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The complex potential of a two-dimensional potential flow combines the velocity potential $u$ with the stream function $v$. For its holomorphic representation, the Cauchy-Riemann equations are $u_x=v_y$ and $u_y=-v_x$.

Use the inner product $⟨u,v⟩=u^{†}v$ on ${\mathbb{C}}^n$. The distinct real eigenvalues permit a full orthonormal eigenvector expansion; the two child solutions establish that expansion and apply it to the proposed quadratic form.

The decomposition is $A=H+iS$. Both matrices are Hermitian: $H^{†}=H$ and $S^{†}=S$, directly from their definitions and complex conjugation of $1/(2i)$. The matrix $H$ is the Hermitian part of a matrix, whereas $S$ is the Hermitian matrix obtained by dividing the skew-Hermitian part by $i$.

The spectral theorem for a Hermitian matrix supplies a unitary matrix $U$ and real diagonal matrix $D=\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_n)$ with $A=UD{U}^{†}$. This single representation gives all three conclusions below.

Let $v_1,\dots ,v_n$ be the linearly independent eigenvectors, and form $V=(v_1\cdots v_n)$. This matrix is invertible and $AV=VD$, where $D=\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_n)$. Hencewhich proves diagonalizability.

If $B=P^{-1}AP$, then $P^{-1}{v}_j$ are linearly independent eigenvectors of $B$ with the same eigenvalues. Explicitly, with $W=P^{-1}V$,Thus matrix similarity preserves diagonalizability.

Here $p=w=1$ and $q=x^2$ on the whole real line. The quantum harmonic oscillator energy form is $Q[\psi ]={\int }_{\mathbb{R}}(|{\psi }^{\prime }{|}^2+x^2|\psi {|}^2)dx$. Admissible trials have a square-integrable weak derivative and finite potential energy; they need not have a classical second derivative everywhere.

Write the energy form as $Q[\psi ]={\int }_{\alpha }^{\beta }(p|{\psi }^{\prime }{|}^2+q|\psi {|}^2)dx$ and the weighted squared norm as $N[\psi ]={\int }_{\alpha }^{\beta }w|\psi {|}^{2}dx$. For real coefficients and appropriate homogeneous boundary conditions, the Sturm-Liouville problem is the stationary-point equation for their Rayleigh quotient $Q/N$.