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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


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