Krylov subspace

In linear algebra, the order-r Krylov subspace generated by an n-by-n matrix A and a vector b of dimension n is the linear subspace spanned by the images of b under the first r powers of A (starting from ${\displaystyle A^{0}=I}$), that is,

The concept is named after Russian applied mathematician and naval engineer Alexei Krylov, who published a paper about it in 1931. The basis for the Krylov subspace is derived from the Cayley–Hamilton theorem which implies that the inverse of a matrix can be found in terms of a linear combination of its powers.

Modern iterative methods for finding one (or a few) eigenvalues of large sparse matrices or solving large systems of linear equations avoid matrix-matrix operations, but rather multiply vectors by the matrix and work with the resulting vectors. Starting with a vector, b, one computes ${\displaystyle Ab}$, then one multiplies that vector by ${\displaystyle A}$ to find ${\displaystyle A^{2}b}$ and so on. All algorithms that work this way are referred to as Krylov subspace methods; they are among the most successful methods currently available in numerical linear algebra.

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