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

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

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.

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