In
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
, the order-''r'' Krylov subspace generated by an ''n''-by-''n''
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
''A'' and a vector ''b'' of dimension ''n'' is the
linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...
spanned by the
images
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of ''b'' under the first ''r'' powers of ''A'' (starting from
), that is,
:
Background
The concept is named after Russian applied mathematician and naval engineer
Alexei Krylov
, birth_date = O.S. (August 15, 1863 N.S.)
, death_date =
, image = Alexey Krylov 1910s.JPG
, image_size = 200px
, caption = Official portrait (1910)
, birth_place = Alatyrsky uezd of Simbirsk Gubernia, Russian ...
, who published a paper about it in 1931.
Properties
*
.
* Vectors
are linearly independent until