ORTHOGONAL MATRICES 10/27/2002 Math 21b, O. KnillTRANSPOSE The transpose of a matrix A is the matrix (AT)ij= Aji. If A is a n × m matrix, then ATis am × n matrix. For square matrices, the transposed matrix is obtained by reflecting the 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) ~x ·~Ay = AT~x · ~y.b) (AB)T= BTAT.c) (AT)T= A.PROOFS.a) Because x · Ay =PjPixiAijyjandATx · y =PjPiAjixiyjthe two expressions arethe same by renaming i and j.b) (AB)kl=PiAkiBil. (AB)Tkl=PiAliBik=ATBT.c) ((AT)T)ij= (AT)ji= Aij.ORTHOGONAL MATRIX. A n × n matrix A is called orthogonal if ATA = 1. The corresponding lineartransformation is called orthogonal.INVERSE. It is easy to invert an orthogonal matrix: A−1= AT.EXAMPLES. The rotation matrix A =cos(φ) sin(φ)− sin(φ) cos(φ)is orthogonal because its column vectors havelength 1 and are orthogonal to each other. Indeed: ATA =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 the columns of the matrix A havelength 1 and are orthogonal. Indeed: 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:A~x · A~y = ~x · ~yProof: this is ahomework assignment: Hint: just look at the properties of the transpose.Orthogonal transformations preserve the length of vectors as wellas the angles between them.Proof. We have ||A~x||2= A~x · A~x = ~x · ~x ||~x||2. Let α be the angle between ~x and ~y and let β denote the anglebetween A~x and A~y and α the angle between ~x and ~y. Using A~x·A~y = ~x·~y we get ||A~x||||A~y|| cos(β) = A~x·A~y =~x · ~y = ||~x||||~y|| cos(α). Because ||A~x|| = ||~x||, ||A~y|| = ||~y||, this means cos(α) = cos(β). Because this propertyholds for all vectors we can rotate ~x in plane V spanned by ~x and ~y by an angle φ to get cos(α +φ) = cos(β + φ)for all φ. Differentiation with respect to φ at φ = 0 shows also sin(α) = sin(β) so that α = β.ORTHOGONAL MATRICES AND BASIS. A linear transformation A is orthogonal if and only if thecolumn vectors of A form an orthonormal basis. (That is what ATA = 1nmeans.)COMPOSITION OF ORTHOGONAL TRANSFORMATIONS. The composition of two orthogonaltransformations is orthogonal. The inverse of an orthogonal transformation is orthogonal. Proof. The propertiesof 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.The composition of a reflections at a plane with a reflection at an other plane is a rotation (the axis of rotationis the intersection of the planes).ORTHOGONAL PROJECTIONS. The orthogonal projection P onto a linear space with orthonormal basis~v1, . . . ,~vnis the matrixAAT, where A is the matrix with column vectors ~vi. To see this just translate theformula P~x = (~v1· ~x)~v1+ . . . + (~vn· ~x)~vninto the language of matrices: AT~x is a vector with components~bi= (~vi· ~x) and A~b is the sum of the~bi~vi, where ~viare the column vectors of A. Orthogonal projections are noorthogonal transformations in general!EXAMPLE. Find the orthogonal projection P from R3to the linear space spanned by ~v1=03415and~v2=100. Solution: AAT=0 13/5 04/5 00 3/5 4/51 0 0=1 0 00 9/25 12/250 12/25 16/25.WHY ARE ORTHOGONAL TRANSFORMATIONS USEFUL?• In Physics, Galileo transformations are compositions of translations with orthogonal transformations. Thelaws of classical mechanics are invariant under such transformations. This is a symmetry.• Many coordinate transformations are orthogonal transformations. We will see examples when dealingwith differential equations.• In the QR decomposition of a matrix A, the matrix Q is orthogonal. Because Q−1= Qt, this allows toinvert A easier.• Fourier transformations are orthogonal transformations. We will see this transformation later in thecourse. In application, it is useful in computer graphics (i.e. JPG) and sound compression (i.e. MP3).• Quantum mechanical evolutions (when written as real matrices) are orthogonal transformations.WHICH OF THE FOLLOWING MAPS ARE ORTHOGONAL TRANSFORMATIONS?:Yes NoShear in the plane.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 ~x 7→ A~x in the plane with A =cosh(α) sinh(α)sinh(α) cosh(α)Yes NoA translation.CHANGING COORDINATES ON THE EARTH. Problem: what is the matrix which rotates a point onearth with (latitude,longitude)=(a1, b1) to a point with (latitude,longitude)=(a2, b2)? Solution: The ma-trix which rotate the point (0, 0) to (a, b) a composition of two rotations. The first rotation bringsthe point into the right latitude, the second brings the point into the right longitude. Ra,b=cos(b) − sin(b) 0sin(b) cos(b) 00 0 1cos(a) 0 − sin(a)0 1 0sin(a) 0 cos(a). To bring a point (a1, b1) to a point (a2, b2), we formA = Ra2,b2R−1a1,b1.EXAMPLE: With Cambridge (USA): (a1, b1) =(42.366944, 288.893889)π/180 and Z¨urich (Switzerland):(a2, b2) = (47.377778, 8.551111)π/180, we get the matrixA =0.178313 −0.980176 −0.08637320.983567 0.180074 −0.01298730.028284 −0.082638
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