**Proof plan.** We proceed by induction on $n$. The key ingredients are: (i) the existence of at least one eigenvalue (guaranteed by the [Fundamental Theorem of Algebra](/theorems/347) over $\mathbb{C}$), (ii) reality of eigenvalues and orthogonality of eigenvectors from the [Spectral Properties of Hermitian Matrices](/theorems/924), and (iii) the observation that the orthogonal complement of an eigenspace is invariant under $A$. **Step 1: Base case.** For $n = 1$, any $1 \times 1$ Hermitian matrix $(a)$ has $a = \overline{a}$, so $a \in \mathbb{R}$. It is already diagonal, and $P = (1)$ is unitary. $\checkmark$ **Step 2: Inductive step.** Let $n \geq 2$ and assume the result holds for all Hermitian matrices of size less than $n$. Let $A$ be an $n \times n$ Hermitian matrix. Since the characteristic polynomial $\chi_A(t)$ has degree $n$, it has at least one root $\lambda_1 \in \mathbb{C}$ by the Fundamental Theorem of Algebra. By the [Spectral Properties of Hermitian Matrices](/theorems/924), $\lambda_1 \in \mathbb{R}$. Let $u_1$ be a unit eigenvector: $Au_1 = \lambda_1u_1$ with $\|u_1\| = 1$. **Step 3: Invariance of the orthogonal complement.** Define $W = u_1^\perp = \{w \in \mathbb{C}^n : u_1^\dagger w = 0\}$. This is an $(n-1)$-dimensional subspace. We claim $A(W) \subseteq W$: for any $w \in W$, \begin{align*} u_1^\dagger(Aw) = (A^\daggeru_1)^\daggerw = (Au_1)^\daggerw = (\lambda_1u_1)^\daggerw = \overline{\lambda_1}\,u_1^\daggerw = \lambda_1 \cdot 0 = 0, \end{align*} using $A^\dagger = A$ and $\overline{\lambda_1} = \lambda_1$. So $Aw \in W$. **Step 4: Apply the inductive hypothesis.** The restriction $A|_W: W \to W$ is represented by an $(n-1) \times (n-1)$ Hermitian matrix (Hermiticity is inherited: if $v, w \in W$, then $v^\dagger(Aw) = (Av)^\daggerw$). By the inductive hypothesis, there exists an orthonormal basis $\{u_2, \ldots, u_n\}$ of $W$ consisting of eigenvectors of $A|_W$, with real eigenvalues $\lambda_2, \ldots, \lambda_n$. **Step 5: Assemble the unitary diagonalisation.** The [set](/page/Set) $\{u_1, u_2, \ldots, u_n\}$ is an orthonormal basis for $\mathbb{C}^n$: the vectors $u_2, \ldots, u_n$ are orthonormal within $W$ by the inductive hypothesis, and $u_1 \perp W$ by construction. Each $u_i$ is an eigenvector of $A$ with real eigenvalue $\lambda_i$. Setting $P = (u_1 \mid u_2 \mid \cdots \mid u_n)$, orthonormality gives $P^\dagger P = I$, so $P$ is unitary, and $P^\dagger AP = \operatorname{diag}(\lambda_1, \ldots, \lambda_n)$. **Step 6: Real symmetric case.** If $A$ is real symmetric, the eigenvalues are real (by Part 1 of the [Spectral Properties of Hermitian Matrices](/theorems/924)). Since $A$ has real entries and $\lambda_i \in \mathbb{R}$, the system $(A - \lambda_i I)v = \mathbf{0}$ has real coefficients, so the eigenspace has a basis of real vectors. Applying Gram-Schmidt within each real eigenspace produces real orthonormal eigenvectors, so $P$ can be chosen real orthogonal ($P^T P = I$).