Information Technology Reference
In-Depth Information
X
2
)
2
.
•
=
[
]−
[
]
Va r
(X)
E
(E
X
a
2
Va r
(X)
.
•
+
=
Va r
(aX
b)
•
+
=
Va r
(X
Y)
Va r
(X)
Va r
(Y )
,if
X
and
Y
are independent.
21.2 Linear Algebra and Matrix Computation
Linear algebra is an important branch of mathematics concerned with the study
of vectors, vector spaces (or linear spaces), and functions with vector input and
output. Matrix computation is the study of algorithms for performing linear algebra
computations, including the study of matrix properties and matrix decompositions.
Linear algebra and matrix computation are useful tools in machine learning and
information retrieval.
21.2.1 Notations
We start with introducing the following notations:
n
:the
n
-dimensional space of real numbers.
•
R
m
×
n
:
A
is a matrix with
m
rows and
n
columns, with real number elements.
•
A
∈
R
n
:
x
is a vector with
n
elements of real numbers. By default,
x
denotes a
column vector, which is a matrix with
n
rows and one column. To denote a row
vector (or a matrix with one row and
n
columns), we use the transpose of
x
, which
is written as
x
T
.
•
x
∈
R
•
x
i
∈
R
:the
i
th element of a vector
x
.
⎡
⎣
⎤
⎦
=
x
1
x
1
x
2
.
x
n
x
n
T
.
x
=
x
2
···
•
a
ij
∈
R
: the element in the
i
th row and
j
th column of a matrix
A
, which is also
written by
A
ij
,
A
i,j
,etc.
⎡
⎣
⎤
⎦
a
11
a
12
···
a
1
n
a
21
a
22
···
a
2
n
A
=
.
.
.
.
.
.
.
a
m
1
a
m
2
···
a
mn
•
a
j
or
A
:
,j
:the
j
th column of
A
.
=
a
1
a
n
.
A
a
2
···
