In review of linear algebra I come across this phenomenon, the Google Book link is this:
What I do not understand is Lax tried to persuade us that "there is another way of parametrizing these matrices, namely by their eigenvectors and eigenvalues.". I do not see how one can do this explicitly in symmetric real matrices.
He argue that "the first eigenvector,corresponding to the largest eigenvalue, depends on $n-1$ parameters; the second...depends on $n-2$ parameters...to the $n-1$st eigenvector that depends on one parameter. " But suppose we solved the characteristic equation and already know the eigenvalues, then solving $AX=\lambda_{1}X$ would involve $\frac{n(n+1)}{2}$ entries in $A$ right? This question is too naive to be post elsewhere, so I venture to post in here.
Similarly I could not get why Lax wrote "...all the way down to the last simple eigenvector that depends on two parameters. The remaining eigenspace is then uniquely determined". What does he meant that "depending on two parameters" and "remaining eigenspace is then uniquely determined?"
Last I also found the "avoidance of crossing" graph he gave to be ambiguous. According to his explanation, "..as $t$ approaches certain values of $t$, a pair of adjacent eigenvalues $a_{1}t$ and $a_{2}t$ appear to be on a collision course; yet at the last minute they turn aside." And the reason is "..a line or curve lying in $N$ dimensional space will in general avoid intersecting a sufrace depending on $N_2$ parameters." My questions are:
- What does it mean "a line or curve will in general avoid intersecting a surface"? This feels totally via intuition and is not a rigorous proof, considering we could have knots, etc in the general space.
- I tried to draw some graphs in Mathematica, but surprising I found the two eigenvalues do collide. I do not know why.
Try the command:
ParametricPlot[{Eigenvalues[{{450, 505}, {458.6, 545.3}} + t {{0.325435, 0.4354334}, {0.4354353, 0.3453453}}], {t, t}}, {t, 1500, -1500}, PlotPoints -> 1500]
(for unknown reason I cannot upload photos of the graph).