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- What is the importance of eigenvalues eigenvectors?
the eigenvalues are real: our instruments tend to give real numbers are results :-) As a more concrete and super important example, we can take the explicit solution of the Schrodinger equation for the hydrogen atom In that case, the eigenvalues of the energy operator are proportional to spherical harmonics:
- All tricks to find eigenvalues in $3\\times 3$ in a faster way?
To check whether your found eigenvalues are correct, simply compare it to the trace of the matrix (as the sum of the eigenvalues equals the trace) Besides these pointers, the method you used was pretty certainly already the fastest there is
- Proof that the trace of a matrix is the sum of its eigenvalues
Trace is preserved under similarity and every matrix is similar to a Jordan block matrix Since the Jordan block matrix has its eigenvalues on the diagonal, its trace is the sum (with multiplicity) of its eigenvalues
- Compute the characteristic equation of a $3 \\times 3$ matrix
The characteristic equation is used to find the eigenvalues of a square matrix A First: Know that an eigenvector of some square matrix A is a non-zero vector x such that Ax = λx Second: Through standard mathematical operations we can go from this: Ax = λx , to this: (A - λI)x = 0
- Real life examples for eigenvalues eigenvectors
Eigenvectors are axes, eigenvalues are distances along these axes Eigenvectors are the base of dimensional reduction techniques like PCA (principal component analysis), extremely useful in situations where we want to reduce the number of dimensions to a more practical one
- Determine a matrix knowing its eigenvalues and eigenvectors
Writing the matrix down in the basis defined by the eigenvectors is trivial It's just $$ M=\left( \begin{array}{ccc} 1 0 0 \\ 0 -2 0 \\ 0 0 2 \end{array} \right) $$ Now, all we need is the change of basis matrix to change to the standard coordinate basis, namely: $$ S = \left( \begin{array}{ccc} 1 1 -1 \\ 0 1 2 \\ -1 1 -1 \\ \end{array} \right) $$ This is just the
- What are the Eigenvalues of - Mathematics Stack Exchange
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- linear algebra - The definition of simple eigenvalue - Mathematics . . .
There seem to be two accepted definitions for simple eigenvalues The definitions involve algebraic multiplicity and geometric multiplicity When space has a finite dimension, the most used is algebraic multiplicity As I am interested in the general case (where the dimension of space can be infinite) I will not speak of the characteristic
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