Question

Could you please provide an example of these 3 decompositions on LAPACK, or just an idea how to use this library to solve them??

Eigen-value decomposition. 
Orthogonal decomposition.
Schur decomposition. 

Answer 1

Examples of eigenvalue problems are vibrations in mechanical systems; the eigenvalues are the natural frequencies and the eigenvectors are the normalized modes of vibration.

It turns out that PageRank is also just a huge eigenvalue decomposition. Page and Brin are billionaires because of it.

I don’t know what’s in LAPACK, but look for Jacobi, Householder, or Lanczos methods.

Orthogonal decomposition can be used to invert a special class of matrix:

http://en.wikipedia.org/wiki/Orthogonal_matrix

Here are the LAPACK docs:

http://www.netlib.org/lapack/lug/node39.html

Schur decomposition is similar to orthogonal decomposition, except for a diagonal matrix in the middle whose values are equal to the diagonal values of the matrix in question:

http://en.wikipedia.org/wiki/Schur_decomposition

I’ve never heard it called Schur decomposition, but here are the LAPACK docs for symmetric, real matricies:

http://www.netlib.org/lapack/lug/node48.html

The latter two are techniques for solving special classes of matricies.