Matrix Similarity

Two matrices are similar if there exists an invertible $P$ such that:

$$\begin{array}{r}A=P^{-1}BP\\ \\ \text{Multiplying both sides:}PA{P}^{-1}=B\end{array}$$

Why care?

They are called similar cuz:

• $A$ and $B$ have the same rank.
• $C_A(\lambda )$ and $C_B(\lambda )$ have the same roots, hence both matrices have the same eigenvalues.
• Super useful: computing powers:

$$\begin{array}{r}{A}^k=P^{-1}{B}^{k}P\text{for }k\ge 1\end{array}$$

• This is especially useful when one of the matrices is diagonal, makes powers so much simpler.