3 4 Day 2 Similar Matrices This is

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 = (v 1, v 2) 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 What assumptions does this method make about v 1 and v 2?

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 = (v 1, v 2) 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 = 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. 7 c 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 R 2 is given by _ I √ 3/2 -1/2 _ A= 1/2 √ 3/2 II) Start with the matrix S=. 8289. 9011. 8289 [ [ Use S-1 AS = B ] B= [ ] 6. 8459 - 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 a 2 + b 2 ≠ 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. "