The eigenvalues and eigenvectors of a matrix A are defined as the nonzero vectors x and scalars k such that

Ax = kx

or equivalently

(A - kI) x = 0

where I is the identity matrix.  The scaling of x is irrelevant, so we will consider two eigenvectors to be the same if they are multiples of each other.

The largest and smallest eigenvalues of a symmetric matrix have special interpretations: they are the maximum and minimum of the Rayleigh quotient

x'Ax / x'x

An n by n matrix will have n eigenvalues (although some may be repeated, i.e., may correspond to several non-collinear eigenvectors).  In particular, a two-by-two matrix has two eigenvalues.  We can find them using the characteristic polynomial: the equation

(A - kI) x = 0

for a nonzero vector x means that (A - kI) has rank less than 2, which means that its determinant is zero.  If we write A as

a	b
c	d

then

det(A - kI) = (a-k)(d-k) - bc = 0

which is a quadratic equation in k.  Simplifying, we have

k^2 - (a+d)k + ad - bc = 0

which has roots at

[(a+d) +- sqrt((a-d)^2 + 4bc)] / 2