matrix
matrix代写 | math – 该题目是一个常规的matrix的练习题目代写, 是比较有代表性的matrix等代写方向
ECE 5253 – Applied Matrix Theory Problem Set 7 Professor Zhong-Ping Jiang
Problem 1.For the matrix
A=
1 0 1
0 2 0
0 0 1
identify the spacepg()and the principal verctors of grade 2. Solution. We have a given matrixAR^3 ^3. So,
det(IA) =det
1 0 1
0 2 0
0 0 1
= 0
which is 1 = 1 2 = 2 We know, (iIA)gp= 0 Now, for= 1,
(IA)^2 = 0
0 0 1
0 1 0
0 0 0
0 0 1
0 1 0
0 0 0
p= 0
0 0 0
0 1 0
0 0 0
p 1
p 2
p 3
=
0
0
0
So, P 2 ( 1 = 1) ={p^1 |p^1 =col(, 0 ,)} Now, for 2 ,
(2IA)^2 = 0
1 0 1
0 0 0
0 0 1
p 1
p 2
p 3
=
0
0
0
So, P 1 ( 2 = 2) ={p^2 |p^2 =col(0,,0)} Hence,
p^1 =
0
p^2 =
0
0
Problem 2. express the following vectors as unique representations of principle vectors found in Problem 1:
x=
2
9
84
, x=
0
9. 3
0
Solution. i) Given:
x=
2
9
84
ExpressXas unique representations of principle vectors found in problem 1, from problem 1,
p^1
0
, p^2 =
0
0
Then,
x= 2
1
0
0
9
0
1
0
+ 84
0
0
1
ii) Given:
x=
0
9. 3
0
Use same method,
x= 9. 3
0
1
0
, where p^2 =
0
0
and = 1
Problem 3.Can you transform the following matrix into a Jordan form:
A=
0
0 0
, 6 = 0
Solution.
A=
0
0 0
IA=
0
0 0
0 0 0
So, null(P 1 ) =< 1 , 0 , 0 >
(IA)^2 =
0
0 0
0 0 0
0
0 0
0 0 0
=
0 0 ^2
0 0 0
0 0 0
So, null(P 2 ) =< 1 , 0 , 0 >;< 0 , 1 , 0 >
(IA)^3 =
0 0 ^2
0 0 0
0 0 0
0
0 0
0 0 0
=
0 0 0
0 0 0
0 0 0
So, null(P 2 ) =< 1 , 0 , 0 >;< 0 , 1 , 0 >;< 0 , 0 , 1 > Hence,
v^1 =
0
0
1
v^2 = (AI)v^1 =
0
0 0
0 0 0
0
0
1
=
0
v^3 = (AI)v^2 =
0
0 0
0 0 0
0
=
^2
0
0
So,
P=
0 ^2
0 0
1 0 0
P^1 =
0 0 1
(^011) 0 ^2 ^120
Therefore,
P^1 AP=
0 0 1
(^01) ^10 ^2 ^120
0
0 0
0 ^2
0 0
1 0 0
=
0 0
1 0
0 1