Abstract
Authors
D.V. Ionescu
Institutul de Calcul
Keywords
?
Paper coordinates
D.V. Ionescu, Reducerea unei forme biliniare la o formă canonică. (Romanian) Stud. Cerc. Mat. (Cluj) 10 1959 45–49.
About this paper
Journal
Studii si Cercetari Matematice
Publisher Name
Academy of the Republic of S.R.
Print ISSN
1220-269X
Online ISSN
MR0130264
google scholar link
??
Paper (preprint) in HTML form
Original text
Rate this translation
Your feedback will be used to help improve Google Translate
REDUCTION OF A BILINEAR FORM TO A CANONICAL FORM
In a note in the Bulletin of the Polish Academy of Sciences, from 1957, M. Altman [1] gave a generalization of Jacobi's method for bilinear forms. We take the liberty of pointing out that we presented at the first scientific session of the Society of Mathematical and Physical Sciences of the RPR, in 1955 [2], a paper containing as a particular case the reduction of a bilinear form to a canonical form. In this note we give another method for obtaining the canonical form.
Let us consider the bilinear form
{:(1)Phi=sum_(i,n=1)^(n)a_(i,k)x_(i)y_(k):}
and either
{:(2)X_(k)=sum_(j=1)^(n)a_(j,k)x_(j)","quadY_(i)=sum_(j=1)^(n)a_(i,j)y_(j):}
wherei,k=1,2,dots,n.
Eitherrmatrix rank||a_(ik)||_(1)^(n)and suppose that the notations were chosen so that the determinants
Eitherrmatrix rank||a_(ik)||_(1)^(n)and suppose that the notations were chosen so that the determinants
{:(3)Delta_(h)=|{:[a_(11),dots,a_(1h)],[*,dots,*],[a_(h1),dots,a_(hh)]:}|:}
not be null, which is always possible, whereh=1,2,dots,r.
In this case it is demonstrated that
{:(4)Phi=(-1)/(Delta_(r))|{:[a_(11),dots,a_(1r),Y_(1)],[*,,*,*],[*,,*,*],[a_(r1),dots,a_(rr),Y_(r)],[X_(1),dots,X_(r),0]:}:}
Applying Gauss' algorithm (see [3]) to solve the linear equations (2) iny_(k)andx_(i), these equations can be written in the form
{:[a_(11)y_(1)+a_(12)y_(2)+dots+a_(1n)y_(n)=Y_(1)],[a_(22)^((1))y_(2)+dots+a_(2n)^((1))y_(n)=Y_(2)^((1))],[a_(n2)^((1))y_(2)+dots+a_(nn)^((1))y_(n)=Y_(n)^((1))],[a_(11)x_(1)+a_(21)x_(2)+dots+a_(n1)x_(n)=X_(1)],[a_(22)^((1))x_(2)+dots+a_(n2)^((1))x_(n)=X_(2)^((1))],[a_(2n)^((1))x_(2)+dots+a_(nn)^((1))x_(n)=X_(n)^((1))]:}
and
a_(ik)^((1))=((|{:[a_(11)a_(1k)],[a_(i1)a_(1k)]:}|))/(Delta_(1)),quadX_(k)^((1))=((|{:[a_(11)a_(1k)],[X_(1)X_(k)]:}|))/(Delta_(1)),quadY_(i)^((1))=((|{:[a_(11)Y_(1)],[a_(i1)Y_(i)]:}|))/(Delta_(1))
where
It is then demonstrated that
Phi=(-1)/(Delta_(r))|{:[a_(11),0,dots,0,Y_(1)],[0,a_(22)^((1)),dots,a_(2r)^((1)),Y_(2)^((1))],[*,*,dots,*,*],[0,a_(r2)^((1)),dots,a_(rr)^((1)),Y_(r)^((1))],[X_(1),X_(2)^((1)),dots,X_(r)^((1)),0]:}|
