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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

1.1algebra

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 $A X = I_{3}$ .

1.2L1L2algebra

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 $κ_{1} (A) = 5 \cdot 1 = 5$ .

1.3L1L2algebra

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 $κ_{\infty} (A) = 4 \cdot 11 / 6 = 22 / 3$ .

1.4algebra

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

1.5L1L2algebra

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 $κ_{1} (D A) = 23 \cdot 1 = 23$ .

2.1step 1.2step 1.3

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.

2.2step 1.5step 1.2algebra

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.

3.1step 2.1step 2.2∎

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.

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Sources

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