Computational Chemistry

AB¼C where C¼

c 1

c 2



Ais the coefficients matrix,Bis the unknowns matrix, andCis the constants
matrix. Now, if we can find a matrixA"^1 such thatA"^1 AB¼B(analogous to the
numbersa"^1 ab¼b) then

A"^1 AB¼A"^1 C i.e: B¼A"^1 C

Thus the unknowns matrix is simply the inverse of the coefficients matrix times
the constants matrix. Note that we multiplied byA"^1 on the left (A"^1 AB¼A"^1 C),
which is not the same as multiplying on the right, which would giveABA"^1 ¼
CA"^1 ; this is not necessarily the same asB.
To see that a matrix can act as an operator consider the vector from the origin to
the pointP(3,4). This can be written as a column matrix, and multiplying it by the
rotation matrix shown transforms it (rotates it) into another matrix:

vector

new, rotated

vector

x

y

x

y

3

4 –4

3

0

1

multiply on the left

by rotation matrix

0

–1

4.3.3.4 Some Important Kinds of Matrices

These matrices are particularly important in computational chemistry:

1. The zero matrix (the null matrix)


2. Diagonal matrices


3. The unit matrix (the identity matrix)


4. The inverse of another matrix


5. Symmetric matrices


6. The transpose of another matrix


7. Orthogonal matrices


8. The zero matrix or null matrix, 0 , is any matrix with all its elements zero.
   Examples:

00

00



000

000



ðÞ 0000

Clearly, multiplication by the zero matrix (when the (a&b)(b&c) mnemonic

permits multiplication) gives a zero matrix.

112 4 Introduction to Quantum Mechanics in Computational Chemistry