I was looking at different equivalence relations we can define on matrices : first of all, say A ~ B iff there exists an invertible P such that A = P-1 BP . If we're working over an algebraically closed field, we get as a simple representative of each equivalence class a matrix in Jordan Normal Form.
42.
Can we just say that each matrix is of the form J \ oplus M _ 1 \ oplus \ cdots \ oplus M _ k where J is a matrix in Jordan Normal Form and the M i are some sort of fundamental irreducible matrices ( in the same way we can split any polynomial over \ mathbb { R } as a product of linear factors and irreducible polynomials of degree 2 ).
43.
There is another way to define a normal form, that like the Frobenius normal form is always defined over the same field " F " as " A ", but that does reflect a possible factorization of the characteristic polynomial ( or equivalently the minimal polynomial ) into irreducible factors over " F ", and which reduces to the Jordan normal form in case this factorization only contain linear factors ( corresponding to eigenvalues ).
44.
If " x " is in the Jordan normal form, then " x " ss is the endomorphism whose matrix on the same basis contains just the diagonal terms of " x ", and " x " n is the endomorphism whose matrix on that basis contains just the off-diagonal terms; " x " u is the endomorphism whose matrix is obtained from the Jordan normal form by dividing all entries of each Jordan block by its diagonal element.
45.
If " x " is in the Jordan normal form, then " x " ss is the endomorphism whose matrix on the same basis contains just the diagonal terms of " x ", and " x " n is the endomorphism whose matrix on that basis contains just the off-diagonal terms; " x " u is the endomorphism whose matrix is obtained from the Jordan normal form by dividing all entries of each Jordan block by its diagonal element.
46.
Where \ lambda _ 1, \ lambda _ 2, \ ldots, \ lambda _ r are the distinct eigenvalues of A, then each \ mu _ i is the algebraic multiplicity of its corresponding eigenvalue \ lambda _ i and A is similar to a matrix J in "'Jordan normal form "', where each \ lambda _ i appears \ mu _ i consecutive times on the diagonal, and each entry above each \ lambda _ i ( that is, on the superdiagonal ) is either 0 or 1; the entry above the first occurrence of each \ lambda _ i is always 0.
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