Continuing Gauss' algorithm, we arrive at the identity
Phi=(-1)/(Delta_(r))|{:[a_(11),0,0,dots,0,Y_(1)],[0,a_(22)^((1)),0,dots,0,Y_(2)^((1))],[0,0,a_(33)^((2)),dots,0,Y_(3)^((2))],[dots,dots,dots,dots,dots,.]:}|
whereY_(h)^((h-1))andX_(h)^((h-1))are the second members of the linear equations
{:[a_(11)y_(1)+a_(12)y_(2)+dots+a_(1n)y_(n)=Y_(1)],[a_(22)^((1))y_(2)+dots+a_(2n)^((1))y_(n)=Y_(2)^((1))],[a_(rr)^((r-1))y_(r)+dots+a_(rn)^((r-1))y_(n)=Y_(r)^((r-1))],[a_(11)x_(1)+a_(21)x_(2)+dots+a_(n1)x_(n)=X_(1)],[a_(22)^((1))x_(2)+dots+a_(n2)^((1))x_(n)=X_(2)^((1))],[a_(rr)^((r-1))x_(r)+dots+a_(nr)^((r-1))x_(n)=X_(r)^((r-1))]:}
and
Gauss's.
obtained by Gauss's algorithm.
It is shown that
Gauss's.
obtained by Gauss's algorithm.
It is shown that
{:(6)X_(k)^((k-1))=(P_(k)(x))/(Delta_(k))","quadY_(i)^((i-1))=(Q_(i)(x))/(Delta_(i))","quada_(hh)^((h-1))=(Delta_(h))/(Delta_(h-1)):}
where
{:(7)P_(k)(x)=|{:[a_(11),dots,a_(1k)],[*,dots,*],[a_(k-1,1),dots,a_(k-1,k)],[X_(1),dots,X_(k)]:}|","quadQ_(i)(y)=|{:[a_(11),dots,a_(1,i-1)Y_(1)],[*,,*],[*,,*],[a_(i1),dots,a_(i,i-1)Y_(i)]:}|:}
Expanding the determinant from identity (5), we obtain the canonical form of the bilinear form (1)
Phi=(X_(1)Y_(1))/(a_(11))+(X_(2)^((1))Y_(2)^((1)))/(a_(22)^((1)))+dots+(X_(r)^((r-1))Y_(r)^((r-1)))/(a_(rr)^((r-1)))
which with the help of formulas (6) and (7) is written in the definitive form
{:(8)Phi=(P_(1)(x)Q_(1)(y))/(Delta_(0)Delta_(1))+(P_(2)(x)Q_(2)(y))/(Delta_(1)Delta_(2))+dots+(P_(r)(x)Q_(r)(y))/(Delta_(r-1)Delta_(r)).:}
It is easy to see thatP_(k)(x)it does not depend onx_(1),x_(2),dots,x_(k-1), thatQ_(i)(y)it does not depend ony_(1),y_(2),dots,y_(i-1)and that
(D((P_(1),dots,P_(r))))/(D((x_(1),dots,x_(r))))=(D((Q_(1),dots,Q_(r))))/(D((y_(1),dots,y_(r))))=Delta_(1)Delta_(2)dotsDelta_(r)!=0,
which proves that linear formsP_(k)(x)andQ_(i)(y)are linearly independent.
When the bilinear form (1) is symmetric, that is,a_(ik)=a_(ki), we haveP_(i)(x)=Q_(i)(x)and formula (8) becomes
{:(9)Phi=(Q_(1)(x)Q_(1)(y))/(Delta_(0)Delta_(1))+(Q_(2)(x)Q_(2)(y))/(Delta_(1)Delta_(2))+dots+(Q_(r)(x)Q_(r)(y))/(Delta_(r-1)Delta_(r)).:}
In the particular case of quadratic shapesx_(k)=y_(k), the previous formula gives Jacobi's classical formula for decomposing a quadratic form into a sum of squares
{:(10)sum_(i,k=1)^(n)a_(ik)x_(i)x_(k)=(Q_(1)^(2)(x))/(Delta_(0)Delta_(1))+(Q_(2)^(2)(x))/(Delta_(1)Delta_(2))+dots+(Q_(r)^(2)(x))/(Delta_(r-1)Delta_(r)):}
If the bilinear form (1) is with complex coefficients and indeterminates and in addition Hermitian, that isa_(ik)= bar(a_(ki)), it can be written in the form
Phi_(1)=sum_(i,k=1)^(n)a_(ik) bar(x_(i))y_(k).
