3.4 Day 2 Similar Matrices
This is a hypercube
Two matrices A and B are similar if a
matrix P exists such that:
AP = PB
Or
This can be thought of as the same linear
transformation with regards to different bases. We
will explain this in detail and provide several
examples of this in section 4.3.
Problem 20
Find the matrix B of the linear transformation T(x) = Ax with
Respect to the basis B = (v1,v2) For practice solve the problem in
3 ways; a) use the formula B = S-1AS
b) Use a commutative diagram
c) And construct B column by column
What assumptions does this method make about v1 and v2?
Problem 20 Solution 1
Problem 20 Solution 2
Problem 20 solution 3
Problem 22
Find the matrix B of the linear transformation T(x) = Ax with
Respect to the basis B = (v1,v2) For practice solve the problem in
3 ways;
a) use the formula B = S-1 AS
b) Use a commutative diagram
c) And construct B column by column
Problem 22 solution 1
Problem 22 Solution 2
Problem 22 Solution 3
Problem 29
Problem 29 Solution
Properties of similarity
An nxn matrix A is similar to A (reflexive)
If A is similar to B then B is similar to A (symmetric)
If A is similar to B and B is similar to C then A is
similar to C (transitive)
Prove Transitive
(other to prove in Homework)
A ~B, B ~ C
Means
Hence APQ = PQC
However as PQ is a matrix
AP = PB
BQ = QC
Multiplication yields
APQ = PBQ
PBQ = PQC
We have proved A ~C
An Example from Special Relativity
Person A is standing still sees a space ship (S) fly in way that it moves along 30
degree arc on a ST diagram.
(a rotate 30 degrees around the origin ).
Part I Write a matrix that describes the rotation.
(note: this is review from chapter 2)
Part II Person B is flying by in a spaceship at .7c sees the path (Person S left a trail
with times indicated). But The axis is tilted as shown. Write a matrix that
describes the rotation from the point of view of person B. A basis for this new axis
is <.8289 , .9011> <.9011, .8289 , >
Part III What is the relationship between the matrices in parts I and II?
Part IV When person S was flying he left a trail of beacons of his path. Will person B
perceive this path as a circular arc on his ST Diagram?
(Is the matrix in part II still a rotation matrix?)
Relativity example solution part 1
Recall a rotation in R2 is given by
_
I
√3/2
_-1/2
A=
1/2 √3/2
II)\ Start with the matrix
S=
.8289
.9011
.9011
. 8289
[
[
Use S-1AS = B
]
]
B=
[
6.8459
- 6.0001
6.0001
-5.1138
]
Relativity example solution part 2
III The matrices are similar.
They represent the same linear transformation with
regard to different basis.
IV no a2 + b2 ≠ 1
(where a and b are components of a column or row
vector)
Homework p .147 19-29 odd, 37
Prove the symmetric and reflexive property for
similarity of matrices
A Mathematician, a Biologist and a Physicist are
sitting in a street cafe, watching people going in and
coming out of the house on the other side of the
street. First they see two people going into the house.
Time passes. After a while they notice three persons
coming out of the house.
The Physicist: "The measurement wasn't accurate".
The Biologists conclusion: "They have reproduced".
The Mathematician: "If exactly 1 person enters the
house then it will be empty again."