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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 skewsymmetric
1 1 1
0 1 1 1 1 1 2 1/√2 1 1/√2 An n x n matrix is skewsymmetric if A=At. Example
Consider W the subset of 3x3 matrices,
formed by the skewsymmetric 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 skewsymmetric
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 A1=At.
If a is an n x n invertible matrix, then At is
also invertible and (At)1=(A1)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 skewsymmetric 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 nonzero 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.
 Fall '08
 MOVSHEV
 Transformations, Vectors, Matrices

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