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 ...

, a reducing subspace

W

of a

linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...

T:V\to V

from a Hilbert space

V

to itself is an

invariant subspace In mathematics, an invariant subspace of a linear mapping ''T'' : ''V'' → ''V '' i.e. from some vector space ''V'' to itself, is a subspace ''W'' of ''V'' that is preserved by ''T''; that is, ''T''(''W'') ⊆ ''W''. General descri ...

T

whose

orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...

W^\perp

is also an invariant subspace of

T.

That is,

T(W) \subseteq W

and

T(W^\perp) \subseteq W^\perp.

One says that the subspace

W

reduces the map

T.

One says that a linear map is reducible if it has a

nontrivial In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or an object which possesses a simple structure (e.g., groups, topological spaces). The noun triviality usually refers to a ...

reducing subspace. Otherwise one says it is irreducible. If

V

is of finite

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...

r

and

W

is a reducing subspace of the map

T:V\to V

represented under

basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...

B

by matrix

M \in\R^

then

M

can be expressed as the sum

M = P_W M P_W + P_ M P_

where

P_W \in\R^

is the matrix of the

orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...

from

V

W

and

P_ = I - P_

is the matrix of the projection onto

W^\perp.

(Here

I \in \R^

is the identity matrix.) Furthermore,

V

has an

orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...

B'

with a subset that is an orthonormal basis of

W

. If

Q \in \R^

is the transition matrix from

B

B'

then with respect to

B'

the matrix

Q^MQ

representing

T

is a

block-diagonal matrix In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original mat ...

Q^MQ = \left \begin A & 0 \\ 0 & B \end \right

with

A\in\R^,

where

d= \dim W

, and

B\in\R^.

References

{{linear-algebra-stub Linear algebra Matrices