APPENDI X A

Reminding Matrices and Determinants

Matrices
Denition
A n m matrix A represents a rectangular table of numbers
1
A
i j
standing like soldiers in n
perfect rows and m columns (index i tells us in which row, and index j tells us in which column
the number A
i j
is located):
A =

A
11
A
12
. . . A
1m
A
21
A
22
. . . A
2m
.
.
.
.
.
.
.
.
.
.
.
.
A
n1
A
n2
. . . A
nm

.
This notation allows us to operate the whole matrices (like troops), instead of specifying
what happens to each number (soldier) separately. If matrices were not invented, then the
equations would be very long and clumsy, instead of short and clear.
Addition
Two matrices A and B may be added, if their dimensions n and m match. The result is matrix
C = A + B (of the same dimensions as A and B), where each element of C is a sum of the
corresponding elements of A and B:
C
i j
= A
i j
+ B
i j
,
e.g.,

1 1
3 4

2 1
2 3

3 0
5 7

.
Multiplying by a Number
A matrix may be multiplied by a number by multiplying every element of the matrix by this
number: cA = B with B
i j
= cA
i j
. E.g., 2

1 1
3 2

2 2
6 4

.
1
If instead of the numbers a matrix contained functions, then everything in this Appendix would remain valid (at
particular values of the variables instead of the functions, we would have their values).
Ideas of Quantum Chemistry, Second Edition. http://dx.doi.org/10.1016/B978-0-444-59436-5.00021-0
2014 Elsevier B.V. All rights reserved.
e1
e2 Appendix A
Matrix Product
A product of two matrices A and B is matrix C, denoted by C = AB; its elements are calculated
using elements of A and B:
C
i j
=
N

k=1
A
i k
B
kj
,
where the number of the columns (N) of matrix A has to be equal to the number of rows in
matrix B. The resulting matrix C has the number of rows equal to the number of rows in A and
the number of columns equal to the number of columns in B. Let us see how it works in an
example. The product AB = C:

A
11
A
12
A
13
A
14
A
21
A
22
A
23
A
24
A
31
A
32
A
33
A
34

B
11
B
12
B
13
B
14
B
15
B
16
B
17
B
21
B
22
B
23
B
24
B
25
B
26
B
27
B
31
B
32
B
33
B
34
B
35
B
36
B
37
B
41
B
42
B
43
B
44
B
45
B
46
B
47

C
11
C
12
C
13
C
14
C
15
C
16
C
17
C
21
C
22
C
23
C
24
C
25
C
26
C
27
C
31
C
32
C
33
C
34
C
35
C
36
C
37

;
e.g., C
23
is the dot product of two vectors, or in matrix notation:
C
23
=

A
21
A
22
A
23
A
24

B
13
B
23
B
33
B
43

= A
21
B
13
+ A
22
B
23
+ A
23
B
33
+ A
24
B
43
.
At this point, note the following:
The result of matrix multiplication depends in general on whether one has AB or BA; i.e.,
in general,
2
AB = BA.
Matrix multiplication satises the following relation (which is easy to check): A(BC) =
(AB)C; i.e., the parentheses do not count and we can write simply: ABC.
Often we will have multiplication of a square matrix A by a matrix B composed of one
column. Then, using the rule of matrix multiplication, we obtain the matrix C in the form
of a single column (with the number of elements identical to the dimension of A):

A
11
A
12
. . . A
1m
A
21
A
22
. . . A
2m
.
.
.
.
.
.
.
.
.
.
.
.
A
m1
A
m2
. . . A
mm

B
1
B
2
.
.
.
B
m

C
1
C
2
.
.
.
C
m

.
2
Note, however, that it may be that AB = BA.
Reminding Matrices and Determinants e3
Transposed Matrix
For a given matrix A, we may dene the transposed matrix A
T
as (A
T
)
i j
= A
j i
.
For example, if A =

1 2
2 3

, then A
T
=

1 2
2 3

.
If matrix A = BC, then A
T
= C
T
B
T
; i.e., the order of multiplication is reversed. Indeed,
(C
T
B
T
)
i j
=

k
(C
T
)
i k
(B
T
)
kj
=

k
C
ki
B
j k
=

k
B
j k
C
ki
= (BC)
j i
= (A
T
)
i j
.
Inverse Matrix
For some square matrices A (which will be called non-singular), we can dene the so-called
inverse matrix as A
1
, which has the property AA
1
= A
1
A = 1, where 1 stands for the unit
matrix:
1 =

1 0 . . . 0
0 1 . . . 0
.
.
.
.
.
.
.
.
.
.
.
.
0 0 . . . 1

. E.g., for the matrix A =

2 0
0 3

we can nd A
1
=

1
2
0
0
1
3

.
For square matrices, A1 = 1A = A.
If we cannot nd A
1
(because it does not exist), then A is called a singular matrix. For
example, the matrix A =

1 1
1 1

is singular. The inverse matrix for A = BCis A

1
= C
1
B
1
.
Indeed, AA
1
= BCC
1
B
1
= B1B
1
= BB
1
= 1.
Adjoint, Hermitian, Symmetric Matrices
If the matrix A is transposed and all its elements are changed to their complex conjugates, then
we obtain the adjoint matrix denoted as A

= (A
T
)

= (A

)
T
. If for a square matrix we have
A

= A, then A is called Hermitian. If A is real, then, of course, A

= A
T
. In addition, if for
a real square matrix A
T
= A, then A is called symmetric. Examples: A =

