MAT211_Lecture16[1]

MAT211_Lecture16[1] - Definition MAT211 Lecture 16...

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Unformatted text preview: Definition MAT211 Lecture 16 Orthogonal transformations and orthogonal matrices A linear transformation from Rn to Rn is called orthogonal if it preserves the length vectors. In symbols, ||T(x)||=||x|| for all x in Rn. The matrix A of an orthogonal transformation is said to be an orthogonal matrix. Two examples: Questions: A rotation in R2 and a reflexion in Rn are orthogonal transformations. Are projections orthogonal transformations? What is the kernel of an orthogonal transformation? Example: Determine whether the matrices are orthogonal 1 1 1 0 1/√2 1 -1 1 1 1/√2 1 0 -2 1/√3 Theorem An orthogonal transformation T in Rn preserves angles; that is for each x and y in Rn the angle between x and y equals the angle between T(x) and T(y). Question: If a transformation preserves angles, is it orthogonal? Example: Determine whether the matrices are orthogonal Theorem A linear transformation from Rn to Rn is orthgonal if and only if the vectors (T(e1), T (e2),.., T(en)) form an orthonormal basis. A matrix A is orthogonal if an only if the columns of A form an orthonormal basis. Theorem 1 1 1 0 1/√2 1 -1 1 1 1/√2 1 0 -2 1/√3 Definition The product of orthogonal matrices is orthogonal. The transpose of an m x n matrix A, denoted At, is the n x m matrix which contains in the i,j entry the j,i entry of A. The inverse of an orthogonal matrix is orthogonal. An n x n matrix is symmetric if A=At. Example: Find At. Determine whether A is symmetric. Determine whether A is skew-symmetric 1 1 1 0 1 -1 1 1 1 2 1/√2 1 1/√2 An n x n matrix is skew-symmetric if A=-At. Example Consider W the subset of 3x3 matrices, formed by the skew-symmetric matrices. Is it a subspace of all 3 x 3 matrices? If so, what is the dimension? (What about the same problem with symmetric instead of skew-symmetric matrices) Theorem Example: Determine whether the matrices are orthogonal 1 /√3 1 /√2 If v and w are column vectors in Rn then the dot product of v and w equals the matrix product v.wt. An n x n matrix is orthogonal if and only if A.At = In. Theorem If A is an n x p matrix, and B is an p x m matrix then (A.B)t=Bt.At. If A is an orthogonal matrix then A-1=At. If a is an n x n invertible matrix, then At is also invertible and (At)-1=(A-1)t. ! 0 1/√2 1 /√3 /√2 ! 1 1/√2 1 /√3 0 -1 /√2 Theorem If u1, u2,..um is an orthonornal basis of a subspace V of Rn then the matrix of the projection onto V is Q.Qt where Q is the matrix with columns u1, u2,..um For any matrix A, rank(A)=rank(At) Review An n x n matrix is orthogonal if and only if A.At = In. A matrix is symmetric if A=At. A matrix is skew-symmetric if A=-At. Example 5.3 35 Find orthogonal transformation T from R3 to such that T(⅔,⅔,⅓)=(0,0,1) Find the matrix of the orthogonal projection of the line in Rn spanned by the vector. (1,1...1) Let A be the matrix of an orthogonal projection. Find A2 in two ways Geometrically Using the formula we saw. Given an example of a non-zero skew symmetric matrix A and compute A2 ...
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This note was uploaded on 09/14/2011 for the course MAT 211 taught by Professor Movshev during the Fall '08 term at SUNY Stony Brook.

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