CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passaudited 2026-08-29

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Condition number depends on the chosen norm and on diagonal rescaling

Statement refuted

The condition number of a linear system is a property of the bare linear relation alone: it does not change with the chosen norm, and diagonally rescaling the equations leaves it unchanged.

Facts & Assumptions

Given: The matrix $A = (\begin{array}{ccc} 1 & 2 & 0 \\ 0 & 3 & 1 \\ 0 & 0 & 4 \end{array})$ , the diagonal matrix $D = diag (10, 1, 1)$ , and the equivalent systems $A x = b$ and $(D A) x = D b$ .

[L1]

$κ_{p} (A) = ∥ A ∥_{p} ∥ A^{- 1} ∥_{p}$ (The condition number kappa_p(A) = ||A||_p ||A^{-1}||_p of a nonsingular linear system).

[L2]

The induced $1$ -norm is the maximum column sum and the induced $\infty$ -norm the maximum row sum (The induced 1-norm is the maximum column sum and the induced infinity-norm is the maximum row sum).

Counterexample

technique · direct

1.1

The inverse of the upper triangular matrix $A$ is $A^{- 1} = (\begin{array}{ccc} 1 & - 2 / 3 & 1 / 6 \\ 0 & 1 / 3 & - 1 / 12 \\ 0 & 0 & 1 / 4 \end{array}),$ obtained by back substitution from $A X = I_{3}$ .

algebra

1.2

By [L2], the column sums of $A$ are $1, 5, 5$ so $∥ A ∥_{1} = 5$ , and the column sums of $A^{- 1}$ are $1, 1, 1 / 2$ so $∥ A^{- 1} ∥_{1} = 1$ ; [L1] gives $κ_{1} (A) = 5 \cdot 1 = 5$ .

L1L2algebra

1.3

By [L2], the row sums of $A$ are $3, 4, 4$ so $∥ A ∥_{\infty} = 4$ , and the row sums of $A^{- 1}$ are $11 / 6, 5 / 12, 1 / 4$ so $∥ A^{- 1} ∥_{\infty} = 11 / 6$ ; [L1] gives $κ_{\infty} (A) = 4 \cdot 11 / 6 = 22 / 3$ .

L1L2algebra

1.4

The rescaled matrix is $D A = (\begin{array}{ccc} 10 & 20 & 0 \\ 0 & 3 & 1 \\ 0 & 0 & 4 \end{array})$ , with inverse $(D A)^{- 1} = A^{- 1} D^{- 1} = (\begin{array}{ccc} 1 / 10 & - 2 / 3 & 1 / 6 \\ 0 & 1 / 3 & - 1 / 12 \\ 0 & 0 & 1 / 4 \end{array})$ .

algebra

1.5

By [L2], the column sums of $D A$ are $10, 23, 5$ so $∥ D A ∥_{1} = 23$ , and the column sums of $(D A)^{- 1}$ are $1 / 10, 1, 1 / 2$ so $∥ (D A)^{- 1} ∥_{1} = 1$ ; [L1] gives $κ_{1} (D A) = 23 \cdot 1 = 23$ .

L1L2algebra

2.1

Steps 1.2 and 1.3 give $κ_{1} (A) = 5 \neq 22 / 3 = κ_{\infty} (A)$ : the same matrix has different condition numbers in different induced norms.

step 1.2step 1.3

2.2

Since $D$ is invertible, the systems $A x = b$ and $(D A) x = D b$ have exactly the same solution set, yet step 1.5 shows $κ_{1} (D A) = 23 \neq 5 = κ_{1} (A)$ : diagonal rescaling changes the condition number even though it changes nothing about the set of solutions.

step 1.5step 1.2algebra

3.1

Steps 2.1 and 2.2 refute both halves of the false statement: condition number depends on the declared norm and on the scaling of the equations.

step 2.1step 2.2∎

Depends on

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Sources

L. N. Trefethen and D. Bau III, Numerical Linear Algebra, Lecture 12 (standard reference, not scraped)