3 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:
4 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.
5 Properties of QNote: Q is a notation to denote that some matrix A is orthogonalQT Q = I (note: this is not normally true for QQT)If Q is square, then QT = Q-1The Columns of Q form an orthonormal basis of RnThe 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)
6 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.
7 Example 1Is the rotation an orthogonal transformation?
8 Solution to Example 1Yes, because the vectors are orthogonal
9 Orthogonal transformations and orthogonal bases A linear transformation R from Rn to Rn is orthogonal if and only if the vectors form an orthonormal basis of RnAn nxn matrix A is orthogonal if and only if its columns form an orthonormal bases of Rn
10 Problems 2 and 4Which of the following matrices are orthogonal?
12 Properties of orthogonal matrices The product AB of two orthogonal nxn matrices is orthogonalThe inverse A-1 of an orthogonal nxn matrix A is orthogonalIf we multiply an orthogonal matrix times a constant will the result be an orthogonal matrix? Why?
13 Problems 6,8 and 10If A and B represent orthogonal matrices, which of the following are also orthogonal?6. -B8. A + B10. B-1AB
14 Solutions to 6, 8 and 10The product to two orthogonal matrices is orthogonalThe inverse of an orthogonal nxn matrix is orthogonal
24 Homework: p odd, allA 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: 10QStudent 1: You’re Welcome(Question: Is 10Q an orthogonal matrix?)
25 Proof – Q preserves orthogonality See next slide for a picture
26 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.