3The transpose of a matrix The transpose of a matrix is made by simply taking the columns and making them rows (and vice versa)Example:
4Properties 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.
5Properties 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)
6Orthogonal 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.
7Example 1Is the rotation an orthogonal transformation?
8Solution to Example 1Yes, because the vectors are orthogonal
9Orthogonal 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
10Problems 2 and 4Which of the following matrices are orthogonal?
12Properties 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?
13Problems 6,8 and 10If A and B represent orthogonal matrices, which of the following are also orthogonal?6. -B8. A + B10. B-1AB
14Solutions to 6, 8 and 10The product to two orthogonal matrices is orthogonalThe inverse of an orthogonal nxn matrix is orthogonal
24Homework: 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?)
25Proof – Q preserves orthogonality See next slide for a picture
26Orthogonal 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.