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§1.3 Determinants

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§1.3(i) Definitions and Elementary Properties

1.3.1 det[ajk]=|a11a12a21a22|=a11a22-a12a21.
1.3.2 det[ajk]=|a11a12a13a21a22a23a31a32a33|=a11|a22a23a32a33|-a12|a21a23a31a33|+a13|a21a22a31a32|=a11a22a33-a11a23a32-a12a21a33+a12a23a31+a13a21a32-a13a22a31.

Higher-order determinants are natural generalizations. The minor Mjk of the entry ajk in the nth-order determinant det[ajk] is the (n-1)th-order determinant derived from det[ajk] by deleting the jth row and the kth column. The cofactor Ajk of ajk is

1.3.3 Ajk=(-1)j+kMjk.

An nth-order determinant expanded by its jth row is given by

1.3.4 det[ajk]==1najAj.

If two rows (or columns) of a determinant are interchanged, then the determinant changes sign. If two rows (columns) of a determinant are identical, then the determinant is zero. If all the elements of a row (column) of a determinant are multiplied by an arbitrary factor μ, then the result is a determinant which is μ times the original. If μ times a row (column) of a determinant is added to another row (column), then the value of the determinant is unchanged.

1.3.5 det[ajk]T =det[ajk],
1.3.6 det[ajk]-1 =1det[ajk],
1.3.7 det([ajk][bjk]) =(det[ajk])(det[bjk]).

Hadamard’s Inequality

For real-valued ajk,

1.3.8 |a11a12a21a22|2(a112+a122)(a212+a222),
1.3.9 det[ajk]2(k=1na1k2)(k=1na2k2)(k=1nank2).

Compare also (1.3.7) for the left-hand side. Equality holds iff

1.3.10 aj1ak1+aj2ak2++ajnakn=0

for every distinct pair of j,k, or when one of the factors k=1najk2 vanishes.

§1.3(ii) Special Determinants

An alternant is a determinant function of n variables which changes sign when two of the variables are interchanged. Examples:

1.3.11 det[fk(xj)],
j=1,,n; k=1,,n,
1.3.12 det[f(xj,yk)],
j=1,,n; k=1,,n.

Vandermonde Determinant or Vandermondian

1.3.13 |1x1x12x1n-11x2x22x2n-11xnxn2xnn-1|=1j<kn(xk-xj).

Cauchy Determinant

1.3.14 det[1aj-bk]=(-1)n(n-1)/21j<kn(ak-aj)(bk-bj)/j,k=1n(aj-bk).

Circulant

1.3.15 |a1a2anana1an-1a2a3a1|=k=1n(a1+a2ωk+a3ωk2++anωkn-1),

where ω1,ω2,,ωn are the nth roots of unity (1.11.21).

Krattenthaler’s Formula

For

1.3.16 tjk=(xj+an)(xj+an-1)(xj+ak+1)(xj+bk)(xj+bk-1)(xj+b2),
1.3.17 det[tjk]=1j<kn(xj-xk)2jkn(bj-ak).

§1.3(iii) Infinite Determinants

Let aj,k be defined for all integer values of j and k, and Dn[aj,k] denote the (2n+1)×(2n+1) determinant

1.3.18 Dn[aj,k]=|a-n,-na-n,-n+1a-n,na-n+1,-na-n+1,-n+1a-n+1,nan,-nan,-n+1an,n|.

If Dn[aj,k] tends to a limit L as n, then we say that the infinite determinant D[aj,k] converges and D[aj,k]=L.

Of importance for special functions are infinite determinants of Hill’s type. These have the property that the double series

1.3.19 j,k=-|aj,k-δj,k|

converges (§1.9(vii)). Here δj,k is the Kronecker delta. Hill-type determinants always converge.

For further information see Whittaker and Watson (1927, pp. 36–40) and Magnus and Winkler (1966, §2.3).