10. Spectral decomposition Consider the spectral decomposition of matrix A=( 14 -2 (-2 3 A = Mjuju|+ Azuzuz with |Ail > 142l where vectors uj and uz are the columns of matrix P in the orthogonal diagonalization of A. Select all statements that are true: 1. u, and uz constitute an orthogonal basis for R2 2. 14.35 is a dominant einvenvalue of aA with one decimal place 3. Uzlly is an invertible matrix 4. none of the aboveThe Dirac delta function in three dimensions can be taken as the improper limit as a - 0 of the Gaussian function D(ox. v, ?) = (20) a exp -(+ + Consider a general orthogonal coordinate system specified by the surfaces u = constant, v = constant, w = constant, with length elements dull. du(V, dwiW in the three perpendicular directions. Show that 8(x - X ) = 8( 1 - 1) 6(0 - 0') 8(1 - W) . UVWV by considering the limit of the Gaussian above. Note that as a - 0 only the inlin itesimal length element need be used for the distance between the points in the exponent.(1 point) The following function is called a Gaussian Function. Gaussian functions are used in statistics, image processing, signal processing, and in mathematics to solve heat and diffusion equations. - (x -b) 2 f(x) = ae 202 Find the first and second derivative of the given Gaussian function when a = 3, b = 1, and c = 3 f' ( x ) = -(x-1) ((2e)(-((x-1)^2)/(9))/(3)) f"(x) = -(12e)^(-((x-1)^2)/(9))(-2x^2+4:(1 mark each) For each of the given matrices, select all decompositions that can be applied to it. You do not have to show your work for this question. 2 ) A = [61 O Diagonalization (i.e., A = PDP-1 with D diagonal) O Schur triangularization O Spectral decomposition O Singular value decomposition b ) B = 0 1.001 Diagonalization O Schur triangularization O Spectral decomposition Singular value decomposition c) C is the 75 x 75 matrix with every entry equal to 1. Diagonalization Schur triangularization O Spectral decomposition Singular value decomposition d) D = 0 1 1 Diagonalization OSchur triangularization Spectral decomposition Singular value decomposition