Eigenbasis

An eigenbasis of a square matrix $A$ is a basis of the vector space consisting entirely of eigenvectors of $A$.

Key Idea

A matrix has an eigenbasis iff it is diagonalizable.

If $\{\overrightarrow{v_1},\dots ,\overrightarrow{v_n}\}$ is an eigenbasis, then:

• Each $\overrightarrow{v_i}$ is an eigenvector of $A$
• The vectors are linearly independent
• They span the space

Connection to Diagonalization

Choosing an eigenbasis gives:

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

Where:

• $P$ is the matrix whose columns are the eigenbasis vectors
• $D$ is a diagonal matrix with the corresponding eigenvalues on the diagonal