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The inverse infinity row-sum bound is uniform and sharp

Example

For a real matrix $B = (b_{i j})$ , define its row-sum matrix norm by $∥ B ∥_{r o w, \infty} : = \max_{i} \sum_{j} ∣ b_{i j} ∣$ . For the matrices $A_{h}$ from Centered finite differences for the zero-Dirichlet boundary-value problem, this norm of $A_{h}^{- 1}$ is at most $1 / 8$ for every $N \geq 1$ ; at $N = 3$ it is exactly $1 / 8$ .

Facts & Assumptions

Given: The row-sum norm just defined, the Dirichlet matrices $A_{h}$ for $N \geq 1$ , and the inverse bound of Discrete maximum principle and uniform inverse bound for the Dirichlet stencil.

[F1]

Centered finite differences for the zero-Dirichlet boundary-value problem defines $A_{h} = h^{- 2} tridiag (- 1, 2, - 1)$ with $h = 1 / (N + 1)$ .

[F2]

Discrete maximum principle and uniform inverse bound for the Dirichlet stencil proves $∥ A_{h}^{- 1} r ∥_{\infty} \leq ∥ r ∥_{\infty} / 8$ for every $r \in R^{N}$ .

[F3]

The triangle inequality gives $∣ x + y ∣ \leq ∣ x ∣ + ∣ y ∣$ for real $x, y$ .

Verification

1.1givenF3algebra

The row-sum expression is nonnegative and equals zero only when every matrix entry is zero; $∥ λ B ∥_{r o w, \infty} = ∣ λ ∣ ∥ B ∥_{r o w, \infty}$ ; and [F3] applied entrywise gives $∥ B + C ∥_{r o w, \infty} \leq ∥ B ∥_{r o w, \infty} + ∥ C ∥_{r o w, \infty}$ . Hence it is a matrix norm.

1.2F1algebra

At $N = 3$ , $h = 1 / 4$ and [F1] gives $A_{h} = 16 T$ with $T = (\begin{array}{ccc} 2 & - 1 & 0 \\ - 1 & 2 & - 1 \\ 0 & - 1 & 2 \end{array})$ . Direct multiplication gives $A_{h}^{- 1} = \frac{1}{64} (\begin{array}{ccc} 3 & 2 & 1 \\ 2 & 4 & 2 \\ 1 & 2 & 3 \end{array})$ , since the displayed integer matrix multiplied by $T$ is $4 I_{3}$ .

2.1givenstep 1.1choosealgebra

Its value equals the induced infinity operator norm $\sup_{∥ z ∥_{\infty} \leq 1} ∥ B z ∥_{\infty}$ : for every such $z$ , each row obeys $∣ (B z)_{i} ∣ \leq \sum_{j} ∣ b_{i j} ∣$ , while a row $i_{0}$ attaining the finite maximum is attained by the vector $z_{j} = sgn (b_{i_{0} j})$ , with $sgn (0) = 0$ .

3.1F2step 2.1

By [F2], $∥ A_{h}^{- 1} z ∥_{\infty} \leq 1 / 8$ whenever $∥ z ∥_{\infty} \leq 1$ . Taking the supremum and using step 2.1 shows $∥ A_{h}^{- 1} ∥_{r o w, \infty} \leq 1 / 8$ for every $N$ .

4.1step 3.1step 1.2algebra∎

The three absolute row sums of this inverse are $6 / 64$ , $8 / 64$ , and $6 / 64$ , so its row-sum norm is $8 / 64 = 1 / 8$ . Along with the uniform bound in step 3.1, this proves the stated example and shows its constant is attained.

Depends on

Used by

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Sources

Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, author-source revision 2025.01.27, Chapter 10, linear.md (standard reference, not scraped)
Benjamin Seibold, MIT OpenCourseWare 18.336, Lecture 5: Finite Difference Approximation, Spring 2009 (standard reference, not scraped)
Benjamin Seibold, MIT OpenCourseWare 18.336, Lecture 6: General Linear Second Order Equation, Spring 2009 (standard reference, not scraped)