Monday, August 26, 2019

algebraic multiplicity and geometric multiplicity

https://people.math.carleton.ca/~kcheung/math/notes/MATH1107/wk10/10_algebraic_and_geometric_multiplicities.html

The algebraic multiplicity of λ is the number of times λ is repeated as a root of the characteristic polynomial.

Let A be an n × n matrix with eigenvalue λ. The geometric multiplicity of λ is the dimension of the eigenspace of λ.

In general, the algebraic multiplicity and geometric multiplicity of an eigenvalue can differ. However, the geometric multiplicity can never exceed the algebraic multiplicity.
It is a fact that summing up the algebraic multiplicities of all the eigenvalues of an $n×n$ matrix $A$ gives exactly $n$If for every eigenvalue of $A$, the geometric multiplicity equals the algebraic multiplicity, then $A$ is said to be diagonalizable. As we will see, it is relatively easy to compute powers of a diagonalizable matrix.