It is shown that in this case we have
P_(i)(x)= bar(Q_(i)(x))
and the decomposition formula (8) becomes
Phi_(1)=( bar(Q_(1))(x)Q_(1)(y))/(Delta_(0)Delta_(1))+( bar(Q_(2)(x))Q_(2)(y))/(Delta_(1)Delta_(2))+dots+( bar(Q_(r)(x))Q_(r)(y))/(Delta_(r-1)Delta_(r))
determinantsDelta_(i)being real.
Whenx_(i)=y_(i), the previous formula gives the decomposition of a Hermitian quadratic form into a sum of squares
Whenx_(i)=y_(i), the previous formula gives the decomposition of a Hermitian quadratic form into a sum of squares
{:(11)sum_(i,k=1)^(n)a_(lk) bar(x)_(i)x_(k)=(|Q_(1)(x)|^(2))/(Delta_(0)Delta_(1))+(|Q_(2)(x)|^(2))/(Delta_(1)Delta_(2))+dots+(|Q_(r)(x)|^(2))/(Delta_(r-1)Delta_(r)).:}
BIBLIOGRAPHIC BOOKS
- A1tman M., A generalisation of Jacobi's method for bilinear forms. Bull. de l'Acad. Polon. des Sci., 1957, tom. V, p. 99-104.
- Ionescu DV, An important identity and the decomposition of a bilinear form into a sum of products. Gazeta matematica și fizica, Series A, 1955, pp. 303-312 (referred to in Reterativnîi Jurnal - Matematika, 1957, ref. no. 1146).
- Gantmacher FR, Matrix Theory, Moscow, 1953, vol. II, p. 28 (lithographic transl. from the 1st Russian).
REDUCTION OF BILINEAR FORM TO CANONICAL FORM (Summary)
In a note from the Bulletin of the Polish Academy of Sciences for 1957, M. Altman [1] gave a generalization of Jacoba's method for bilinear forms. At the first scientific session of the PHP Society for Mathematical and Physical Sciences in 1955 [2], a paper was presented containing a special case of reducing a bilinear form to a conical form. This note presents a new method for obtaining the canonical form.
First, identity (5) is proved for the bilinear form (1), which implies the canonical form (8), whereP(x),Q(x)AndDelta_(i)are given by formulas (7) and (3). As a special case, we obtain formula (10), which represents the classical Jacobi decomposition for a quadratic form, as well as formula (11), which represents the decomposition of a Hermitian quadratic form into a sum of squares.
REDUCTION FROM A BILINERIC FORM TO A CANONICAL FORM
(Summary)
In a note in the Bulletin of the Polish Academy of Sciences, dated 1957, Mr. Altman [1] gave a generalization of Jacobi's method for bilinear forms. I would like to point out that I presented a paper at the first scientific session of the Society of Mathematical and Physical Sciences of the Polish People's Republic (RPR) in 1955 [2], which includes as a special case the reduction of a bilinear form to a canonical form, and in this note I give a summary of the method I used.
We first demonstrate, for the bilinear form (1), the identity (5), from which the canonical form (8) follows, theP_(i)(x),Q_(i)(y)AndDelta_(i)given formulas (7) and (3). We obtain as a special case formula (10) which is the classical Jacobi decomposition of the quadratic form and formula (11) which is the decomposition of a Hermitian quadratic form into a sum of squares.