3.4 Day 2 Similar Matrices

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3.4 Day 2 Similar Matrices
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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.

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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?
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Problem 20 Solution 1
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Problem 20 Solution 2
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Problem 20 solution 3
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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
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Problem 22 solution 1
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Problem 22 Solution 2
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Problem 22 Solution 3
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Problem 29
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Problem 29 Solution
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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)

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Prove Transitive (other to prove in Homework)

• A B, B C
• Means
• AP PB
• BQ QC
• Multiplication yields
• APQ PBQ
• PBQ PQC

Hence APQ PQC However as PQ is a matrix We
have proved A C
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• 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 lt.8289 , .9011gt lt.9011, .8289 ,
  gt
• 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?)

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Relativity example solution part 1

• Recall a rotation in R2 is given by
• I v3/2 -1/2
• 1/2 v3/2
• II)\ Start with the matrix
• .8289 .9011
• .9011 . 8289

_

_
A

S

Use S-1AS B
6.8459 6.0001 - 6.0001 -5.1138
B
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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)

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Homework p .147 19-29 odd, 37Prove 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."