Computational Chemistry

2. A diagonal matrix is asquarematrix that has all its off-diagonal elements zero;
   the (principal) diagonal runs from the upper left to the lower right.
   Examples:

20

04

 300

060

001

0

@

1

A

000

000

000

0

@

1

A

3. The unit matrix or identity matrix 1 orIis a diagonal matrix whose diagonal
   elements are all unity. Examples:

ð 1 Þ

10

01

 100

010

001

0

@

1

A

Since diagonal matrices are square, unit matrices must be square (but zero

matrices can be any size). Clearly, multiplication (when permitted) by the unit

matrix leaves the other matrix unchanged:1A¼A1¼A.

4. The inverseA"^1 of another matrixAis the matrix that, multipliedA, on the
   left or right, gives the unit matrix:A"^1 A¼AA"^1 ¼1.Examples:

IfA¼

12

34



then A"^1 ¼

" 21

3 = 2 " 1 = 2



Check it out.

5. A symmetric matrix is a square matrix for whichaij¼ajifor each element.
   Examples:

14

43



a 12 ¼a 21 ¼ 4

271

735

154

0

@

1

A a 12 ¼a 21 ¼ 7 ;etc:

Note that a symmetric matrix is unchanged by rotation about its principal

diagonal. The complex-number analogue of a symmetric matrix is aHermitian

matrix(after the mathematician Charles Hermite); this hasaij¼aji*, e.g. if

element (2,3)¼aþbi, then element (3,2)¼a"bi, the complex conjugate of

element (2,3);i¼

pffi

"1. Since all the matrices we will use are real rather than

complex, attention has been focussed on real matrices here.

6. The transpose (ATorA ̃) of a matrixAis made by exchanging rows and columns.
   Examples:

IfA¼

23

47



then AT¼

24

37



IfA¼

216

172



then AT¼

21

17

62

0

@

1

A

4.3 The Application of the Schr€odinger Equation to Chemistry by H€uckel 113