[barbudorojo]

An example of eigenvalues: Suppose you take a polynomial of x and then you change x for -x, you obtain a new polynomial. Some polynomials doesn't change they are associated to eigenvalue 1, other change sign they are associated to eigenvalue -1, those are the only ways in which a polynomial gets transformed into a multiple of itself when changing sign. And what happens with the others polynomials? Well, you can write any of them as a sum of those with even exponents (those that doesn't change when changing the sign of x) and those with odd exponents (those are the ones that change sign), so any polynomial can be expressed as the sum of an even polynomial and an odd polynomial in a unique way. In the same way, taken the conjugate of a complex number the part that doesn't change is the real part (associate to eigenvalue 1) and the part that change sign is the imaginary part (associate to eigenvalue -1) and every complex number is written as the sum of its real and imaginary part. Take a matrix and computes its transpose, the matrix can be expressed as a sum of a symmetric matrix (corresponding to eigenvalue 1, doesn't change with the transpose operator) and an antisymmetric matrix (change the sign, associated to eigenvalue -1). Finally take for example any function of x and consider the function obtained when you change x to -x, then any function can be expressed in a unique way as the sum of a even function (corresponding to eigenvalue 1, that is doesn't change with that transformation) and an odd function (associate to eigenvalue -1, that is change sign). For example our familiar function exponential of x is the sum of the hyperbolic cosiness and the hyperbolic sinus.

This way you are near the Euler Formula:

e^x = cosh(x) + sinh(x) (real case)

e^it = cos(t) + i sin(t) (complex case)

I must add that in this example the transformation satisfies that applied two times is the identity, that is called involution A^2=1 and the only eigenvalues K are those that K^2=-1 that is 1 and -1.