Tagged Questions

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Find solutions for an differential equation system

I have a differential equation system $x_1'(t) = -x_2(t)$ $x_2'(t) = -x_1(t)$ I see that I can write $\dot{x} = Ax$ where $A = \begin{pmatrix}0 & -1 \\ -1 & 0\end{pmatrix}$ The complete ...
475 views

Why integration operator has no eigen values?

Let $V$ be the vector space of all functions from $\mathbb R$ into $\mathbb R$ which are continuous. Let $T$ be the linear operator on $V$ defined by $$(Tf)(x) = \int_0^x f(t) dt$$ Prove that ...
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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.
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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.
121 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 ...
499 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 ...
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871 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 ...
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. ...