Matrix Operations.pdf
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Matrix Operations
Yu Jiangsheng
Institute of Computational Linguistics
Peking University
September 26, 2002
Top i cs
1. Matrix Multiplication
2. Solving System of Linear System
3. Inverting Matrixes
4. Symmetric Positive-definitive Matrixes and
Least-squares Approximation
5. Winograd Theorem and AHU Theorem
1
Matrix
Amatrix
A
is, in fact, a sequence of arrays:
a
11
a
12
···
a
1
n
=
a
ij
m×n
a
21
a
22
···
a
2
n
A
=
.
a
m
1
a
m
2
···
a
mn
Note
From the viewpoint of transformation,
Ax
describes the rotation of
x
around 0 and
linear stretching.
2
Singular Matrix
linearly independent
→
row rank and column
rank
→
rank
→
full rank
↔
existence of in-
verse (nonsingular)
Definition 1
A
null vector
for a matrix
A
m×n
is a nonzero vector
x
such that
Ax
=0.
Homework 1
Amatrix
A
has full column rank
iff it has no null vector.
Homework 2
A square matrix is singular iff
it has null vector.
3
Determinant of Matrix
−
1) matrix
A
[
ij
]
obtained
by deleting the
i
th row and the
j
th column
of
A
.
Definition 3
The
determinant
of
A
n×n
is de-
j
=1
(
−
1)
1+
j
a
1
j
det(
A
[1
j
]
)f
n>
1
if
n
=1
det(
A
)=
(1)
Property 1
A
n×n
is singular iff det(
A
)=0.
4
Definition 2
The
ij
th
minor
of matrix
A
n×n
is the (
n
−
1)
×
(
n
fined recursively in terms of minors by
a
11
n
Plik z chomika:
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