Definition of $\exp(A)$ in terms of spectral decomposition.

Question

I am read this question Plugging a matrix multiplied by an imaginary number in the exponential function. Here the question'author defined $$\exp(A) := \sum_{1\le k\le n}\exp(\lambda_k) P_k$$

What reference contain this definition "exp is defined in terms of the spectral decomposition"?

Answer 1

Why $\exp A=P^{−1}(\exp D)P$?

Because plugging $A=P^{-1}DP$ into $$\exp A=\sum_{n=0}^\infty \frac{1}{n!} A^n \tag1$$ and simplifying $(P^{-1}DP)^n = P^{-1}D^n P$, one obtains $$\exp A=P^{-1}\left(\sum_{n=0}^\infty \frac{1}{n!} D^n \right) P = P^{−1}(\exp D)P \tag2$$