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$$mt1 = \pmatrix{a& b& c\\b& d& e\\c& e&h}$$

For $mt1^T.mt2.mt1$ = d, where mt1 is an orthogonal matrix, mt2 is a real symmetric matrix and d is a diagonal matrix with eigenvalues of mt2 on the diagonal. How to find mt1 such that it is orthogonal and contains three variables using Mathematica. Any suggestion is appreciated!

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    $\begingroup$ This is true if mt1 is a matrix of eigenvectors of mt2. However, why do you think this is true for an arbitrary orthogonal matrix? $\endgroup$
    – Daniel Huber
    Commented May 17 at 15:59
  • $\begingroup$ @DanielHuber: Check the edits. $\endgroup$
    – SciJewel
    Commented May 18 at 5:51
  • $\begingroup$ mt1 is found via Eigensystem. This in turn can be parametrized by the coefficients of the characteristic polynomial of mt2. These are found as In[362]:= coeffs = Rest[Reverse[CoefficientList[-CharacteristicPolynomial[m, t], t]]] Out[362]= {-a - d - h, -b^2 - c^2 + a d - e^2 + a h + d h, c^2 d - 2 b c e + a e^2 + b^2 h - a d h} $\endgroup$
    – Daniel Lichtblau
    Commented Jun 17 at 19:09
  • $\begingroup$ @DanielLichtblau: Can you please write this as a proper answer in the following? $\endgroup$
    – SciJewel
    Commented Jun 19 at 0:12
  • $\begingroup$ Actually I'm finding I can only represent the d values in this way. $\endgroup$
    – Daniel Lichtblau
    Commented Jun 19 at 18:04

1 Answer 1

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You can use Eigensystem

In the following ev is the desired matrix. I leave it to the user to choose how to express and use ev. You can use ToRadical for Root objects. I will not print ev as it rather ugly.

The following finds ev and confirms orthogonal.

m = {{a, b, c}, {b, d, e}, {c, e, h}};
{val, ev} = {DiagonalMatrix[#1], Transpose[#2]} & @@ Eigensystem[m];
ev . Inverse[ev] // FullSimplify

yields: {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}

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  • $\begingroup$ orthogonal and "contains three variables" $\endgroup$
    – SciJewel
    Commented May 18 at 8:06
  • $\begingroup$ @SciJewel thank you. I think you need more constraints (that you have not specified). As such I cannot offer more advice than to try and give us updates $\endgroup$
    – ubpdqn
    Commented May 18 at 8:13

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