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Proof of Affine Invariance for Newton's Method

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The Formal Theorem

Let f:Rn→Rn f: \mathbb{R}^n \to \mathbb{R}^n be a differentiable function and let A∈Rn×n A \in \mathbb{R}^{n \times n} be an invertible matrix. Consider the transformed function g(x)=Af(x) g(x) = Af(x) . Newton's method iteration for f(x)=0 f(x) = 0 is defined as xk+1=xk−[Df(xk)]−1f(xk) x_{k+1} = x_k - [Df(x_k)]^{-1}f(x_k) . The method is affine invariant if the sequence {xk} \{x_k\} generated by applying Newton's method to g(x) g(x) is identical to that generated for f(x) f(x) . This holds because:
xk−[Dg(xk)]−1g(xk)=xk−[ADf(xk)]−1Af(xk)=xk−[Df(xk)]−1A−1Af(xk)=xk−[Df(xk)]−1f(xk) x_k - [Dg(x_k)]^{-1}g(x_k) = x_k - [A Df(x_k)]^{-1} A f(x_k) = x_k - [Df(x_k)]^{-1} A^{-1} A f(x_k) = x_k - [Df(x_k)]^{-1} f(x_k)

Analytical Intuition.

Imagine you are navigating a landscape defined by f(x)=0 f(x) = 0 . Newton's method is your compass, calculating the local slope to find the root. Now, suppose you view this landscape through a distorted lens, a linear transformation A A . From a standard perspective, this lens stretches, rotates, and scales your world, making the paths look completely different. However, the remarkable beauty of Newton's method lies in its inherent 'geometry-blind' nature. Because the derivative Dg(x) Dg(x) captures the exact same transformation A A that warps the function g(x) g(x) , the matrix A A effectively cancels itself out in the update step. Whether you are walking in the original coordinate system or the warped one, the 'relative' step you take toward the root remains invariant. The method doesn't care about the coordinate system; it only cares about the underlying manifold structure. By staying invariant to affine transformations, Newton's method proves it is a coordinate-independent solver, making it robust against arbitrary scaling of your variables.
CAUTION

Institutional Warning.

Students often struggle to see how the matrix A A cancels, forgetting that for the inverse of a product, (AB)−1=B−1A−1 (AB)^{-1} = B^{-1}A^{-1} . They mistakenly try to invert A A and Df Df individually without considering the order of operations in the matrix product.

Academic Inquiries.

01

Why is affine invariance important in practice?

It ensures that Newton's method performs identically regardless of the units of measurement or the scaling of your input variables, meaning you do not need to precondition your system manually.

02

Does this invariance hold for non-linear transformations?

No. Affine invariance is restricted to linear transformations (Ax+b Ax + b ). If the coordinate transformation is non-linear, the Jacobian structure changes, and the method loses its invariance.

Standardized References.

  • Definitive Institutional SourceNocedal, J. & Wright, S. J., Numerical Optimization.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Proof of Affine Invariance for Newton's Method: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/fundamentals-of-optimization/proof-of-affine-invariance-for-newton-s-method

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