What is the geometric interpretation of the formula $A^k = PD^kP^{-1}$?

A matrix is diagonalizable if and only if it possesses a full set of linearly independent eigenvectors (an eigenbasis). The primary benefit is computational efficiency: calculating high powers of the matrix (e.g., $A^{100}$) becomes trivial because you simply raise the diagonal entries (eigenvalues) to that power, rather than performing repeated matrix multiplication.

Conditions: Matrix has $n$ linearly independent eigenvectors in $n$-dimensional space; Change of basis matrix $P$ formed by eigenvectors is invertible

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.

To compute the $k$-th power of a diagonal matrix, you raise each diagonal entry to the power of $k$. This avoids the complexity of repeated general matrix multiplication.

Conditions: The matrix is diagonal.; The exponent $k$ is a positive integer.