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Les matrices :
formes de représentation et pratiques opératoires (1850-1930).

Frédéric Brechenmacher - Centre Alexandre Koyré

Encart 3

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


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 = A1 + A2+…+Ak =

where Ai is of order ni and rank ri. Let the row vectors of A span the space S, let the first n1 two vectors of A span S1, the next n2 row vectors of A span S2, etc. […] Thus S is the supplementary sum S1+S2+…+Sk of the subspaces Si. [...] A subspace S0 of S is said to be an invariant space of the matrix A if, for every vector Φ of S0, it is true that A.Φ is in S0. […]  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, …,Sk

where each Si is of dimension ri and has the basis σi, σi, …, σiri. Let P be the matrix whose column vectors are


If each space Si is an invariant space of the matrix A, then

P-1AP = B1+B2+…+Bk

where Bi is matrix of order ri

[…] In general M can be written

where each submatrix Bi has ri  rows and columns.


THEOREM 64. Let A be any n ×n matrix with elements in a field F, and let

m(x) ) m1(x)m2(x)…mk(x)

be its minimum function expressed as a product of polynomials which are relatively prime as pairs. Let the null space of mi(A) be of rank ri. then A is similar to a direct sum


where Bi is of order ri, and the minimum function of Bi is mi(x).

[…] Example :

Let us choose Φ = (1,0,0,0). Then

P = (Φ, AΦ, A²Φ, A3Φ] =

[…] We find that


P-1AP =


which is the companion matrix of the minimum equation

m(x) = x4+x²+1=0

The second canonical form can be obtained from the matrix

   B1 =
B2 =


Q1 =
Q2 =


Q1-1B1Q1 =
Q2-1B2Q2 =


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