If $D$ is a diagonal matrix, then raising it to a power $k$ is equivalent to raising each diagonal entry to the same power:

$$ D^k = \text{diag}(d_1^k, d_2^k, \dots, d_n^k) $$

This works because diagonal matrix multiplication is element-wise on the diagonal.

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🧩 Example

Let:

$$ D = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{bmatrix} \Rightarrow D^2 = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 25 \end{bmatrix} $$

Each diagonal entry is squared independently.

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🔍 Why This Matters

• Diagonal matrices are trivially exponentiated
• Used in diagonalization: $$ A = P D P^{-1} \Rightarrow A^k = P D^k P^{-1} $$
• Powers of matrices become easy once diagonalized
• Matrix exponentials simplify: $$ e^{D t} = \text{diag}(e^{d_1 t}, e^{d_2 t}, \dots) $$

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✅ Validator Summary

Operation	Result
$D^k$	Raise each diagonal entry to power $k$
$A^k$ via diagonalization	$P D^k P^{-1}$
Matrix exponential $e^{D t}$	Apply exponential to each diagonal entry

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🧠 Related Concepts

• Diagonalization
• Eigenvalue powers
• Matrix exponentials
• Spectral decomposition