101–110 of 167 posts
Does anyone know of an example of a simple physical system where eigenvalues have a physical interpretation? The examples I know of are all in quantum mechanics, which is a bit abstract for me.
I was blown away in my Digital Signal Processing (DSP) class that eigen "values" exist for certain systems in the form of "waves". Basicaly you put in a wave made from multiple sine and/or cosine waves through some function f(x) and the output is STILL a wave, though its frequency, amplitude and phase might change. Technicaly if I remember correctly this applies to all complex exponentials, since those can be rewritt…
Maybe I'm missing what's interesting about this, but a function like f(z) = 5z + 2 would output a wave with changed amplitude and phase when z = sin(x). That doesn't seem that interesting to me, so f(z) must have some other interesting properties?
Basicaly there exist some functions into which you can feed in sound waves and the output is guaranteed to still be a sound wave.
If you'd feed in a sound wave and if the function would corrupt it you would not be able to do any digital signal processing, since the output must be a wave.
Sound(wave) in -> Sound(wave) out, guaranteed to always be true.
Does anyone know of an example of a simple physical system where eigenvalues have a physical interpretation? The examples I know of are all in quantum mechanics, which is a bit abstract for me.
Markov probability matrix where the entries are probabilities of some physical event happening. The the eigenvectors will be the long term stable state probabilities.
I was blown away in my Digital Signal Processing (DSP) class that eigen "values" exist for certain systems in the form of "waves". Basicaly you put in a wave made from multiple sine and/or cosine waves through some function f(x) and the output is STILL a wave, though its frequency, amplitude and phase might change. Technicaly if I remember correctly this applies to all complex exponentials, since those can be rewritt…
Maybe I'm missing what's interesting about this, but a function like f(z) = 5z + 2 would output a wave with changed amplitude and phase when z = sin(x). That doesn't seem that interesting to me, so f(z) must have some other interesting properties?
f needs to be linear, but the function in your example is not linear.
However, there are quite interesting linear functions. Example: f(x(t)) = x(t-2) + 4dx/dt - \int_0^t 2x(s) ds
Earlier quoted context omitted.
Maybe I'm missing what's interesting about this, but a function like f(z) = 5z + 2 would output a wave with changed amplitude and phase when z = sin(x). That doesn't seem that interesting to me, so f(z) must have some other interesting properties?
It's even worse than you describe it! f needs to be linear, but the function in your example is not linear. However, there are quite interesting linear functions. Example: f(x(t)) = x(t-2) + 4dx/dt - \int_0^t 2x(s) ds
Earlier quoted context omitted.
In fact functions are just infinite-dimensional vectors. Almost all of the theory goes through unchanged. This is the basic idea of functional analysis.
Or rather size-of-their-domain-dimensional?
Does anyone know of an example of a simple physical system where eigenvalues have a physical interpretation? The examples I know of are all in quantum mechanics, which is a bit abstract for me.
> Does anyone know of an example of a simple physical system where eigenvalues have a physical interpretation? Yep, vibration modes. Vibration frequencies represent their eigenvalues while the shape that the structural system exhibits when subjected to said vibration corresponds to it's eigenvector. If a structural system is modelled as a linear elastic system it's possible to apply an eigendecomposition of that syst…
Actually, trying to understand how eigenmodes and eigenfrequencies — which I understand well — relate to eigenvalues and eigenvectors.
Earlier quoted context omitted.
> Does anyone know of an example of a simple physical system where eigenvalues have a physical interpretation? Yep, vibration modes. Vibration frequencies represent their eigenvalues while the shape that the structural system exhibits when subjected to said vibration corresponds to it's eigenvector. If a structural system is modelled as a linear elastic system it's possible to apply an eigendecomposition of that syst…
Does this relate to the normal modes or eigenmodes of a system? Actually, trying to understand how eigenmodes and eigenfrequencies — which I understand well — relate to eigenvalues and eigenvectors.
Yes. The eigenvalues and eigenvectors of an undamped harmonic oscillator are respectively the vibration frequency and vibration mode.
One major class of structural analysis techniques is modal analysis, which determines the vibration modes and corresponding frequencies of specific structural systems subjected to particular boundary conditions.