August 23, 2010
Introduction to Matrix Algebra
Definitions: Elementary Matrices
One of the most important forms of algebra used in statistics is matrix algebra, or more generally, linear algebra.
A
matrix
is simply an array of numbers.
Here are three examples of matrices:
A =
1
2
5
3
B =
4
3
1
5
2
5
C =
4
2
3
1
The numbers in the matrices are the
elements
of the matrix. The elements are identified by their rows first, then
columns. Connecting the general forms below to the specific matrices above, we note that a
11
= 2, b
21
= 3, b
32
= 5
and c
31
= 2.
A
(2x2)
=
21
11
a
a
22
12
a
a
B
(3x2)
=
31
21
11
b
b
b
32
22
12
b
b
b
C
(4x1)
=
41
31
21
11
c
c
c
c
The
dimension
of a matrix refers to the number of its rows and columns. Matrix A has dimension (2x2) and is
designated A
(2x2)
whereas B has dimension (3x2) and is designated B
(3x2)
. In general if any matrix M has n – rows
and k – columns, we denote it as M
(nxk)
. Matrices with one column (e.g., matrix C, above) are referred to as
“vectors.”
If you interchange the rows and columns of a matrix, you have its
transpose
. The transposes of the matrix B and
the vector C, denoted as B
T
and C
T
, respectively, are:
B
T
(2x3)
=
5
2
5
4
3
1
C
T
= [ 1
3
2
4]
A
square matrix
has the same number of columns as rows. Matrix A (above) is a square matrix. If, for each off
diagonal element of a square matrix, a
ij
= a
ji
, then the matrix is said to be a
symmetric matrix
. For example, the
matrix
S
(3x3)
=
6
5
3
5
4
2
3
2
1
is symmetric because s
21
= s
12
= 2, s
31
= s
13
= 3 and s
32
= s
23
= 5. Notice that symmetric matrix (S) is equal to its
transpose (S
T
). Two special symmetric matrices are the variancecovariance matrix (S) and the correlation
matrix (R).
In a square matrix such as S, the elements s
11
, s
22
and s
33
are the diagonal elements. A square matrix where all the
offdiagonal elements are zero is a
diagonal matrix
. An example is the following diagonal matrix, D:
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(3x3)
=
6
0
0
0
4
0
0
0
1
A special diagonal matrix is the
identity matrix
, where all the diagonal elements are equal to 1. Identity matrices
often function like “1’s” in scalar algebra (i.e., 1*4 = 1;
4/1 = 4;
4 * 1/4 = 1). The following is a (3x3) identity
matrix:
I
(3x3)
=
1
0
0
0
1
0
0
0
1
Elementary Matrix Operations: Addition and Multiplication
Matrices can be added and subtracted, multiplied and divided, but only if certain conditions are met. To
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 Spring '08
 Staff
 Linear Algebra, Invertible matrix

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