ON MATRICES WHOSE COEFFICIENTS ARE FUNCTIONS OF A SINGLE VARIABLE

Transcription

1 ON MATRICES WHOSE COEFFICIENTS ARE FUNCTIONS OF A SINGLE VARIABLE BY J. H. M. WEDDERBURN 1. The methods usually* employed i reducig to its ormal form a matrix whose coefficiets are polyomials i a variable X are of such a ature that it is ot at all obvious how they ca be exteded whe polyomials are replaced by aalytic fuctios, t The object of this ote is to show that the mai theorems o elemetary factors are ot restricted to matric polyomials but apply without appreciable modificatio to aalytic matric fuctios. As vectors are employed freely i the sequel, a short explaatio of the otatio used is ecessary. A vector, x = ( 1, 2,,»), is a ordered set of coefficiets: two vectors are equal if, ad oly if, their correspodig coefficiets are equal. The sum of two vectors, x = ( 1,&,,») ad y = ( Vi, V2,, i\ ) is defied as x 4- y = ( 1 4- j?i, 2 4-»?2,, 4- y ) The product of ay umber p ito a vector x is defied as pa; = (p i, p 2,, p ). Except i this elemetary case, multiplicatio of vectors will ot be used. A set of liearly idepedet vectors is called a basis. Ay vector ca be expressed liearly i terms of the elemets of a basis ad, i particular, if e< is the vector for which, = 1,, = 0 (j 4= *) > we have x = SÏ, e,. If A is a matrix (ars) ad x = S,e,-, the vector 2<~?(2^îa#&)«i is deoted by Ax. We evidetly have A(xi + x2) = Axi + Ax2, (A4- B)x = Ax4- Bx, A Bx = (AB)x. A matrix is completely determied whe its actio o liearly idepedet vectors is give. If the determiat of a matrix is ot zero, it trasforms a set of liearly idepedet vectors ito a liearly idepedet set; while if its rak is r ( r < ), there is a uique complex of vectors every elemet of * See for istace Bôcher, Itroductio to Higher Algebra, p t To avoid frequet repetitio, a matrix whose coefficiets are fuctios of a variable, X, will be called a matric fuctio of X. 328

2 1915] ON CERTAIN MATRICES 329 which is aihilated by the matrix. The followig lemma is a immediate cosequece of this. Lemma I. If xi, x2,, x is a set of vector fuctios which are holomorphic ad liearly idepedet for all values of X lyig i a give regio R, the determiat of the matrix defied by xr = P(X)er (r = 1,2, -,«) does ot vaish for ay value of's. i R. It may be remarked that the matrix P ( X ) is holomorphic i R ad, sice P (X) * does ot vaish, P-1 (X) is also holomorphic i that regio. Also, sice er = P-i^Xr, (r = 1,2,,«), the fixed basis ei, e2,, e may be replaced by ay set of vector fuctios which are holomorphic ad liearly idepedet throughout R. 2. I the reductio of a matrix to its ormal form, we require the followig extesio of a well-kow algebraic theorem. Lemma II. ///(X) ad o (X) are two fuctios which are holomorphic i R ad have o factor^ i commo i that regio, there exist two fuctios, holomorphic i R, such that p(x)/(x) + o(x)</(x) =1. If we expad l/[/(x)o(x)] Leffler series % we get i terms of its pricipal parts i a Mittag- (1) ir(~^-) = P(X) + G(X) + 0(X), where P(X) ad G(X) are the parts of the series arisig from the zeros of /(X) ad o (X), respectively, which lie i R, ad <j> (X) is a fuctio which is holomorphic i R. If we ow put p(\) = g(\)[g(\)+<p(\)], a(x) =/(X)P(X), both p(x) ad a(x) are holomorphic i R ad, o multiplyig both sides of (1) by/(x)o(x), we have 1 =/(X) - 7(X)[G(X)+<MX)] + a(x) -/(X)P(X) as required by the Lemma. = /(X)p(X) + 0(X)a(X), * The determiat of the matrix P ( X ) is deoted by P ( X ). 11, e., the two fuctios have o commo zeros i R. Fuctios which are holomorphic ad owhere zero i R play the same rôle as costats do i the algebraic theory ad will therefore ot be regarded as factors i this paper. % The series used here is a special case of the "geeralized Mittag-Leffler theorem." See Osgood, Fuktioelheorie vol. 1 (secod editio), p. 540, or Mittag-Leffler, Acta Mathem a t i c a, vol. 4 (1884), p. 8.

