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I think this might salvage the situation: Assume that you do have a path parameterized by $t$ and that it is stationary for a while any time that the multiplicity of an eigenvalue increases. In the case $n=2$ the eigenvectors are (usually) perpendicular so we could represent them as four points on the unit circle separated by $\frac{\pi}{2}$ radians . Imagine time as an axis so the eigenvectors form four black paths traveling up a cylinder. Any time the matrix becomes a scalar multiple of the identity matrix you suddenly have the solid unit circle. As long as this is a band of some width you can arrange to leave the band in the appropriate configuration.

With larger $n$ and even more general matrices I think it would be about the same. So one point is that it can not be a function of merely where you are, but also where you were and where you will be next.

A related problem is constructive versions of the Fundamental Theorem of Algebra ( cribbed from a paper by Fred Richman which I recommend.) Let $\mathbb{A} \subset \mathbb{C}$ be the field of algebraic numbers (roots of polynomials with rational coefficients) Consider degree $n+1$ monic polynomials $x^{n+1}+\sum_0^na_iz^.i$ x^{n+1}+\sum_0^na_iz^i$They can be parameterized by their coefficient vectors$\mathbf{a}=(a_0,a_1,\cdots,a_n) \in \mathbb{A}^n$and by their "list" of roots$\boldsymbol\alpha=(\alpha_0,\cdots,\alpha_n) \in \mathbb{A}^n$ordered somehow. There is an obvious continuous map (uniformly bicontinuous on bounded sets) in one direction$\boldsymbol\alpha \to \prod(z-\alpha_i).$prod(z-\alpha_i)$ i.e. extract the coefficients using elementary symmetric functions. Is there a continuous mapping in the other? Read the paper (which gets into Dedikind cuts, extension to all of $\mathbb{C}$ and other matters.) As I recall, the correct target for in the space of roots should instead be multisets of algebraic numbers with an appropriate metric. A motivating example is $z^2-b$ with $b$ real. For $b$ close to $0$ we have a sudden shift from the two roots spanning a horizontal line to a vertical one.

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I think this might salvage the situation: Assume that you do have a path parameterized by $t$ and that it is stationary for a while any time that the multiplicity of an eigenvalue increases. In the case $n=2$ the eigenvectors are (usually) perpendicular so we could represent them as four points on the unit circle separated by $\frac{\pi}{2}$ radians . Imagine time as an axis so the eigenvectors form four black paths traveling up a cylinder. Any time the matrix becomes a scalar multiple of the identity matrix you suddenly have the solid unit circle. As long as this is a band of some width you can arrange to leave the band in the appropriate configuration.

With larger $n$ and even more general matrices I think it would be about the same. So one point is that it can not be a function of merely where you are, but also where you were and where you will be next.

A related problem is constructive versions of the Fundamental Theorem of Algebra ( cribbed from a paper by Fred Richman which I recommend.) Let $\mathbb{A} \subset \mathbb{C}$ be the field of algebraic numbers (roots of polynomials with rational coefficients) Consider degree $n+1$ monic polynomials $x^{n+1}+\sum_0^na_iz^.i$ They can be parameterized by their coefficient vectors $\mathbf{a}=(a_0,a_1,\cdots,a_n) \in \mathbb{A}^n$ and by their "list" of roots $\boldsymbol\alpha=(\alpha_0,\cdots,\alpha_n) \in \mathbb{A}^n$ ordered somehow. There is an obvious continuous map (uniformly bicontinuous on bounded sets) in one direction $\boldsymbol\alpha \to \prod(z-\alpha_i).$ Is there a continuous mapping in the other? Read the paper (which gets into Dedikind cuts, extension to all of $\mathbb{C}$ and other matters.) As I recall, the correct target for the roots should instead be multisets of algebraic numbers with an appropriate metric. A motivating example is $z^2-b$ with $b$ real. For $b$ close to $0$ we have a sudden shift from the two roots spanning a horizontal line to a vertical one.