Diagonalisable

A linear transformation is said to be diagonalisable if it’s corresponding transformation matrix is diagonal. Any choice of basis can be used to achieve this.

Diagonalising Linear Transformations §

Diagonalisable Linear Transformations ^formula1

Let $T:V\to V$ be a linear transformation that maps the vector space $V$ onto itself. Then, $T$ is diagonalisable if and only if there exists a basis, $\mathcal{B}$ for $V$ such that $[T{]}_{\mathcal{B}}$ is a diagonal matrix

We can use eigenvectors to also check this:

Diagonalisable using Eigenvectors (Linear Transformations) ^formula2

If $T:V\to V$ is a linear transformation mapping $V$ onto itself, it is only diagonalisable if there exists a basis, $\mathcal{B}$ consisting of only eigenvectors (of any eigenvalue) of $T$:

$$\mathcal{B}=\{({\vec{v}}_{\lambda }{)}_1,({\vec{v}}_{\lambda }{)}_2,\dots \}$$

Diagonalisable Matrices §

The concept of diagonalisable linear transformations extends to matrices as well, since every linear transformation has an associated transformation matrix.

Diagonalisable Matrices ^formula2

An $n\times n$ matrix, $\mathbf{A}$ is diagonalisable if and only if it can be represented as:

$$\mathbf{A}=\mathbf{PD}{\mathbf{P}}^{-1}$$

where $\mathbf{P}$ is an $n\times n$ invertible matrix and $\mathbf{D}$ is an $n\times n$ diagonal matrix. Then $\mathbf{P}$ is said to diagonalise $\mathbf{A}$

Matrix Powers §

The operation of matrix powers can be used on diagonalisable matrices to simplify chained linear transformations:

Matrix Power on a Diagonalisable Matrix

Let $[T]$ be the transformation matrix of the linear transformation $T:U\to V$. If $[T]$ is diagonalisable, then:

$$[T]=\mathbf{PD}{\mathbf{P}}^{-1}$$

And it’s matrix power is given by (for $n\ge 1$):

$$[T{]}^n=\mathbf{P}{\mathbf{D}}^{\mathbf{n}}{\mathbf{P}}^{-1}$$

Which is the same as repeatedly transforming.

#todo Application in Markov Chains (Genetics)!