Do i just reduce the matrix with the eigenvalues put in the matrix?

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- Jul 28th 2010, 10:59 AMadam_leedsDimesnsion of an eigenspace
Do i just reduce the matrix with the eigenvalues put in the matrix?

- Jul 28th 2010, 11:07 AMAckbeet
The eigenspace is defined as the space of vectors spanned by the eigenvectors. Its dimension is equal to the sum of the geometric multiplicities of all the eigenvectors. So, I would find the eigenvectors! After you've done that, you should be able to tell, by inspection, what the dimension of the eigenspace is.

- Jul 29th 2010, 03:17 AMHallsofIvy
Note that a number may be a multiple root of the eigenvalue equation (for example, $\displaystyle \lambda^2- 2\lambda+ 1= (\lambda- 1)^2= 0$ has $\displaystyle \lambda= 1$ as a double root). That multiplicity is the "algebraic multiplicity". The algebraic multiplicity is always greater than or equal to the "geometric multiplicity", the number of independent eigenvectors corresponding to the eigenvalue, that Ackbeet refers to.