﻿ The Decomposition of the Vector Space Associated with a Matrix in Terms of the Null Spaces Associated with the Eigenvalues of that Matrix.
San José State University

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The Decomposition of the Vector Space
Associated with a Matrix in Terms of
the Null Spaces Associated with the
Eigenvalues of that Matrix.

## BACKGROUND

A vector space V consists of a mathematical field F, a nonempty set of elements, called vectors, and two operations, addition of vectors and scalar multiplication of vectors by elements from F.

The operations must satisfy the following conditions

• Associativity of addition: For any three vectors A, B and C belonging to V

A + (B+C) = (A+B) + C

• Commutivity of addition: For any two vectors A and B from V

A + B = B + A

• Existence of an additive identity: There is a vector O belonging to V such that for all A in V

A + O = A

• Existence of additive inverses: For all A in V there exists in V a vector B such that

A + B = O

Such a vector is denoted as −A

The above conditions constitute requiring V to be an abelian group under addition. There are additional properties for scalar multiplication.

• Distributivity over vector addition: For any scalar k in F and any two vectors A and B in V

k(A+B) = kA + kB

• Distributivity over scalar addition: For any two scalars, k and h, in F and any vector A in V

(k+h)A = kA + hA

• Associativity over scalar multiplication: For any two scalars, k and h, in F and any vector A in V

(kh)A = k(hA)

• Multiplicative identity of F is scalar multiplicative identity: If 1 is the multiplicative identity of F then for all A in V

1(A) = A

The conventional examples of vector spaces are ordered n-tuples of elements from F, but there are other examples such a sets of functions and sets infinite sequences.

## The Set of Subspaces of a Vector Space

A subset S of V which is a vector space is a vector subspace of V. If S is closed under vector addition and scalar multiplication then it is a vector space because the other conditions such as associativity are satisfied in S as a result of it being a subset of V.

The set of subspaces SV of V includes V itself and the set consisting of just the zero vector, {0}. If U and W belong to SV then P∩Q belongs to SV and it is the greatest lower bound of members of SV which contain both U and W.

There is an operation that represents roughly the addition of two vector subspaces. It is

#### U+W = {A+B: A∈U and B∈W} or, equivalently NλNλNλ U+W = {aA+bB: a∈F, b∈F, A∈U, B∈W}

U+W is also a vector subspace and it is the least upper bound of subspaces containing both U and W. The dimensions of U+W is (dim(U)+dim(W)−dim(U∩W)).

The set of vector subspaces is a partially ordered set. Furthermore for every pair of members of SV there is a least upper bound and there is a greatest lower bound. The name given to such a mathematical structure is lattice. Thus SV is a lattice.

## The Null Space Decomposition of the Vector Space for a Matrix

Let M be an n×n matrix of complex elements. The eigenvalues of M are found as the roots of the polynomial equation]

#### det(M−λI) = 0

The number of roots counting their multiplicities is n; i.e., if mj is the multiplicity of λj then Σmj=n.

The null space of a matrix Q is the set of all vectors that map into the zero vector

#### QX = 0

Now consider the matrices M−λI where λ is an eigenvalue of M. Let Nλ be the null space of (M−λI). The dimension of Nλ is the multiplicity of λ as an eigenvalue. Therefore the sum of the dimensions of the null spaces associated with the eigenvalues of M is equal to n, the dimension of vector spaces of ordered n-tuples of complex numbers. Thus the null spaces of the eigenvalues of M constitute a decomposition of the vector space associated with M.