Re: Solving matrix equations

*To*: mathgroup at smc.vnet.net*Subject*: [mg114608] Re: Solving matrix equations*From*: Ray Koopman <koopman at sfu.ca>*Date*: Fri, 10 Dec 2010 02:29:36 -0500 (EST)*References*: <idqcrv$iv0$1@smc.vnet.net>

On Dec 9, 3:00 am, florian.mau... at schott.com wrote: > Hi everybody, > > can anyone help me in solving the following question: > > For a symmetric 4x4 matrix m which is of rank 4-1=3 there exist > 4-1=3 vectors vi (v1, v2, v3; each vector vi consisting of four > elements) which solve the equations > > vi.m.vj==1 (where i=j) > vi.m.vj==0 (where i#j) > > m={{435.525, -272.311, -107.660, -55.554}, {-272.311, > 441.083, -109.543, -59.229}, {-107.660, -109.543, > 244.850, -27.647}, {-55.554, -59.229, -27.647, 142.430}} > > How to calculate the vectors vi? I was told I can find the vectors > vi by application of the Gram-Schmidt orthogonalization procedure > (i.e. "Orthogonalize") but the vectors caculated with Orthogonalize > do not fullfil the above equations. > > Thanks in advance for your support > > Many regards > > Mr.Mason Transpose[{e,u} = Eigensystem[m]] v = Most@u / Sqrt[Most@e]; Chop[v.m.Transpose[v]] == IdentityMatrix[3] {{710.656, {0.701909, -0.712229, 0.0049454, 0.00537469}} {367.759, {-0.454622, -0.44173, 0.761601, 0.134751}} {185.473, {-0.225039, -0.218185, -0.412238, 0.855461}} {7.10543*^-14, {-0.5, -0.5, -0.5, -0.5}}} True