An Eigenvector of Matrix A is a non-zero column Vector that satisfies where is a scalar. The value of is the eigenvalue corresponding to the eigen vector.
This corresponds to a vector that is an invariant vector under the Linear Transform transform . The corresponding eigenvalue can be thought of as the magnitude scale factor for the eigenvector.
However the definition can be applied to any linear operator on any vector space as they can be represented as a matrix. E.g. for a linear operator there is an eigen vector that satisfies .
Characteristic Equation
The solutions of are the eigenvalues of . To find the eigen-vectors of we simply solve .
Eigenspace, Multiplicity
Given an eigen value of . The algebraic multiplicity of denoted the number of times the root appears in the characteristic equation of e.g. the eigen vector has
Cayley-Hamilton Theorem
All matrices satisfy their own characteristic equation
This theorem can be used to find various variations of M such as and by multiplying or dividing through by and using the fact that
Diagonalisation and Eigen Basis
The motivation is to compute higher powers of an arbitrary matrix but this is very compute heavy.
However for certain matrices which are diagonal e.g. a square matrix such that every element except those which lie on the leading diagonal are , we can compute high powers very easily - simply just raise each element to the power we are trying to compute and you are done.
Using this fact, for matrices with real valued eigen vectors we can perform a Change of Basis so that the eigen vectors are the basis vectors. So given the change of basis matrix , which is simply the eigen vectors as column vectors we compute in terms of the eigen-basis as .
The special reason why eigen vectors as basis is that under transform they simply scale. This gives the property that for basis eigen-vectors , under the transform so writing this as a matrix in terms of we a guaranteed to get a diagonal matrix
So to find a high power of a matrix we perform the change of basis to an eigne basis yielding a diagonal matrix. Then raise it easily to a high power then undo the change of basis.