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The Euclidean logarithmic norm of a complex matrix
Definition
Let and . In the standard complex inner product from The standard formulas on and on are inner products, the adjoint matrix is the conjugate transpose by In orthonormal bases, the matrix of the adjoint is the conjugate transpose of the matrix. The Hermitian part is . It is self-adjoint in the sense of Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space.
Define the Euclidean logarithmic norm by , where denotes the largest eigenvalue of the Hermitian matrix . The item The Euclidean logarithmic norm is a real Rayleigh-quotient maximum ↗ establishes that this eigenvalue exists, is real, and equals . This is a real growth bound, not a matrix norm.
Depends on
- The adjoint $T^*:W\to V$ is characterised by $\langle Tv,w\rangle_W=\langle v,T^*w\rangle_V$
- Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes
- The vector space $M_{m \times n}(F) := F^{\,m \times n}$ of $m$ by $n$ matrices over a field, with entrywise operations
- The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||
- Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space
- The standard formulas $\langle x,y\rangle=\sum_{k<n}x_k y_k$ on $\mathbb R^n$ and $\sum_{k<n}x_k\overline{y_k}$ on $\mathbb C^n$ are inner products
- In orthonormal bases, the matrix of the adjoint is the conjugate transpose of the matrix
Used by
- Certified-action implementation of exponential Euler and ETD2 Algorithm
- A nonnormal action grows transiently while its Arnoldi residual certificate remains valid Example
- A one-dimensional Arnoldi projection misses a slow mode of a stiff diagonal matrix Example
- Certified action bounds control one exponential-integrator step Lemma
- The Euclidean logarithmic norm is a real Rayleigh-quotient maximum Lemma
- The logarithmic norm bounds forward and reverse matrix exponentials Lemma
- Arnoldi action error is the semigroup-weighted residual integral Theorem
Dependency tree · two levels
25 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Nicholas J. Higham, What Is the Logarithmic Norm? (standard reference, not scraped)