Appendix C: Computation of Eigenvalues and Eigenvectors, and Principal
Component Analysis
C.1 Introduction
The AHP and principal component analysis (PCA) require calculating eigenvalues
and eigenvectors of a matrix. This appendix presents an Excel-based method to calculate
the eigenvalue and eigenvector and briefly discusses the PCA technique.
C.2 Eigenvalues and eigenvectors of a matrix
Consider a matrix A  (aij )nn . Let I be an n  n identity matrix, and  be an
eigenvalue of A . The eigenvalue is the solution of Eq. (1):
| B || A   I | 0 .
(1)
where symbol “ | . | ” means the determinant of a matrix.
Generally, Eq. (1) has n roots for  . The largest eigenvalue (in terms of absolute
value) is called the principal eigenvalue. In the AHP setting, we have max  n . The
following two relations are useful for finding all the eigenvalues
n
n
n
   a , 
i 1
i
i 1
ii
i
i 1
| A | .
(2)
Let U  (u1 ,..., un )T is an n -dimensional column vector and  be an eigenvalue
of A . U is the eigenvector of matrix A if
AU  U .
(3)
The principal eigenvector of a matrix is an eigenvector corresponding to the principal
eigenvalue. The number of eigenvectors associated with Eq. (3) is infinite. To find a
specific eigenvector, we may take one of ui ' s as 1, i.e., uk  1 ( 1  k  n ). In this case,
we can obtain (u1 , ..., 1, ..., un )T from Eq. (3). This eigenvector can be further normalized
into a unit vector through the following
ui'  ui /
n
u
j 1
2
j
.
(4)
In the AHP setting, U is the principal eigenvector, and the weights or priorities are
given by
n
u  ui /  u j .
'
i
(5)
j 1
C.3 Using Excel Functions to find the eigenvalue and eigenvector of a matrix
The following Excel functions are useful for finding the eigenvalue and eigenvector
of a matrix.
 Mdeterm( A ) returns the determinant of matrix A .
 Minverse( A ) returns the inverse matrix of matrix A .
 Mmult(array 1, array 2) returns the matrix product of two arrays. The result is an
array with the same number of rows as array 1 and the same number of columns as
array 2. The number of columns in array 1 must be the same as the number of rows
in array 2.
When using Minverse( A ) or Mmult(array 1, array 2), first select the range of the
outcome, enter the formula as an array formula, then press F2, and finally press
CTRL+SHIFT+ENTER.
Example C.1: Calculate the principal eigenvalues and principal eigenvectors of the
matrices shown in Tables C.1 and C.2.
Table C.1 A 4  4 matrix
i \ j
1
2
3
4
1
1
1/5
1/8
1/6
2
5
1
1/2
1
3
8
2
1
1
4
6
1
1
1
Table C.2 A 3  3 matrix
i \ j
1
2
3
1
1
2
1/2
2
1/2
1
1/3
3
2
3
1
The principal eigenvalue of the matrix given by Table C.1 equals 4.0407, and the
principal eigenvector normalized by Eq. (5) is ( ui ,1  i  4 ) = (0.0494, 0.2476, 0.3944,
0.3086). The principal eigenvalue of the matrix given by Table C.2 equals 3.0092, and the
principal eigenvector is ( ui ,1  i  3 ) = (0.2970, 0.1634, 0.5396).
C.4 Principal Component Analysis
Consider the situations where several variables are simultaneously observed. There
can be correlations among the variables but the relationships among the variables may be
implicit. To make the analysis clearer and simpler, the PCA uses orthogonal and linear
transformations to transform the original variables into new and linearly uncorrelated
variables, which are called principal components. The principal components are mutually
independent and do not contain overlapped information. Through neglecting one or more
unimportant principal components, the relationships among the original variables can be
approximately represented by a few important principal components without losing too
much information. As such, the PCA has been widely used for data compression and data
explanation.
Consider a set of n observations of p random variables. The observations can be
presented in the form of a matrix as below:
 x11 ... x1 p 


X   ... ... ...  .
 xn1 ... xnp 


(6)
X is called the sample matrix.
Since the original variables are usually measured using different scales or units (e.g.,
temperature and pressure), it is necessary to standardize their observations by the
following transformation:
yij 
xij   j
sj
,1  j  p
(7)
where  j and s 2j are sample average and sample variance, respectively, and given by
j 
Let
A  [aij ,1  i, j  p ]
1 n
1 n
2
x
s

