
Mathematical Formulation
Consider a transformation matrix , such that it transforms a vector into . Thus, we can write,
Recall that 2 vectors directed along same direction are simply scalar multiples of each other. e.g. and (Check their direction). We can write .
Now, suppose there exists a scalar , such that
Now this is interesting, because and are now related via. a matrix and a scalar (a number) . i.e. as well as .
Thus,
So,
Since RHS is a null vector, this forms a homogeneous system, which will have nontrivial solutions, when . On expanding the determinant, we will get values of , if is of the order . These values are known as the eigenvalues. In German, eigen means special.

More About Eigenvalues
I) The determinant is known as the characteristic determinant and the polynomial obtained on expanding the determinant is known as the characteristic polynomial. If matrix is of the order , the degree of polynomial is and hence, the matrix has eigenvalues, which may be distinct or identical.
The set of eigenvalues is known as the spectrum.
II) Trace
III)
This implies, if any one of the eigenvalues is , then is .
IV) Eigenvalues of are , where is a nonnegative integer.
V) Eigenvalues of and are same.
VI) Eigenvalues of are , where is any number.
VII) If is symmetric, then its eigenvalues are real.

Eigenvectors
Corresponding to each of the eigenvalues , there will be a vector, known as eigenvector. This is obtained by solving the system . As stated earlier, .
This forms a system of homogeneous equations, and to get , we solve the system.

Properties of Eigenvectors
These are the vectors, whose direction does not change under the transformation .
I) For an eigenvalue , if is an eigenvector, then is also an eigenvector.
II) If the eigenvalues are distinct, the eigenvectors are linearly independent.
III) If is symmetric, then the eigenvectors corresponding to 2 distinct eigenvalues are orthogonal.

Use of Eigenvalues and Eigenvectors
There are many applications of eigenvalues and eigenvectors. Finding natural frequencies of a system with multiple degrees of freedom is an example which is a typical mechanical engineering application. The equations of motion form a system of the kind . The eigenvalues of tell us its natural frequencies. The corresponding eigenvectors indicate the mode shapes.

Cayley Hamilton Theorem
It states that every matrix satisfies its characteristic equation, i.e. if , or , then
This theorem can be used to find higher powers of a matrix as well as its inverse.