the.com/eigenvalues
the numbers that tell you what a matrix stretches, not where it points.
means a scalar that shows how much a matrix scales a special vector without changing its direction.
from from german eigen, meaning own or characteristic — coined because these values belong intrinsically to the transformation itself, not to any chosen coordinate system.
invariant identitysame eigenvalues no matter what basis you pick
trace shortcuttheir sum always equals the matrix trace
determinant linktheir product equals the determinant exactly
complex allowedreal matrices can still have complex eigenvalues
for instance
google pagerank — ranks web pages via the dominant eigenvector of a link matrix
quantum energy levels — schrodinger equation eigenvalues give an atom's allowed energies
vibration modes — bridge and building resonances are eigenvalues of stiffness matrices
principal component analysis — largest eigenvalues pick out the directions data varies most