Orthogonal Matrices 3/9/2001 Math 21b, O. KnillHomework for Monday, March 11, 2001: Section 4.3, 2,6,8,18*,20,34cdefg*DEFINITION The transpose of a matrix A is the matrix (AT)ij= Aji. If A is a n × m matrix,then ATis a m × n matrix. For square matrices, the transposed matrix is obtained by reflectingthe matrix at the diagonal.EXAMPLES The transpose of a vector A =123is the row vector AT=1 2 3 .The transpose of the matrix1 23 4is the matrix1 32 4.A PROPERTY OF THE TRANSPOSE.a) If x, y are two vectors, then x · Ay = ATx · y. b) (AB)T= BTAT. c) (AT)T= A.a) Because x · Ay =PjPixiAijyjand ATx ·y =PjPiAjixiyjthe two expressions are the sameby renaming i and j.b) (AB)kl=PiAkiBil. (AB)Tkl=PiAliBik= ATBT.c) ((AT)T)ij= (AT)ji= Aij.DEFINITION. A n × n matrix A is called orthogonal if ATA = 1. The corresponding lineartransformation is called orthogonal.EXAMPLES. The rotation matrix A =cos(φ) sin(φ)− sin(φ) cos(φ)is orthogonal becauseATA =cos(φ) sin(φ)− sin(φ) cos(φ)·cos(φ) − sin(φ)sin(φ) cos(φ)=1 00 1.A reflection at a line is an orthogonal transformation because ATA =cos(2φ) sin(2φ)sin(2φ) − cos(2φ)·cos(2φ) − sin(2φ)sin(2φ) − cos(2φ)=1 00 1.FACTS. An orthogonal transformation preserves the dot product: Ax · Ay = x · y. Proof: we usethe above property of the transpose, we see that the left hand side is ATAx · y and because of theorthogonality property, this is x · y.Consequently, an orthogonal transformation preserves the length of vectors as well as the anglesbetween them. The reason is that We have ||Ax||2= Ax · Ax = x · x ||x||2. Let α be the anglebetween x and y and β denote the angle between Ax and Ay. Using a) and the property of the dotproduct, we get ||Ax||||Ay|| cos(β) = Ax · Ay = x · y = ||x||||y|| cos(β). Because of b), this meanscos(alpha) = cos(β).ORTHOGONAL MATRICES AND BASIS. A linear transformation A is orthogonal if andonly if the column vectors of A form an orthonormal basis. (That is what ATA = 1 means.)COMPOSITION OF ORTHOGONAL TRANSFORMATIONS. The composition of twoorthogonal transformations is orthogonal. The inverse of an orthogonal transformation is orthog-onal. Proof. The properties of the transpose give (AB)TAB = BTATAB = BTB = 1 and(A−1)TA−1= (AT)−1A−1= (AAT)−1= 1.EXAMPLES. The composition of two reflections at a line is a rotation.The composition of two rotations is a rotation.ORTHOGONAL PROJECTIONS. The orthogonal projection P onto a linear space with basisv1, . . . , vnis the matrix AAT, where A is the orthogonal matrix with column vectors vi.To see this just translate the formula P x = (w1· x)w1+ . . . + (wn· x)wninto the language ofmatrices: ATx is a vector with components bi= (wi· x) and Ab is the sum of the biwi, where wiare the column vectors of A.WHY DO WE CARE ABOUT ORTHOGONAL TRAFOS?• Galileo transformations in physics are compositions of translations with orthogonal transfor-mations.• Many coordinate transformations are orthogonal transformations.• In the QR decomposition of a matrix A, the matrix Q is orthogonal. Because Q−1= Qt,this allows to invert A easier.• Many transformations which have applications (i.e. Fourier transformation) are orthogonaltransformations.• Quantum mechanical evolutions (when written as real matrices) are orthogonal transforma-tions.WHICH OF THE FOLLOWING MAPS ARE ORTHOGONAL?:•Yes NoShear•Yes NoProjection in three dimensions onto a plane.•Yes NoReflection in two dimensions at the origin.•Yes NoReflection in three dimensions at a plane.•Yes NoDilation with factor 2.•Yes NoThe Lorenz boost in the plane.•Yes NoA
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