Solution to Problem 74-14: A generalization of the Vandermonde determinant

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Solution to Problem 74-14: A generalization of the

Vandermonde determinant

Citation for published version (APA):

Lossers, O. P. (1975). Solution to Problem 74-14: A generalization of the Vandermonde determinant. SIAM Review, 17(4), 694-695. https://doi.org/10.1137/1017084

DOI:

10.1137/1017084

Document status and date: Published: 01/01/1975

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694 PROBLEMS AND SOLUTIONS

term in thebracket; thus

/al’n =o \/Z,,

E

tn+l =0

\

TJ

/a[,l(1 ,)

Z

(/aln+l

--/’n+2)2

=0 [a[,n

It is clear that ([1

--a[,2)2//,1

vanishes only if P(X=O)+ P(X 1)=1, and

/Zl(1

-/Zl)

only if/x,

,

sothatequalityholdsonlyifP(X O) P(X 1)

.

Also solved by M. L. J. HAUTUS (Technische Hogeschool Eindhoven, the

Netherlands), A. A. JAGERS (Technische Hogeschool Twente, Enschede, the Netherlands), T. G. KURTZ(University ofWisconsinmMadison) and the prop-oser.

Problem 74-14,AGeneralization

_of

the VandermondeDeterminant,by S. VENIT

(California State Universityat Los Angeles).

Evaluate then n determinantD(m, n)

JArs[,

where n>m-> 0and

(n r)!x

Ars

l:<s-<m+l, l<=r<=n+l-s

(n+l-r-s)!’

x_,.,

m+

l

<s<=n,

l

<=r<=n,

Ar

0, 1-<s=<m+l, n+l-s<r<=n.

Solution by O. P. LOSSERS (Technological University, Eindhoven, the

Netherlands).

Since the jth column, 1

=<

j

=<

m

+

1, is the (j- 1)th derivative of the first column,itfollows that

D(m, n)=

_H

VM(yl, ym+l, X2, Xn-m)

t=l Yl=Y2 Ym+l=xi

where VM(yl,"

,

ym+l, X2," X,,-m)is the Vandermonde determinant, i.e.,

VM(yl, ym+,,X2,""",X,,_,,,)

Fl

(y,-y,)"

[I

(y-x,).

_II

(Xp-X,).

<=i<j<=m+l <--_k=<m+l 2<=p<q<n-m 2<=ln-m

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PROBLEMS AND SOLUTIONS 695

TheonlynonzerocontributionstoD(m, n)comefromapplyingallderivativesto

thefirst ofthe three factors. Furthermore,

l-I

(y, y,) =(-1)"‘"+’,/2

(t- 1)!.

1Ni<j<=m+l Yl Y2 Ym =Xl

Hencewe obtain theresult

m+l D(m, n)=(-)+l’/

II

(t-). (x,-xl)+’

H

(x,-xo)

t=l 2NlNn--m 2Np<qNn--m m+l

=(-)m+"/

_H

(t-1).

_H

(Xl-Xl)

_"

_H

(x,-xo).

t=l 2NlNn--m lNp<qn--m

Also solved by T. FOREGGER (Bell Telephone Eaboratories), C. GVES

(Michigan Technological University), A. A. JAGERS (Technische Hogeschool

Twente,Enschede,theNetherlands), D. J.KEITMA (MassachusettsInstitute of

Technology), O. G. RUENR (Michigan Technological University)and the prop-oser.

Problem 74-16, A Matrix Problem, by C. R. CRAWFORD (Erindale College,

Ontario,Canada).

IfA denotes a given nonsingularsquare matrix (realorcomplex) oforder

n rs, prove that there exist nonzero column vectors x=(x,,x2,’’’,x,)r and

b (bl, b2,""", b,)r_{such that} (1) Ax=b, (2) x,=

Y

x, i=1 i=r+l Xi i=n-r+l (3)

bkr+l

bk+,_

b+

(k 0, 1,...,s-1).

The problem arose in power system engineering in the study of mutual

impedance in a coil with relatively high voltage and current but with constant

frequency.

Solution by THOMAS FOREGGER (Bell Laboratories, Murray Hill, New

Jersey).

The condition (2) can be rewritten as a system of (s-1) homogeneous

equations in the n variables x,,...,x,. The condition (3) can be written as a systemofs(r- 1)homogeneousequationsinthe n variablesx,.

,

x,. Thuswe

have a homogeneous system of s-1

+

s(r- 1) n-1 equations in n variables.

Thesystemmusthaveanonzero solutionx, andsinceA isnonsingular,b Ax is

also nonzero.

Also solved by J. V. BAXLEY (Wake Forest University), A. J. BOSCH

(Technological University,Eindhoven, theNetherlands), J. M. BROWNandD.A.

Voss (Western Illinois University), C. DAVIS (University of Toronto), A. A. DIAZ (WhiteSands MissileRange),C.GiveNs(Michigan Technological