eigenvalue

If an eigenvector is a special direction the matrix does not rotate, the eigenvalue is the exact factor by which it stretches that direction. We write it as lambda, and the whole relationship reads A*v = lambda*v: acting with the matrix is the same as just multiplying by the single number lambda.

The size and sign of lambda tell the story. lambda > 1 stretches the vector longer; 0 < lambda < 1 shrinks it; a negative lambda flips it to point the opposite way; and lambda = 0 means the matrix collapses that direction to zero, so v sits in the null space.

Example: for A = [[2,0],[0,3]], A*(1,0) = (2,0) = 2*(1,0), so lambda = 2 for the direction (1,0). To find eigenvalues in general you solve det(A - lambda*I) = 0.