# §1.3 Determinants

## §1.3(i) Definitions and Elementary Properties

1.3.1
1.3.2

Higher-order determinants are natural generalizations. The minor of the entry in the th-order determinant is the ()th-order determinant derived from by deleting the th row and the th column. The cofactor of is

1.3.3

An th-order determinant expanded by its th row is given by

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 , then the result is a determinant which is times the original. If times a row (column) of a determinant is added to another row (column), then the value of the determinant is unchanged.

1.3.5
1.3.6
1.3.7

For real-valued ,

1.3.8

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

1.3.10

for every distinct pair of , or when one of the factors vanishes.

## §1.3(ii) Special Determinants

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

1.3.13

### ¶ Circulant

where are the th roots of unity (1.11.21).

For

1.3.16
1.3.17

## §1.3(iii) Infinite Determinants

Let be defined for all integer values of and , and denote the determinant

1.3.18

If tends to a limit as , then we say that the infinite determinant converges and .

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

1.3.19

converges (§1.9(vii)). Here 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).