Generalized eigenspace invariant?

Thread starter mind0nmath
Start date Jun 7, 2008
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generalized Invariant

In summary, a generalized eigenspace invariant is a set of vectors associated with a specific eigenvalue of a matrix that forms a subspace. It differs from a regular eigenspace by including generalized eigenvectors, and it is important in various scientific fields for efficient computation and understanding of complex systems. These invariants can be calculated using the Jordan canonical form of a matrix and a matrix can have multiple generalized eigenspace invariants associated with different eigenvalues.

Jun 7, 2008

mind0nmath

Hey,
Is the generalized eigenspace invariant under the operator T? Let T be finite dimensional Linear operator on C(complex numbers).
My understanding of the Generalized Eigenspace for the eigenvalue y is:
"All v in V such that there exists a j>=1, (T-yIdenitity)^j (v) = 0." plus 0.
thanks

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Jun 7, 2008

matt grime

Science Advisor

Homework Helper

What does invariant mean? Try to prove it.

Jun 8, 2008

mind0nmath

got it. thanks.

Related to Generalized eigenspace invariant?

1. What is a generalized eigenspace invariant?

A generalized eigenspace invariant is a mathematical concept used in linear algebra to describe the set of all vectors that are associated with a specific eigenvalue of a matrix. These vectors form a subspace, known as the generalized eigenspace, and are invariant under certain transformations of the matrix.

2. How is a generalized eigenspace invariant different from a regular eigenspace?

A regular eigenspace only contains eigenvectors, while a generalized eigenspace also includes generalized eigenvectors. Generalized eigenvectors are necessary for matrices that do not have a complete set of eigenvectors, as they can still be used to diagonalize the matrix.

3. What is the importance of generalized eigenspace invariants in scientific research?

Generalized eigenspace invariants are important in various scientific fields, particularly in quantum mechanics and signal processing. They allow for the efficient computation of eigenvalues and eigenvectors, which are crucial in understanding complex systems and data.

4. How are generalized eigenspace invariants calculated?

To calculate the generalized eigenspace invariants, one can use the Jordan canonical form of a matrix. This form breaks down a matrix into blocks, with each block corresponding to a different generalized eigenspace. The dimensions of these blocks determine the size of the generalized eigenspace invariant.

5. Can a matrix have multiple generalized eigenspace invariants?

Yes, a matrix can have multiple generalized eigenspace invariants, each associated with a different eigenvalue. This is because a matrix can have multiple eigenvectors for a single eigenvalue, and each of these eigenvectors can be extended to form a generalized eigenvector.