Physical Chemistry Third Edition

B Some Useful Mathematics 1251

The trace of thenbynidentity matrix is equal ton. The trace is sometimes called the

spur, from a German word that means track or trace.

A matrix in which all the elements below the diagonal elements vanish is called an

upper triangular matrix. A matrix in which all the elements above the diagonal elements

vanish is called alower triangular matrix, and a matrix in which all the elements except

the diagonal elements vanish is called adiagonal matrix. The matrix in which all of

the elements vanish is called thenull matrixor thezero matrix. Thetransposeof a

matrix is obtained by replacing the first column by the first row of the original matrix,

the second column by the second row of the original matrix, and so on. The transpose

ofAis denoted byA ̃(pronounced “A tilde”).

(A ̃)ijAji (B-94)

If a matrix is equal to its transpose, it is asymmetric matrix.

Thehermitian conjugateof a matrix is obtained by taking the complex conjugate

of each element and then taking the transpose of the matrix. If a matrix has only

real elements, the hermitian conjugate is the same as the transpose. The hermitian

conjugate is also called theadjoint(mostly by physicists) and theassociate(mostly

by mathematicians, who use the term “adjoint” for something else). The hermitian

conjugate is denoted byA†.

(A†)ijA∗ji (B-95)

A matrix that is equal to its hermitian conjugate is said to be ahermitian matrix.

Anorthogonal matrixis one whose inverse is equal to its transpose. IfAis orthog-

onal, then

A−^1 A ̃ (orthogonal matrix) (B-96)

Aunitary matrixis one whose inverse is equal to its hermitian conjugate. IfAis unitary,

then

A−^1 A†A ̃∗ (unitary matrix) (B-97)

Determinants A square matrix has a quantity associated with it that is called a deter-

minant. IfAis a square matrix, we denote its determinant by det(A). When explic-

itly written, it contains the same elements as the matrix, but is written with vertical

lines at the left and right. If the elements of a matrix are constant, its determinant is

equal to a constant, which can be evaluated as follows:A2 by 2 determinant has the

value

∣

∣

∣

∣

A 11 A 12

A 21 A 22

∣

∣

∣

∣A^11 A^22 −A^12 A^21 (B-98)

Larger determinants can be evaluated byexpanding by minors, as follows:

1. Pick a row or a column of the determinant. Any row or column will do, but one with
   zeros in it will minimize the work.


2. The determinant equals a sum of terms, one for each element in the row or column.
   Each term consists of an element of the chosen row or column times theminorof
   that element, with an assigned sign that is either positive or negative. The minor of
   an element in a determinant is the determinant that is obtained by deleting the row
   and the column containing that element. The minor of annbyndeterminant is an