can be less than or equal to the algebraic multiplicity, but it can never be greater. If the geometric multiplicity is less than the algebraic multiplicity, the matrix is not diagonalizable.

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