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Mathematics
Linear and Abstract Algebra
General form of symmetric 3x3 matrix with only 2 eigenvalues
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[QUOTE="odietrich, post: 5470470, member: 585568"] Considering Orodruin's suggestion in more detail, I found that I can write the symmetric matrix ##\textbf{S}=\begin{pmatrix}s_1&s_2&s_3\\s_2&s_4&s_5\\s_3&s_5&s_6\end{pmatrix}## as ##v\textbf{1} + (u-v) (\textbf{v}_1\otimes\textbf{v}_1) = v\textbf{1} + (u-v) \begin{pmatrix}rr&rs&rt\\rs&ss&st\\rt&st&tt\end{pmatrix}## where the (first) eigenvector be ##\textbf{v}_1=(r,s,t)^T## with ##r^2+s^2+t^2=1##, so: $$\textbf{S} = v\textbf{1} + (u-v) \begin{pmatrix}r^2&rs&r\sqrt{1-r^2-s^2}\\rs&s^2&s\sqrt{1-r^2-s^2}\\r\sqrt{1-r^2-s^2}&s\sqrt{1-r^2-s^2}&1-r^2-s^2\end{pmatrix}$$ or (all together): $$\begin{pmatrix}s_1&s_2&s_3\\s_2&s_4&s_5\\s_3&s_5&s_6\end{pmatrix} = \begin{pmatrix} v+(u-v)r^2 & (u-v)rs & (u-v)r\sqrt{1-r^2-s^2}\\ (u-v)rs & v+(u-v)s^2 & (u-v)s\sqrt{1-r^2-s^2}\\ (u-v)r\sqrt{1-r^2-s^2} & (u-v)s\sqrt{1-r^2-s^2} & v+(u-v)(1-r^2-s^2)\end{pmatrix}.$$ Thus, I have expressed the symmetric matrix ##\textbf{S}## by a matrix parametrized by ##(u,v,r,s)##. Now, I would like to (partially) invert this and find some dependencies between the six parameters ##(s_1, \ldots, s_6)## based on this result. In theory, it must be possible to express e.g. ##s_3## and ##s_5## by the other matrix entries ##s_1, s_2, s_4, s_6##. I wonder if these dependencies can be found be staring long enough at these two matrices ... UPDATE (just to clarify): The last paragraph basically means that I would like to express e.g. ##s_3=(u-v)r\sqrt{1-r^2-s^2}## by an appropriate combination of the terms ##s_1, s_2, s_4, s_6##, i.e. by combining ##v+(u-v)r^2##, ##(u-v)rs##, ... [/QUOTE]
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Mathematics
Linear and Abstract Algebra
General form of symmetric 3x3 matrix with only 2 eigenvalues
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