# Engineering Analysis/Diagonalization

## Similarity[edit]

Matrices *A* and *B* are said to be similar to one another if there exists an invertable matrix *T* such that:

If there exists such a matrix *T*, the matrices are similar. Similar matrices have the same eigenvalues. If *A* has eigenvectors *v _{1}*,

*v*..., then

_{2}*B*has eigenvectors

*u*given by:

## Matrix Diagonalization[edit]

Some matricies are similar to diagonal matrices using a **transition matrix**, *T*. We will say that matrix *A* is diagonalizable if the following equation can be satisfied:

Where *D* is a diagonal matrix. An *n × n* square matrix is diagonalizable if and only if it has *n* linearly independent eigenvectors.

## Transition Matrix[edit]

If an *n × n* square matrix has *n* distinct eigenvalues λ, and therefore *n* distinct eigenvectors *v*, we can create a transition matrix *T* as:

And transforming matrix X gives us:

Therefore, if the matrix has *n* distinct eigenvalues, the matrix is diagonalizable, and the diagonal entries of the diagonal matrix are the corresponding eigenvalues of the matrix.

## Complex Eigenvalues[edit]

Consider the situation where a matrix *A* has 1 or more complex conjugate eigenvalue pairs. The eigenvectors of *A* will also be complex. The resulting diagonal matrix *D* will have the complex eigenvalues as the diagonal entries. In engineering situations, it is often not a good idea to deal with complex matrices, so other matrix transformations can be used to create matrices that are "nearly diagonal".

## Generalized Eigenvectors[edit]

If the matrix *A* does not have a complete set of eigenvectors, that is, that they have *d* eigenvectors and *n - d* generalized eigenvectors, then the matrix *A* is not diagonalizable. However, the next best thing is acheived, and matrix *A* can be transformed into a Jordan Cannonical Matrix. Each set of generalized eigenvectors that are formed from a single eigenvector basis will create a jordan block. All the distinct eigenvectors that do not spawn any generalized eigenvectors will form a diagonal block in the Jordan matrix.