北京邮电大学国际学院线性代数讲义Lecture 04
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7
Elementary Matrices
Lecture 4
Elementary Matrices & Partitioned Matrices
1
Overview
This lecture includes two parts. In the first part, we will use matrix multiplications to solve a linear system, rather than row operations. These special matrices are called elementary matrices. In the second part, we will think of a matrix as being composed of a number of sub-matrices. To do this, is always very useful. These sub-matrices are called as blocks.
2
Part One
Elementary Matrices
3
Equivalent Systems
Given an m n linear system Ax b , we can obtain an equivalent system by multiplying both sides of the equation by a nonsingular m m matrix M
E1 E2 Ek is nonsingular and
( E1 E2
Finish.
1 Ek )1 Ek
1 1 E2 E1
5
Elementary Matrices
Type I. Type I elementary matrices are obtained by interchanging two row of I .
4
Equivalent Systems
Theorem: If A and B are nonsingular n n matrices, then AB is also nonsingular and ( AB)1 B1 A1 . Proof.
( B 1 A1 ) AB B 1 ( A1 A) B B 1 B I ( AB )( B 1 A1 ) A( BB 1 ) A1 AA1 I It follows by induction that if E1 , , Ek are all nonsingular then the product
0 1 0 Example: E1 1 0 0 is an elementary matrix of type I. Let A be a 3 3 0 0 1 matrix. Then
0 1 0 a11 a12 a13 a21 a22 a23 a E1 A 1 0 0 a a a a a 22 23 12 13 11 21 0 0 1 a 31 a32 a33 a31 a32 a33 a11 a12 a13 0 1 0 a12 a11 a13 a AE1 a21 a22 a23 1 0 0 a a 22 21 23 a 31 a32 a33 0 0 1 a32 a31 a33
Ux c E2 E1 A and c Ek E2 E1b . The new system will be equivalent
E1 is nonsingular.
to the original provided that M Ek
The following theorem shows that product of nonsingular matrices is nonsingular.
Ax b MAx Mb
since these two system have same solutions. To obtain an equivalent system that is easy to solve, we can apply a sequence of nonsingular matrices E1 , , Ek to both sides of the equation Ax b and obtain where U Ek
6wenku.baidu.com
Elementary Matrices
Type II. Type II elementary matrices are obtained by multiplying a row of I by a nonzero constant.
1 0 0 Example: E 2 0 1 0 is an elementary matrix of type II. Then 0 0 3 a12 a13 1 0 0 a11 a12 a13 a11 0 1 0 a a a a a a 22 23 22 23 21 21 0 0 3 a 31 a32 a33 3a31 3a32 3a33 a11 a12 a13 1 0 0 a11 a12 3a13 a a a a 0 1 0 a 3 a 22 23 22 23 21 21 a 31 a32 a33 0 0 3 a31 a32 3a33
Elementary Matrices
Lecture 4
Elementary Matrices & Partitioned Matrices
1
Overview
This lecture includes two parts. In the first part, we will use matrix multiplications to solve a linear system, rather than row operations. These special matrices are called elementary matrices. In the second part, we will think of a matrix as being composed of a number of sub-matrices. To do this, is always very useful. These sub-matrices are called as blocks.
2
Part One
Elementary Matrices
3
Equivalent Systems
Given an m n linear system Ax b , we can obtain an equivalent system by multiplying both sides of the equation by a nonsingular m m matrix M
E1 E2 Ek is nonsingular and
( E1 E2
Finish.
1 Ek )1 Ek
1 1 E2 E1
5
Elementary Matrices
Type I. Type I elementary matrices are obtained by interchanging two row of I .
4
Equivalent Systems
Theorem: If A and B are nonsingular n n matrices, then AB is also nonsingular and ( AB)1 B1 A1 . Proof.
( B 1 A1 ) AB B 1 ( A1 A) B B 1 B I ( AB )( B 1 A1 ) A( BB 1 ) A1 AA1 I It follows by induction that if E1 , , Ek are all nonsingular then the product
0 1 0 Example: E1 1 0 0 is an elementary matrix of type I. Let A be a 3 3 0 0 1 matrix. Then
0 1 0 a11 a12 a13 a21 a22 a23 a E1 A 1 0 0 a a a a a 22 23 12 13 11 21 0 0 1 a 31 a32 a33 a31 a32 a33 a11 a12 a13 0 1 0 a12 a11 a13 a AE1 a21 a22 a23 1 0 0 a a 22 21 23 a 31 a32 a33 0 0 1 a32 a31 a33
Ux c E2 E1 A and c Ek E2 E1b . The new system will be equivalent
E1 is nonsingular.
to the original provided that M Ek
The following theorem shows that product of nonsingular matrices is nonsingular.
Ax b MAx Mb
since these two system have same solutions. To obtain an equivalent system that is easy to solve, we can apply a sequence of nonsingular matrices E1 , , Ek to both sides of the equation Ax b and obtain where U Ek
6wenku.baidu.com
Elementary Matrices
Type II. Type II elementary matrices are obtained by multiplying a row of I by a nonzero constant.
1 0 0 Example: E 2 0 1 0 is an elementary matrix of type II. Then 0 0 3 a12 a13 1 0 0 a11 a12 a13 a11 0 1 0 a a a a a a 22 23 22 23 21 21 0 0 3 a 31 a32 a33 3a31 3a32 3a33 a11 a12 a13 1 0 0 a11 a12 3a13 a a a a 0 1 0 a 3 a 22 23 22 23 21 21 a 31 a32 a33 0 0 3 a31 a32 3a33