The notation is that of (1.2.58). For :
1.3.1 | |||
For :
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
1.3.4 | |||
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 | ||||
The determinant of an upper or lower triangular, or diagonal, square matrix is the product of the diagonal elements .
For real-valued ,
1.3.8 | |||
1.3.9 | |||
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.
An alternant is a determinant function of variables which changes sign when two of the variables are interchanged. Examples:
1.3.11 | |||
; , | |||
1.3.12 | |||
; . | |||
1.3.13 | |||
1.3.14 | |||
1.3.15 | |||
where are the th roots of unity (1.11.21).
For
1.3.16 | |||
1.3.17 | |||
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.
Square matices can be seen as linear operators because for all and , the space of all -dimensional vectors.
The adjoint of a matrix is the matrix such that for all . In the case of a real matrix and in the complex case .
Real symmetric () and Hermitian () matrices are self-adjoint operators on . The spectrum of such self-adjoint operators consists of their eigenvalues, , and all . The corresponding eigenvectors can be chosen such that they form a complete orthonormal basis in .
Let the columns of matrix be these eigenvectors , then , and the similarity transformation (1.2.73) is now of the form . For Hermitian matrices is unitary, and for real symmetric matrices is an orthogonal transformation.
For self-adjoint and , if , see (1.2.66), simultaneous eigenvectors of and always exist.
Assuming is an orthonormal basis in , any vector may be expanded as
1.3.20 | |||
. | |||
Taking norms,
1.3.21 | |||
which is Parseval’s equality.