Change of Coordinate Matrix

$Q=[I_v{]}_{{\beta }^{\prime }}^{\beta }$ is the change of coordinate matrix

Properties

• For any $v\in V$, $[v{]}_{\beta }=Q[v{]}_{{\beta }^{\prime }}$
  • $Q$ changes ${\beta }^{\prime }$ coordinates into $\beta$ coordinates
  • $Q^{-1}$ changes $\beta$ coordinates into ${\beta }^{\prime }$ coordinates
• $Q$ is invertible

Application

• $T:V\to V$ linear operator. $[T{]}_{{\beta }^{\prime }}=Q^{-1}[T{]}_{\beta }Q$

$A,B\in M_{n\times n}(F)$. $B$ is similar to $A$ if there exists invertible $Q$ such that $B=Q^{-1}AQ$