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Two-column dominant-subspace condition

Let a real symmetric matrix have an orthonormal eigenbasis

w_{1}, \dots, w_{n}

, with

∣ λ_{1} ∣ \leq \dots \leq ∣ λ_{n - 2} ∣ < ∣ λ_{n - 1} ∣ = ∣ λ_{n} ∣.

Write two starting vectors as

u = \sum_{i} b_{i} w_{i}, v = \sum_{i} c_{i} w_{i} .

For two-column subspace iteration to recover

span (w_{n - 1}, w_{n})

, the two projections onto that dominant subspace must be independent. The exact condition is

det (b_{n - 1} b_{n} c_{n - 1} c_{n}) = b_{n - 1} c_{n} - b_{n} c_{n - 1} \neq = 0.

Requiring each of the four coefficients to be nonzero does not imply this determinant condition.

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