• On the contrary, if the matrix depends upon two or more parameters, the eigenvalues are at most Lipschitz when crossing happens, and the eigenvectors cannot be chosen continuously. A typical example is $$(s,t)\mapsto\begin{pmatrix} s & t \\ t & -s \end{pmatrix},$$ whose eigenvalues are $\pm\sqrt{s^2+t^2}$. Up to the shift by $I_2$, Quas' example is just a Lipschitz (but not piecewise $C^1$) C^1$section of this two-parameters example, and it inherits its lack of continuous selection of eigenvectors. • Likewise, if analyticity is dropped, a$C^\infty$-example by Rellich shows that eigenvectors need not be continuous functions of a single parameter. Of course, Quas' example can be recast as a$C^\infty$one, by flatening the parametrisation at$t=0$, say by replacing$t$by$s$such that$t={\rm sgn}(s)\cdot e^{-1/s^2}$. Side remark: Kato's result is only local. If the domain is not simply connected, it could happen that a global continuous selection of eigenvectors is not possible. This is classical in the exemple above if you restrict to the unit circle$s^2+t^2=1$; then the eigenvalues$\pm1$are global continuous functions, but when following an eigenvector, it experiences a flip$v\mapsto -v$as one makes one turn. 2 added 139 characters in body The example given by Anthony Quas reveals a phenomenon discussed in Kato's book Perturbation Theory for Linear Differential Operators. The point is the following: • If the symmetric matrix depends smoothly ($C^k$or analytic) analytically upon one parameter, then you can follow smoothly analytically its eigenvalues and its eigenvectors. Notice that this requires sometimes that the eigenvalues cross. When this happens, the largest eigenvalues, as the maximum of smooth functions, is only Lipschitz. • On the contrary, if the matrix depends upon two or more parameters, the eigenvalues are at most Lipschitz when crossing happens, and the eigenvectors cannot be chosen continuously. A typical example is $$(s,t)\mapsto\begin{pmatrix} s & t \\ t & -s \end{pmatrix},$$ whose eigenvalues are$\pm\sqrt{s^2+t^2}$. Up to the shift by$I_2$, Quas' example is just a Lipschitz (but not$C^1$) section of this two-parameters example, and it inherits its lack of continuous selection of eigenvectors. • Likewise, if analyticity is dropped, a$C^\infty$-example by Rellich shows that eigenvectors need not be continuous functions of a single parameter. Side remark: Kato's result is only local. If the domain is not simply connected, it could happen that a global continuous selection of eigenvectors is not possible. This is classical in the exemple above if you restrict to the unit circle$s^2+t^2=1$; then the eigenvalues$\pm1$are global continuous functions, but when following an eigenvector, it experiences a flip$v\mapsto -v$as one makes one turn. 1 The example given by Anthony Quas reveals a phenomenon discussed in Kato's book Perturbation Theory for Linear Differential Operators. The point is the following: • If the symmetric matrix depends smoothly ($C^k$or analytic) upon one parameter, then you can follow smoothly its eigenvalues and its eigenvectors. Notice that this requires sometimes that the eigenvalues cross. When this happens, the largest eigenvalues, as the maximum of smooth functions, is only Lipschitz. • On the contrary, if the matrix depends upon two or more parameters, the eigenvalues are at most Lipschitz when crossing happens, and the eigenvectors cannot be chosen continuously. A typical example is $$(s,t)\mapsto\begin{pmatrix} s & t \\ t & -s \end{pmatrix},$$ whose eigenvalues are$\pm\sqrt{s^2+t^2}$. Up to the shift by$I_2$, Quas' example is just a Lipschitz (but not$C^1$) section of this two-parameters example, and it inherits its lack of continuous selection of eigenvectors. Side remark: Kato's result is only local. If the domain is not simply connected, it could happen that a global continuous selection of eigenvectors is not possible. This is classical in the exemple above if you restrict to the unit circle$s^2+t^2=1$; then the eigenvalues$\pm1$are global continuous functions, but when following an eigenvector, it experiences a flip$v\mapsto -v\$ as one makes one turn.