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isometries

# isometries - 1 Math 1b Practical Isometries March 4 2009 We...

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1 Math 1b Practical Isometries March 4, 2009 We have explained that a linear transformation of R n to itself is an isometry when its matrix is an orthogonal matrix. [The converse is also true.] An example of an isometry is the reﬂection through a subspace U of R n . The reﬂection of x through U is defined as follows: if x = u + w with u U and w U , then the reﬂection reﬂ U ( x ) is simply u w . It follows that reﬂ U ( x ) = 2 proj U ( x ) x . The proofs of the following two theorems are short, but understanding and reading them will be diﬃcult. We will not do all details in class. Theorem 1. Every n by n orthogonal matrix Q is the product of at most n matrices of reflections through hyperplanes. Proof: Recall that a hyperplane H is an ( n 1)-dimensional subspace. We will use R H to denote the matrix [reﬂ H ] of the reﬂection through H . So R H is an orthogonal matrix and R 2 H = I . Let U = { x : Q x = x } be the eigenspace of Q corresponding to eigenvalue 1, i.e.

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isometries - 1 Math 1b Practical Isometries March 4 2009 We...

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