0
votes
1answer
12 views

When finding the frequencies of normal modes, can you have a negative frequency?

Do you simply just consider the positive solutions? I tried a google search but didn't find anything quickly. The work I am studying is Lagrangian systems.
0
votes
0answers
17 views

how does PCA components change upon the addition of new data?

How does the PCA components change on addition of new data? e.g. d(PCA1(x))/d(var(x))? I am looking for any mathematical formulae and proof. Since it would be easy to understand.
5
votes
1answer
114 views

Prove that an eigenvector is the maximum of a symmetric matrix

Let $f : S^{n-1} \rightarrow \mathbb{R}, x \mapsto x^TAx$ ( A is a symmetric matrix), then an eigenvector $\xi$ of A is a local maximum of this function. We are supposed to prove this in 6 steps and ...
8
votes
5answers
371 views

Matrices with eigenvalues 0 and 1

How can you describe all $2\times 2$ matrices whose eigenvalues are 0 and 1? My attempt: I know that 0 and 1 has to be solutions of the characteristic polynomial. And I've considered some examples ...
1
vote
2answers
77 views

Eigenvalues and IVPs

So I have this question: Solve the initial value problem: $$\begin{align*} \frac{dx_1}{dt}&=3x_3-2x_4,\\ \frac{dx_2}{dt}&=-2x_3+3x_4,\\ \frac{dx_3}{dt}&=3x_1-2x_2,\\ ...
0
votes
1answer
329 views

Bessel Function in Sturm-Liouville problem

I have the following Sturm-Liouville problem: $$\frac{d^2 y}{dx^2}+\lambda x^2y=0,$$ where $y(0)=0$ and $y(1)=0$. I have solved this using MAPLE and found the exact solution to be: ...
1
vote
1answer
752 views

Mahalanobis Distance using Eigen-Values of the Covariance Matrix

Given the formula of Mahalanobis Distance: $D^2_M = (\mathbf{x} - \mathbf{\mu})^T \mathbf{S}^{-1} (\mathbf{x} - \mathbf{\mu})$ If I simplify the above expression using Eigen-value decomposition ...
0
votes
1answer
39 views

Prove diagonalizability of a continuous transform

We've been going over diagonalization in my linear algebra class, but we've only been dealing with matrices—nothing too complicated. All of a sudden this problem came along and blindsided me: Let V ...
0
votes
1answer
93 views

Eigenfunction of $(a(x) f^{II})^{II}= - \lambda^2f$

I need the eigenfunctions $f$ and eigenvalues $\lambda$ of $(a(x) f^{II}(x))^{II}= - \lambda^2f$ for a given $a(x)$. For $a(x)$ constant the solution is a combination of sin, cos, sinh and cosh. ...