**Complex Matrices**

In general, a square complex matrix *A* is similar to a block diagonal matrix

where each block *J*_{i} is a square matrix of the form

So there exists an invertible matrix *P* such that *P-1AP* = *J* is such that the only non-zero entries of *J* are on the diagonal and the superdiagonal. *J* is called the **Jordan normal form** of *A*. Each *J*_{i} is called a Jordan block of *A*. In a given Jordan block, every entry on the super-diagonal is 1.

Assuming this result, we can deduce the following properties:

- Counting multiplicity, the eigenvalues of
*J*, therefore*A*, are the diagonal entries. - Given an eigenvalue λ
_{i}, its**geometric multiplicity**is the dimension of Ker(*A*− λ_{i}**I**), and it is the number of Jordan blocks corresponding to λ_{i}. - The sum of the sizes of all Jordan blocks corresponding to an eigenvalue λ
_{i}is its**algebraic multiplicity**. *A*is diagonalizable if and only if, for every eigenvalue λ of*A*, its geometric and algebraic multiplicities coincide.- The Jordan block corresponding to λ is of the form λ
**I**+*N*, where*N*is a nilpotent matrix defined as*N*_{ij}= δ_{i,j−1}(where δ is the Kronecker delta). The nilpotency of*N*can be exploited when calculating*f*(*A*) where*f*is a complex analytic function. For example, in principle the Jordan form could give a closed-form expression for the exponential exp(*A*).

Read more about this topic: Jordan Normal Form

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### Famous quotes containing the word complex:

“Power is not an institution, and not a structure; neither is it a certain strength we are endowed with; it is the name that one attributes to a *complex* strategical situation in a particular society.”

—Michel Foucault (1926–1984)