5.3 Orthogonal
Transformations
This picture is from
knot theory
Recall
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:
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.
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 Qx=b 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)
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.
Example 1
Is the rotation an orthogonal transformation?
Solution to Example 1
Yes, because the vectors are orthogonal
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
Problems 2 and 4
Which of the following matrices are
orthogonal?
Solutions to 2 and 4
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?
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
Solutions to 6, 8 and 10
The product to two orthogonal matrices is orthogonal
The inverse of an orthogonal nxn matrix is orthogonal
Properties of the transpose
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
Proof of transpose properties
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
Solutions to 14,16,18
Problems 22 and 24
A and B are arbitrary nxn matrices. Which of
the following must be symmetric?
22. BBT
24. ATBA
Solutions to 22 and 24
ATBA
Problem 36
Find and orthogonal matrix of the form
_
[ ]
2/3
2/3
1/3
1/√2
_
-1/√2
0
a
b
c
Problem 36 Solution
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: You’re Welcome
(Question: Is 10Q an orthogonal matrix?)
Proof – Q preserves orthogonality
See next slide for a picture
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.