( xij   j ) 2 .
,


j
ij
n  1 i 1
n i 1
denote
the
covariance
(8)
matrix
of
Matrix
Y  [ yij ,1  i  n,1  j  p ] , which is actually the correlation matrix of Matrix X .
Generally, Matrix A has p positive eigenvalues. Let  j ,1  j  p , denote the
eigenvalues with:
1  2  ...   p  0 .
(9)
Let V j = (v1 j ,..., v pj )T denote any one of the eigenvectors associated with  j . The
unit eigenvector U j  (u1 j ,..., u pj )T is given by
uij  vij /
p
v
i 1
2
ij
.
(10)
As such, we can obtain p mutually independent and orthogonal unit eigenvectors.
Define the j -th principal component as
p
Pj   yi uij .
(11)
i 1
Clearly, u ij reflects the importance of Yi for Pj and is called the loading or weight.
For a particular data point ( y1 ,..., y p ), the value of Pj obtained from Eq. (11) is called
the component (or factor) score.
The sum of the diagonal elements of a covariance matrix represents the total variance
of the variables. The total variance equals the sum of all the eigenvalues (see Eq. (2)), i.e.,
each eigenvalue has a contribution to the total variance. The first principal component has
the greatest contribution; the second principal component has the second greatest
contribution; and so on. The contribution of the j -th principal component to the total
variance is given by
p
c j   j /  l .
(12)
l 1
The cumulative contribution of the first j principal components to the total variance is
given by
j
C j   cl .
(13)
l 1
The information that a principal component contains can be represented by its
contribution to the total variance. The information represented by the original variables
can be approximated by the first k principal components if Ck 1  0.9 and Ck  0.9 .
This is because the variance values in the other principal components are smaller than
10% and may be dropped without losing too much information.
Each of the first k principal components can be further interpreted through
examining the symbol and magnitude of u ij , and the physical or engineering meaning of
X j . For example, if u1J is positive and largest among ( u1 j ,1  j  p ), and X J is
larger-the-better, then
P1
mainly represents the effect of
X J , and is also
larger-the-better.
Example C.2: The data shown in Table C.3 are the performance scores of 34
candidates. All the four criteria are larger-the-better. The problem is to find principal
components.
Table C.3 Data for Example C.2
i
x1
x2
x3
x4
i
x1
x2
x3
x4
1
57
60
90
128
18
47
63
76
103
2
50
64
80
138
19
55
61
62
118
3
40
55
124
111
20
53
59
72
106
4
55
64
92
69
21
61
64
59
108
5
51
60
100
111
22
62
58
62
95
6
41
61
107
122
23
48
58
74
101
7
55
66
85
112
24
56
62
61
123
8
53
64
88
98
25
45
69
59
137
9
64
67
73
115
26
54
60
57
144
10
61
56
83
99
27
39
56
68
116
11
48
60
89
112
28
46
49
71
114
12
44
62
89
112
29
45
59
61
135
13
49
70
74
104
30
46
67
71
118
14
56
62
72
94
31
51
66
64
134
15
46
65
79
119
32
43
55
70
135
16
55
64
70
105
33
44
66
98
113
17
49
60
80
113
34
50
57
66
125
From Table C.3, we have the correlation matrix shown in Table C.4. The eigenvalues
of the correlation matrix are 1.4178, 1.2831, 0.9148 and 0.3843, respectively. The first
three principal components have a total contribution of 90.4%, and hence we take k  3 .
Table C.5 shows their unit eigenvectors.
Table C.4 Correlation matrix
x1
x2
x3
x4
x1
1
0.1818
-0.3246
-0.3052
x2
0.1818
1
-0.0840
-0.0097
x3
-0.3246
-0.0840
1
-0.2859
x4
-0.3052
-0.0097
-0.2859
1
Table C.5 Unit eigenvectors
P1
P2
P3
u1
0.7194
0.2130
0.2748
u2
0.4222
-0.0031
-0.9018
u3
-0.5221
0.5632
-0.3038
u4
-0.1779
-0.7984
-0.1375
Table C.6 shows the correlation coefficients between principal components and
original variables. From the table, we have the observations:

P1 , P2 and P3 mainly reflect the effects of X1 , X 4 and X 2 .

The correlation coefficients in Table C.6 are proportional to the eigenvectors in Table
C.5 with the proportional coefficient being
i .
Table C.6 Correlation coefficients between principal components and original variables
x1
P1
P2
P3
0.8566
0.2412
0.2629
x2
0.5027
-0.0035
-0.8626
x3
-0.6217
0.6379
-0.2906
x4
-0.2118
-0.9044
-0.1315