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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. 

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
A + (B+C) = (A+B) + C
A + B = B + A
A + O = A
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.
k(A+B) = kA + kB
(k+h)A = kA + hA
(kh)A = k(hA)
1(A) = A
The conventional examples of vector spaces are ordered ntuples of elements from F, but there are other examples such a sets of functions and sets infinite sequences.
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 S_{V} of V includes V itself and the set consisting of just the zero vector, {0}. If U and W belong to S_{V} then P∩Q belongs to S_{V} and it is the greatest lower bound of members of S_{V} which contain both U and W.
There is an operation that represents roughly the addition of two vector subspaces. It is
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 S_{V} there is a least upper bound and there is a greatest lower bound. The name given to such a mathematical structure is lattice. Thus S_{V} is a lattice.
Let M be an n×n matrix of complex elements. The eigenvalues of M are found as the roots of the polynomial equation]
The number of roots counting their multiplicities is n; i.e., if m_{j} is the multiplicity of λ_{j} then Σm_{j}=n.
The null space of a matrix Q is the set of all vectors that map into the zero vector
Now consider the matrices M−λI where λ is an eigenvalue of M. Let N_{λ} be the null space of (M−λI). The dimension of N_{λ} 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 ntuples 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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