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Les matrices :
formes de représentation et pratiques opératoires (1850-1930).
Frédéric Brechenmacher - Centre Alexandre Koyré
Extraits de la démonstration du théorème de Jordan
dans le traité de Mac Duffee de 1943.
Let A and B be two square matrices, A of order r and B of order s. the matrix
A+B= |
of order r+s is called their direct sum. […] More generally, let us suppose that the n×n matrix A of rank r can be written
A = A_{1} + A_{2}+…+A_{k} = |
_{} |
where A_{i }is of order n_{i} and rank r_{i}. Let the row vectors of A span the space S, let the first n_{1} two vectors of A span S_{1}, the next n_{2} row vectors of A span S_{2}, etc. […] Thus S is the supplementary sum S_{1}+S_{2}+…+S_{k} of the subspaces S_{i}. [...] A subspace S_{0} of S is said to be an invariant space of the matrix A if, for every vector Φ of S_{0}, it is true that A.Φ is in S_{0}. […] Much of the importance of invariant spaces derive from the following result
LEMMA . Let the total vector space S be the supplementary sum of the subspaces
S, S, …,S_{k}
where each S_{i} is of dimension r_{i} and has the basis σ_{i}, σ_{i}, …, σ_{iri}. Let P be the matrix whose column vectors are
(31). _{}
If each space S_{i} is an invariant space of the matrix A, then
P^{-1}AP = B_{1}+B_{2}+…+B_{k}
where B_{i} is matrix of order r_{i}
[…] In general M can be written
_{}
where each submatrix B_{i} has r_{i}_{ } rows and columns.
[…]
THEOREM 64. Let A be any n ×n matrix with elements in a field F, and let
m(x) ) m_{1}(x)m_{2}(x)…m_{k}(x)
be its minimum function expressed as a product of polynomials which are relatively prime as pairs. Let the null space of m_{i}(A) be of rank r_{i}. then A is similar to a direct sum
B_{1}+B_{2}+…+_{k}
where B_{i} is of order r_{i}, and the minimum function of B_{i} is m_{i}(x).
[…] Example :
_{}
Let us choose Φ = (1,0,0,0). Then
P = (Φ, AΦ, A²Φ, A^{3}Φ] = |
_{} |
[…] We find that
P^{-1}AP = |
_{,} |
which is the companion matrix of the minimum equation
m(x) = x^{4}+x²+1=0
The second canonical form can be obtained from the matrix
B_{1} = |
_{} | , |
B_{2} = |
_{} |
Let
Q_{1} = |
_{} | , | Q_{2} = |
_{} |
Then
Q_{1}^{-1}B_{1}Q_{1} = |
_{} | , | Q_{2}^{-1}B_{2}Q_{2} = |
_{} |
are the companion matrices of the equations obtained from the respective factors x²+x+1 and x²-x+1 of the minimum function of A. Then A is similar to the matrix
_{}