Recipe

Compute a function of a diagonalizable matrix

Use when: you have a diagonalizable matrix $A$ and a function $f$ defined at each of $A$ 's eigenvalues, not necessarily a polynomial

Goal: the matrix $f(A)$ , built by applying $f$ to $A$ 's eigenvalues and conjugating back

Taught in Functions of a diagonalizable matrix

Quick recipe

Diagonalize $A$
Apply $f$ entrywise to $D$ 's diagonal
Conjugate back: $f(A)=Pf(D)P^{-1}$
Check $f(A)$ against a known special case

1
Find $A$ 's eigenvalues and a full set of independent eigenvectors, then assemble $P$ (eigenvectors as columns) and $D$ (matching eigenvalues on the diagonal) so $A=PDP^{-1}$ .

Example

For $A=\begin{pmatrix}4&1\\2&3\end{pmatrix}$ : eigenvalues $5,2$ with eigenvectors $(1,1),(1,-2)$ , so $D=\text{diag}(5,2)$ and $P=\begin{pmatrix}1&1\\1&-2\end{pmatrix}$ .
- $A$ has a full set of $n$ independent eigenvectors — continue; $A$ is genuinely diagonalizable.
- $A$ is missing an independent eigenvector for some eigenvalue — stop; this construction needs section 7's Jordan-block version instead.
Why: this section's whole definition of $f(A)$ is built directly on top of a diagonalization, so nothing else below is possible without one.
2
Compute $f(\lambda_i)$ for each eigenvalue on $D$ 's diagonal, leaving every off-diagonal entry at $0$ .

Example

For $f(x)=\sqrt x$ and eigenvalues $5,2$ : $f(D)=\text{diag}(\sqrt5,\sqrt2)$ .

⚡ Shortcut

If $f$ is a polynomial, this whole procedure agrees exactly with ordinary polynomial evaluation; evaluate $f(A)$ directly by Horner's method instead of building a diagonalization at all.

Why: this is literally the definition of $f(A)$ from this section, applying the ordinary numerical function once per eigenvalue.
3
Compute $P^{-1}$ , then multiply $P$ , $f(D)$ , and $P^{-1}$ together in that order to undo the diagonalization.

Example

$P^{-1}=\begin{pmatrix}2/3&1/3\\1/3&-1/3\end{pmatrix}$ , so $\sqrt A=P\,\text{diag}(\sqrt5,\sqrt2)\,P^{-1}$ , a genuinely non-diagonal matrix.

Why: this conjugation is what moves the answer back out of the eigenvector basis and into $A$ 's own original coordinates.
4
Verify the construction on a case where the answer is independently known, such as squaring $\sqrt A$ back to confirm it reproduces $A$ exactly.

Example

$(\sqrt A)^2=A=\begin{pmatrix}4&1\\2&3\end{pmatrix}$ exactly, confirming the square-root construction above.

Why: this catches an arithmetic slip in the eigenvector, diagonal, or conjugation steps before the answer is trusted.

Common mistakes

Applying $f$ to the wrong diagonal entries by mismatching eigenvalue order between $D$ and the eigenvector columns of $P$ .
Multiplying in the wrong order, such as $P^{-1}f(D)P$ instead of $Pf(D)P^{-1}$ .
Using this construction on a matrix that isn't actually diagonalizable, silently producing a wrong answer.
Forgetting that $f$ must be defined at every eigenvalue, such as $\sqrt{\cdot}$ at a negative real eigenvalue.
Assuming $f(A)$ only makes sense for a polynomial $f$ , missing the whole point of this section's generalization.