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Diagonalization simplifies matrix powers because raising a diagonal matrix to a positive integer power only requires raising each diagonal entry to that power, which is computationally simpler than repeated general matrix multiplication. By expressing a matrix $A$ as $PDP^{-1}$, the power $A^k$ becomes $PD^kP^{-1}$, where the adjacent $P^{-1}P$ factors cancel out, reducing the complexity of the calculation.

Conditions: The matrix $A$ is diagonalizable, meaning $A = PDP^{-1}$ with $P$ invertible.; The exponent $k$ is a positive integer.

Diagonalization simplifies matrix powers because raising a diagonal matrix to a positive integer power only requires raising each diagonal entry to that power, which is computationally simpler than repeated general matrix multiplication. By expressing a matrix $A$ as $PDP^{-1}$, the power $A^k$ becomes $PD^kP^{-1}$, where the adjacent $P^{-1}P$ factors cancel out, reducing the complexity of the calculation.

Conditions: The matrix $A$ is diagonalizable, meaning $A = PDP^{-1}$ with $P$ invertible.; The exponent $k$ is a positive integer.