Exercise 509Linear algebra › Matrices and linear systemsL1Added 3 Nov 2026

Powers of a diagonal matrix

#diagonal matrix#powers#induction

Statement

Let $D = \operatorname{diag}(\lambda_1,\ldots,\lambda_n)$ . Show that $D^k = \operatorname{diag}(\lambda_1^k,\ldots,\lambda_n^k)$ for all $k \ge 1$ .

Hint

Prove by induction on $k$ , using the fact that the product of two diagonal matrices is diagonal.

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Solution

Base case. For $k=1$ , $D^1 = D = \operatorname{diag}(\lambda_1,\ldots,\lambda_n)$ .

Inductive step. Assume $D^k = \operatorname{diag}(\lambda_1^k,\ldots,\lambda_n^k)$ . Then
$D^{k+1} = D^k D = \operatorname{diag}(\lambda_1^k,\ldots,\lambda_n^k)\operatorname{diag}(\lambda_1,\ldots,\lambda_n) = \operatorname{diag}(\lambda_1^{k+1},\ldots,\lambda_n^{k+1}).$

Conclusion. The formula holds for all $k \ge 1$ .

Powers of a diagonal matrix

Statement

Solution

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