What is the importance of eigenvalues/eigenvectors?
2011年2月23日 · 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:
How to intuitively understand eigenvalue and eigenvector?
The effect of the action by this matrix is high for large (absolute) eigenvalues and less for small (absolute) eigenvalues. Hence, the directions/vectors along which this action is high are called the principal directions or principal eigenvectors. The corresponding eigenvalues are called the principal values.
Are the eigenvalues of $AB$ equal to the eigenvalues of $BA$?
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Inverse matrix’s eigenvalue? - Mathematics Stack Exchange
2023年12月2日 · What is a necessary and sufficient condition for all the eigenvalues of a non-symmetric matrix to be positive? 1 Why is the complex eigenvalue $\lambda = a - bi$ only usable in this theorem?
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.
What is the difference between "singular value" and "eigenvalue"?
2017年4月21日 · I've heard the term "singular value" applied to any value for which $(A-\lambda I)^{-1}$ either does not exist or is not continuous, while eigenvalues refer only to those values for which $(A-\lambda I)^{-1}$ does not exist. $\endgroup$
Proof that the trace of a matrix is the sum of its eigenvalues
2013年10月31日 · 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.
Eigenvalues of $A$ and $A - Mathematics Stack Exchange
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Finding the rank of the matrix directly from eigenvalues
$\begingroup$ @LGezelis The restriction that an eigenvector need not be 0 is not necessary with the way I defined the terms, and, I want $0$ to be an eigenvector, so I can define the eigenspace as the set of all eigenvectors and it will be a subspace.
A simple explanation of eigenvectors and eigenvalues with 'big …
2011年5月3日 · Eigenvectors and eigenvalues play a role when we want to compress tire age and tire wear since they are so plain. Imagine a 2d scatterplot with points on a straight diagonal line. This 2 dimensional straight line can be compressed into one dimension without much data loss.