Using mathematica to reduce a system of n equations and m variables

Hi guys,



I have a system of six equations and thirteen variables. I would like to reduce it to seven free parameters. I used Reduce to solve it but it is taking really long (one full day and no output yet) and I think there must be a better way. Here is my code

Code:

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Y = {{y1}, {y2}, {y3}}

T = {{t11, t12, t13}, {t21, t22, t23}, {t31, t32, t33}}

K = k*IdentityMatrix[3]



M = Y.Transpose[Y] + T.Transpose[T] - K %This is a symmetric matrix



Reduce[M[[1]][[1]]==0&&M[[1]][[2]]==0&&M[[1]][[3]]==0&&M[[2]][[2]]==0&&M[[2]][[3]]==0&&M[[3]][[3]]==0, Reals]

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What should I do to speed this up? I'm sure if I spent a few hours, I could do it by hand but surely, there is a simple way to do it in Mathematica or other software? Thanks!!