Unformatted text preview: × 4, where you can have orthogonal matrices which are simultane-ously rotations about diﬀerent axes, which cannot be written as a rotation about a single axis. 3. (3 points) Show that every matrix A ∈ M n × n with determinant 1 has a unique fac-torization A = RH where R is an orthogonal matrix with determinant 1 (ie, R T = R-1 ) and H is a symmetric, positive-deﬁnite matrix with determinant 1 (that is, H T = H and h x,Hx i ≥ 0 for all x ∈ R n with equality if and only if x = 0). You may assume that ev-ery real symmetric matrix is real diagonalizable–a fact which we will hopefully discuss in class. 4. (3 points) Consider R n with its standard inner product. Let U and T be symmetric n × n matrices. Suppose that UT = TU . Show that there exists an orthonormal basis for R n consisting of eigenvectors for both U and T . In other words, we may simultaneously diagonalize U and T . 1...
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This note was uploaded on 10/18/2011 for the course S 2010D taught by Professor Peters during the Spring '10 term at Columbia.
- Spring '10