**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

### Other articles related to "complex matrices, complex, matrices":

**Complex Matrices**

... If instead M is a 2n×2n matrix with

**complex**entries, the definition is not standard throughout the literature ... Other authors retain the definition (1) for

**complex matrices**and call

**matrices**satisfying (2) conjugate symplectic ...

### Famous quotes containing the word complex:

“The money *complex* is the demonic, and the demonic is God’s ape; the money *complex* is therefore the heir to and substitute for the religious *complex*, an attempt to find God in things.”

—Norman O. Brown (b. 1913)