1 + i 3 2i
2 + i 3 i

;
A
T
=

1 + i 2 + i
3 2i 3 i

; A

1 i 2 i
3 + 2i 3 + i

.
Matrix A =

1 i
i 2

represents an example of a Hermitian matrix because A

= A. Matrix
A =

1 5
5 2

is a symmetric matrix.
Unitary and Orthogonal Matrices
If for a square matrix A we have A

= A
1
, then A is called a unitary matrix. If B is Hermitian,
then the matrix exp (i B) is unitary, where we dene exp (i B) by using the Taylor expansion:
e4 Appendix A
exp (i B) = 1 + i B +
1
2!
(i B)
2
+ . Indeed, [exp (i B)]

= 1 i B
T
+
1
2!
(i B
T
)
2
+ =
1 i B +
1
2!
(i B)
2
+ = exp (i B), while exp (i B) exp (i B) = 1.
If A is a real unitary matrix A

= A
T
, then it is called orthogonal with the property
A
T
= A
1
. For example if A =

cos sin
sin cos

, then A
T
=

cos sin
sin cos

= A
1
.
Indeed, AA
T
=

cos sin
sin cos

cos sin
sin cos

1 0
0 1

.
Determinants
Denition
For any square matrix A = {A
i j
}, we may calculate a number called its determinant and denoted
by det A or |A|. The determinant is computed by using the Laplace expansion
det A =
N

i
(1)
i +j
A
i j

A
i j
=
N

j
(1)
i +j
A
i j

A
i j
,
where (N is the dimension of the matrix). Here, the result does not depend on which column
j has been chosen in the rst expression or which row i is the second expression. The symbol

A
i j
stands for the determinant of the matrix, which is obtained fromA by removing the i th row
and the j th column. Thus, we have dened a determinant (of dimension N) by saying that it is
a certain linear combination of determinants (of dimension N 1). It is sufcient, then, to tell
what we mean by the determinant that contains only one number c (i.e., having only one row
and one column); this is simply det c c.
For example, the matrix A =

1 0 1
2 2 4
3 2 3

;
and the determinant
det A =

1 0 1
2 2 4
3 2 3

= (1)
1+1
1

2 4
2 3

+ (1)
1+2
0

2 4
3 3

+(1)
1+3
(1)

2 2
3 2

2 4
2 3

2 2
3 2

= (2 (3)4 (2)) (2 (2)2 3) = 2 + 10 = 12.

In particular,

a b
c d

= ad bc.
Reminding Matrices and Determinants e5
By repeating (i.e., expanding

A
i j
, etc.) the Laplace expansion again and again, we arrive
nally at a linear combination of products of the elements:
det A =

P
(1)
p

P[A
11
A
22
A
NN
],
where the permutation operator

P pertains to the second indices (shown in bold), and p is the
parity of the permutation

P.
Slater Determinant
In this book, we will most often be dealing with determinants of the matrices, whose ele-
ments are functions, not numbers. In particular, the most important will be the so-called Slater
determinants. A Slater determinant for the N electron system is built of the functions called
spinorbitals
i
( j ), i = 1, 2, . . . , N, where the symbol j means the space and spin coordinates
(x
j
, y
j
, z
j
,
j
) of electron j :
(1, 2, . . . , N) =
1

N!

1
(1)
1
(2) . . .
1
(N)

2
(1)
2
(2) . . .
2
(N)
.
.
.
.
.
.
.
.
.
.
.
.

N
(1)
N
(2) . . .
N
(N)

.
After that, the Laplace expansion gives
(1, 2, . . . , N) =
1

N!

P
(1)
p

P[
1
(1)
2
(2)
N
(N)],
where the summation is over N! permutations of the N electrons,

P stands for the permutation
operator that acts on the arguments of the product of the spinorbitals [
1
(1)
2
(2)
N
(N)],
and p is the parity of the permutation

P (i.e., the number of the transpositions that change
[
1
(1)
2
(2)
N
(N)] into

P[
1
(1)
2
(2)
N
(N)].
All the properties of determinants also pertain to the Slater determinants.
Some Useful Properties
det A
T
= det A.
From the Laplace expansion, it follows that if one of the spinorbitals is composed of two
functions
i
= + , then the Slater determinant is a sum of the two Slater determinants,
one with instead of
i
, and the second with instead of
i
.
If we add to a row (column) any linear combination of other rows (columns), the value of
the determinant does not change.
e6 Appendix A
If a row (column) is a linear combination of other rows (columns), then det A = 0. In
particular, if two rows (columns) are identical, then det A = 0. Conclusion: in a Slater
determinant, the spinorbitals have to be linearly independent; otherwise, the Slater deter-
minant equals zero.
If in a matrix A we exchange two rows (columns), then det A changes the sign. Conclusion:
the exchange of the coordinates of any two electrons leads to the change of the sign of the
Slater determinant (the Pauli exclusion principle).
det (AB) = det A det B.
From the Laplace expansion, it follows that multiplying the determinant by a number is
equivalent to multiplying an arbitrary row (column) by this number. Therefore, det (cA) =
c
N
det A, where N is the matrix dimension.
3
If matrix U is unitary then det U = exp (i ), where is a real number. This means that if
U is an orthogonal matrix, then det U = 1.
3
Note that to multiply a matrix by a number, we have to multiply every element of the matrix by this number.
However, to multiply a determinant by a number means to multiply by this number of one row (column).