Question

Where we used eigen values?

Answer 1

Answer:

analysis of financial data

Step-by-step explanation:

Answer 2

Answer:

A zillion different things. Eigenvalues are ubiquitously useful throughout math and science.

In matrix theory, the trace of a matrix is the sum of its eigenvalues and the determinant is the product of its eigenvalues.

The determinant can tell you about the volume of a little region transformed said matrix, whether the matrix is invertible, and the integral of that determinant can tell you about the volume of a region. That determinant also pops up as an invariant called the Norm in algebraic number theory and a useful creature called the Jacobian in differential and algebraic geometry.

The trace gives you a family of dear friends of mine called ‘characters’ that pop up all over representation theory, harmonic analysis, and number theory. They do everything from help understand theorems about the arithmetic of finite fields to write down analogues of waves so that you can use the Fourier transform. Those characters even correspond with the particles in the standard model!

What about the eigenvalues themselves? Well, they give you a great way of understanding the geometry of a linear transformation. Real distinct eigenvalues? Your matrix is diagonalizable. So you know in some basis its just a bunch of rescalings. Complex eigenvalues? Then there’s a rotation in there that you can compute.

What if you look at the eigenvalues of a differential operator? Then you get solutions to the equation by finding the corresponding eigenvectors! Got a (nice) dynamical system or a Markov chain? Then the Frobenius-Perron theorem guarantees you a unique, maximal real eigenvalue and your system is going to trend towards it as time goes on. What if you’re interested in quantum physics? Then the discrete eigenvalues of the Hamiltonian give you the possible energy levels of your quantum states and the eigenvectors themselves are the wave function of the state!

The moral of the story is that if you can phrase a problem in terms of matrices then often the eigenvalues are not only easily computable from the matrix, but they often tell you about important aspects of whatever problem you are studying. They provide both information in the eigenvalue itself and a great way to sort decompose the vectors in your space in a meaningful way.

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