5.3 Orthogonal Transformations

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5.3 Orthogonal Transformations

• This picture is from
• knot theory

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Recall
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The transpose of a matrix

• The transpose of a matrix is made by simply
  taking the columns and making them rows (and vice
  versa)
• Example

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Properties of orthogonal matrices(Q)

• An orthogonal matrix (probably better name would
  be orthonormal). Is a matrix such that each
  column vector is orthogonal to ever other column
  vector in the matrix. Each column in the matrix
  has length 1.
• We created these matrices using the Gram
  Schmidt process. We would now like to explore
  their properties.

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Properties of Q

• Note Q is a notation to denote that some matrix
  A is orthogonal
• QT Q I (note this is not normally true for
  QQT)
• If Q is square, then QT Q-1
• The Columns of Q form an orthonormal basis of Rn
• The transformation Qxb preserves length (for
  every x entered in the equation the resulting b
  vector is the same length. (proof is in the book
  on page 211)
• The transformation Q preserves orthogonality
• (proof on next slide)

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Orthogonal transformations preserve orthogonality

• Why? If distances are preserved then an angle
  that is a right angle before the transformation
  must still be right triangle after the
  transformation due to the Pythagorean theorem.

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Example 1

• Is the rotation an orthogonal transformation?

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Solution to Example 1

• Yes, because the vectors are orthogonal

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Orthogonal transformations and orthogonal bases

1. A linear transformation R from Rn to Rn is
   orthogonal if and only if the vectors form an
   orthonormal basis of Rn
2. An nxn matrix A is orthogonal if and only if its
   columns form an orthonormal bases of Rn

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Problems 2 and 4

• Which of the following matrices are orthogonal?

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Solutions to 2 and 4
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Properties of orthogonal matrices

• The product AB of two orthogonal nxn matrices is
  orthogonal
• The inverse A-1 of an orthogonal nxn matrix A is
  orthogonal
• If we multiply an orthogonal matrix times a
  constant will the result be an orthogonal matrix?
  Why?

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Problems 6,8 and 10

• If A and B represent orthogonal matrices, which
  of the following are also orthogonal?

6. -B
8. A B
10. B-1AB
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Solutions to 6, 8 and 10
The product to two orthogonal matrices is
orthogonal The inverse of an orthogonal nxn
matrix is orthogonal
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Properties of the transpose
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Symmetric Matrix

• A matrix is symmetric of AT A
• Symmetric matrices must be square.
• The symmetric 2x2 matrices have the form

If a AT -A, then the matrix is called skew
symmetric
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Proof of transpose properties
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Problems 14,16,18

• If A and B are symmetric and B is invertible.
  Which of the following must be symmetric as well?

14. B 16. A B
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Solutions to 14,16,18
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Problems 22 and 24

• A and B are arbitrary nxn matrices. Which of the
  following must be symmetric?

22. BBT 24. ATBA
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Solutions to 22 and 24
ATBA
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Problem 36
Find and orthogonal matrix of the form
_ 2/3 1/v2 a
_ 2/3 -1/v2 b 1/3 0
c

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Problem 36 Solution
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• Homework p. 218 1-25 odd, 33-37 all

A student was learning to work with Orthogonal
Matrices (Q) He asked his another student to help
him learn to do operations with them Student 1
What is 7Q 3Q? Student 2 10Q Student 1
Youre Welcome (Question Is 10Q an orthogonal
matrix?)
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Proof Q preserves orthogonality
See next slide for a picture
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Orthogonal Transformations and Orthogonal
matrices

• A linear transformation T from Rn to Rn is called
  orthogonal if it preserves the lengths of
  vectors.
• ? ? T(x) x
• ?
• If T(x) is an orthogonal transformation then we
  say that A is an orthonormal matrix.