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The formula $A^k = PD^kP^{-1}$ represents a sequence of three geometric operations: first, a change of basis using $P^{-1}$ to move to the eigenbasis; second, a diagonal scaling using $D^k$ which stretches or shrinks along the eigenvector directions; and third, a reverse change of basis using $P$ to return to the original coordinate system.

Conditions: $A = PDP^{-1}$ is a valid diagonalization.; $k$ is a positive integer.

The formula $A^k = PD^kP^{-1}$ represents a sequence of three geometric operations: first, a change of basis using $P^{-1}$ to move to the eigenbasis; second, a diagonal scaling using $D^k$ which stretches or shrinks along the eigenvector directions; and third, a reverse change of basis using $P$ to return to the original coordinate system.

Conditions: $A = PDP^{-1}$ is a valid diagonalization.; $k$ is a positive integer.