3 330 J. H. M. wedderbur: [July 3. Lemma III. If pi(x), pt(x),,p(x) are fuctios which are holomorphic ad have o factor commo to all i R, ad Xi, x2,, x is a set of vector fuctios which are holomorphic ad liearly idepedet for all values of X i R, there exists a matric fuctio P (X), holomorphic i R,for which (i) P (X) ] 4= 0/or ay X i R, ad (ii) P(X)xi = pi xi 4- P2 x p x. Assume that the lemma is true for bases of order less tha : the, if I is the H.C.F. of pi, p2,, p-i ad p'r = pr/l, (r = 1,2,, 1), there exists a matric fuctio, Q0, relative to the basis a-i, x2,, x-i which is holomorphic ad ever sigular i R ad is such that Qo xi = p\xi4- Let Qi be the matrix defied relatively p'2x24-4- pli a- _i. to the basis xi, x2,, a;» by QlX = X, QlXr = QoXr ( T = 1, 2,, - 1 ), ad set x'r = Qixr (r = 1,2,,). Sice I ad p have o commo factor i R, we ca, as i Lemma II, fid two fuctios, a ad ß, which are holomorphic i R ad are such that* al 4- ßp = 1. If therefore Q2 is the matrix defied by Q2 r[ = lx[ 4- px', Q x' = - ßx'2 4- ax', Q2 x'r = x'r (r = 2,...,-l), we have IQ2I = 1, ad P(X) Q2Q1 satisfies the coditios of the lemma sice Pa;i = Q2 Qi xi = Q2 x\ = lx[ 4- p x' = YlprXr. 1 Sice the lemma is obviously true for = 1, the required result follows immediately by iductio. 4. Theorem. If A(X) is a matric fuctio of rak r which is holomorphic i a regio R, there exist two matric fuctios, P(X) ad Q(X), which are holomorphic ad o-sigular i R, ad are such that Pi(X) P2(X) P(X)A(X)Q(X) = Eri\) *If p = 0, thef = 1 = a.

4 1915] ON CERTAIN MATRICES 331 where Pi (X), ET(\) are fuctios of X which are holomorphic i R ad are such that Es is a factor of Et whe s < t(s, t = 1,2,,r). This theorem is obviously true whe = 1 so that we may make its proof deped o iductio. We assume therefore that it is true for matrices of order less tha. If the rak of A is less tha, there is at least oe vector fuctio x, holomorphic i R, for which Ax = 0 i R; ad we ca isure that x itself does ot vaish by removig ay commo factor from its coefficiets whe expressed i terms of ei, e2,, e. By Lemma III, there is a matric fuctio, Qo, holomorphic ad o-sigular i R, for which a; = Qoe, whece AQ0e = 0: the elemets i the last colum of AQo are therefore all zero. Cosiderig ow the cojugate matrix Qó A', we see i the same maer that there is a matric fuctio P'0 holomorphic ad o-sigular i R, which is such that the coefficiets i the last colum of Q0 A' are also all zero. It follows the that P0 AQ0 has zeros both i the last colum ad i the last row, ad may therefore be regarded as a matrix of order 1 relative to the basis ei, e2,, e _i. There are therefore by hypothesis two matric fuctios, Pi ad Qi, of order 1 which are holomorphic ad o-sigular i R ad are such that Pi P0 AQ0 Qi has the desired ormal form with regard to the basis ei, e2,, e_i: ad we have oly to exted Pi ad Qi to the origial basis, ei, e2,, e, by addig the coditios Pi e = e = Qi e i order to have A i the required ormal form with regard to this basis. Assume ow that the rak of A is. If the coefficiets of A have a H.C.F., /(X), we may write A(\) =/(X)^4i(X): therefore, sice multiplyig P (\) A (\) Q (\) by a scalar factor still leaves it i the ormal form, we may, without loss of geerality, assume that there is o commo factor. If x = ( i, 2,, ) is a costat vector, Ax ca oly vaish for values of X for which A = 0 : but the values of X for which Ax = 0 are ecessarily cotiuous fuctios of the 's ad, sice the coefficiets of A have o commo factor i R, there are therefore o values of X i R for which Ax = 0 for every costat vector x; hece there is some costat vector, x, for which Ax + 0 for ay value of X i P. Let X be a costat matrix whose first colum cosists of the coordiates of this vector, the remaiig coefficiets beig so chose as to make the determiat of X ot zero. The coefficiets i the first colum of AX are the the coefficiets of Ax ad therefore have o factor i commo. Let AXei = 2ari(X)er. By Lemma III, we ca fid a matric fuctio P71 (X), holomorphic ad o-sigular i R, which is such that whece Pi AXei = d. Pi"1 ei = XXi(X)er, If, therefore,

5 332 j. h. m. wedderbur: o certai matrices Pi^4Xer = c ei 4-^2crtea (r = 2, 3,, ), 8=2 ad Qi is the matrix defied by ei = Qiei, er - Criei = Qier (r = 2, 3,, ), the Qi is a matric fuctio, holomorphic i R, for which Qi = 1 ad Pi AXQi ex = ex, Pi AXQX er = Pi AX ( et cri ex ) =2 cts es»=2 (r =2,3,,). All the coefficiets i the first row ad colum of Pi AXQi are therefore zero, except the first, which is 1. Strikig out this row ad colum, we have a matrix of order 1, ad, as we have assumed the theorem true for matrices of order less tha, this matrix ca be reduced to the required form by meas of two matric fuctios P2 ad Q2 which are defied relative to the basis e%, e3,, e ad are holomorphic ad o-sigular i R. If we exted these matrices to the basis ei, e2,, e as above by addig the coditios P2 ei = ei = Q2 ei, the matrix P2 Pi AXQi Q2 has the form required by the theorem sice i multiplyig by P2 ad Q2 the first row ad colum of Pi AXQi remai ualtered; the iductio is therefore complete.