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A complex square matrix $A$ is a normal matrix if it commutes with its conjugate transpose:By unitary diagonalization of a normal matrix, there are a unitary matrix $U$ and a diagonal matrix $\Lambda =\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_N)$ such thatThe matrix 2-norm is invariant under multiplication by unitary matrices, soFor a diagonal matrix,with equality on a coordinate vector belonging to an eigenvalue of greatest modulus. Hencethe matrix 2-norm of a normal matrix.

Let $D\in {\mathbb{R}}^{M\times M}$ be the Toeplitz antisymmetric tridiagonal matrix with $D_{m,m+1}=1$, $D_{m,m-1}=-1$, and zero diagonal. Fixed homogeneous endpoint values give the homogeneous updateThe hint gives the eigenvaluesBecause each $d_j$ is purely imaginary, $1-(\mu /4)d_j\ne 0$ for every real $\mu$, so the left matrix is invertible. The two matrices are polynomials in $D$ and therefore have the same orthonormal eigenvectors. The amplification matrixhas eigenvaluesThe numerator and denominator are complex conjugates, so $|q_j|=1$. Moreover, the common orthonormal eigenbasis makes $Q$ a normal matrix. By the matrix 2-norm of a normal matrix,for every $n$. Thus the Crank-Nicolson centered-advection scheme on a finite interval is unconditionally stable: