Diagonalisation of a Linear Transformation

Topics: Algebra - Linear Transformation

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(definition)

We say that $T:V\to V$, a linear transformation, with $V$ a vector space on $\mathbb{K}$, is diagonalisable if there exists a basis $\beta$ for $V$ such that $[T{]}_{\beta }$ is a diagonal matrix.

A matrix $A\in M_{n\times n}(\mathbb{K})$ is diagonalisable if $A$ is similar to a diagonal matrix. That is, if there exists a $Q$ such that:

$$A=Q^{-1}BQ$$

…where $B$ is a diagonal matrix. $Q$ can be found by means of eigenvalues and eigenvectors.

Equivalences §

(theorem, important)

Let $T:V\to V$ with $V$ a vector space on $\mathbb{K}$ with finite dimension.

Then, the following statements are equivalent:

1. $T$ is diagonalisable
2. There exists a basis $\beta$ for $V$ such that $[T{]}_{\beta }$ is diagonalisable
3. The matrix $[T{]}_{\gamma }$ is diagonalisable for any basis $\gamma$ of $V$

(corollary)

A matrix $A$ is diagonalisable if and only if $L_A$ is diagonalisable.