# Show that applying a function $f(x)$ to the matrix $A$ gives $f(A) = V f(D)V^{−1}$ where $A = V DV^{−1}$. [closed]

Recall that a matrix $$A$$ has an eigen-decomposition $$A = V DV^{−1}$$, where $$D$$ is diagonal and $$V$$ is the matrix of eigenvectors.

(a) Applying a function $$f(x)$$ to the matrix $$A$$ gives

$$f(A) = V f(D)V^{−1}$$

where $$f(D) = \operatorname{diag}[f(\lambda_1), f(\lambda_2), \ldots]$$ for eigenvalues $$\lambda_1, \lambda_2, \ldots$$

(i) Show that equation (1) holds for a quadratic function $$f(x) = a_0 + a_1x + a_2x^2$$ .

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• Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Please take moment to give this posting a read to learn how to ask a good question. May 14 at 3:50
• The very first sentence of the question is false, as not every matrix has such a decomposition. May 14 at 3:56