1
MATRIX ALGEBRA
Why is it useful?

simple kvariables OLS expressions (and estimators other than OLS)

any data set you deal with is a big matrix

useful at the programming stage (i.e., for Gauss, Matlab – not for Stata)

it’s elegant!
A
matrix
is a rectangular set of elements ordered in rows and columns,
RC
R
C
C
C
R
a
a
a
a
a
a
a
a
...
...
...
...
....
...
...
...
1
2
22
21
1
12
11
A
In this matrix there are R rows and C columns.
If R=C the matrix is a
square
one, otherwise is a
rectangular
one. The diagonal of a square
matrix (\) is called “main (or principal) diagonal” (only square matrices have a main diagonal).
Sometimes this is indicated as
diag
(
A
). The (/) diagonal of a square matrix is called “opposite
diagonal”.
The generic element of the matrix is
a
ij
.
The first subscript (i) indexes the row where this element
is located. The second subscript indexes the column (j) where this element is located. So
a
25
is
the element in the 2
nd
row and 5
th
column of
A
. The “elements” of the matrix can be everything –
even abstract objects, letters, etc – but in econometrics they are numbers.
A matrix is said to be symmetric if
a
ij
=
a
ji
.
The dimension of a matrix is defined by the number of rows and columns it has. So a matrix with
R rows and C columns has dimension (R
C) (written under A – but can be omitted if not
needed). The dimension of a matrix
A
is indicated as
dim
(
A
).
A
vector
is a particular type of matrix with either one row (in which case is called a row vector)
or one column (column vector). A vector (with no denomination) is conventionally assumed to
be a column vector. Examples
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2
1
21
11
1
1
12
11
1
...
...
R
R
C
C
d
d
d
b
b
b
D
B
B
is a row vector and
D
a column vector.
A
scalar
is a particular type of matrix with just one row and one column, i.e., a singleton.
Some (apparently silly) properties of matrices/remarks about matrices
1) Suppose you have two matrixes
A
and
B
with generic elements
a
ij
and
b
ij
. Then
A
=
B
iff
a
ij
=
b
ij
for all i,j
.
2) Suppose you have two matrixes
A
and
B
with generic elements
a
ij
and
b
ij
. You can add them
or subtract them (
B
A
) iff they have the same dimension.
1
The resulting matrix
C
has a
generic element
ij
ij
ij
b
a
c
. Example. Define
2
6
5
8
4
2
5
6
2
1
,
1
4
0
7
,
3
2
5
1
B
A
B
A
C
B
A
However, A+C or AC are not defined (and neither are B+C or BC).
3)
A
is always a feasible operation as long as
is a scalar. The resulting matrix
B
=
A
will
have generic element
b
ij
=
a
ij
.
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 Spring '11
 Pistaferri,L
 Linear Algebra, Econometrics, Matrices, ... ..., Diagonal matrix, nn

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