Let $S(t)$ be a mapping on $[0,1]$ into the space of symmetric real matrix with usual topology, if $S$ is continuous on $[0,1]$,
then can we show the existence of continuous mappings $Q(t), \Lambda(t)$ where $Q(t)$ is real orthogonal and $\Lambda(t)$ is real diagonal such that $S(t)=Q(t)\Lambda(t)Q(t)^T$?
For each $t$, $S(t)$ can be diagonalized into this form, but how to show all these $Q(t)$ and $\Lambda(t)$ can be continuous? Any hint is much appreciated!
Oops, i see the conclusion is false when $S$ has repeated roots and a nasty counter example is found in wiki https://en.wikipedia.org/wiki/Eigenvalue_perturbation
Since eigenvalues are roots of $\det(S(t) - x I)$ with coefficients varying continuously, eigenvalues must be continuous (See https://en.wikipedia.org/wiki/Geometrical_properties_of_polynomial_roots#Continuous_dependence_on_coefficients).
Regarding eigenvectors, its a little tricky since for identity matrix, you can practically choose any unitary matrix for the diagonal decomposition and its not unique.
As mentioned in one of the comments, if eigenvalues are not repeated then the solution to $(S(t) - \lambda(t)I)x = 0$ is 1-dimensional and is a continuous function of $t$. So unitary matrix is also continuous. But if repeated to show the existence by choosing from the eigenspace such that it is continuous, we need to show that the solution as $t \rightarrow t^*$, $\lambda_1(t) \rightarrow \lambda$ and $\lambda_2(t) \rightarrow \lambda$ for $(S(t) - \lambda_i(t)I)v_i(t) = 0$ then eigenspace must tend to limit of $\langle v_1(t),v_2(t) \rangle$. I think the worst case will happen if $\lambda_1(t) = \lambda_2(t)$ happens infinitely often but not always in the neighbourhood of $t^*$.
Try, $$S(t) = t \times \begin{pmatrix} \cos(2\pi/ t) & \sin(2\pi/ t)\\ \sin(2\pi/ t) & 2\cos(2\pi/ t) \end{pmatrix}$$
which case eigen values $e_1$, $e_2$ for $t = 1/N$. Check for $t$ near $0$.
Numerical Calculations: $$t = 1/(N+0.25)$$ $$S(t) = t \begin{pmatrix} \cos(2\pi/ t) & \sin(2\pi/ t)\\ \sin(2\pi/ t) & 2\cos(2\pi/ t) \end{pmatrix}$$ $\lambda_1(t) = -1$, $v_1(t) = [-1,1]$
$$t = 1/(N-0.25)$$ $$S(t) = t \begin{pmatrix} \cos(2\pi/ t) & \sin(2\pi/ t)\\ \sin(2\pi/ t) & 2\cos(2\pi/ t) \end{pmatrix}$$ $\lambda_1(t) = -1$, $v_1(t) = [1,1]$
Now increase $N$ to a large value to prove the discontinuity at $0$.