Eigenvector matrix decomposition and basis sets

For a given N x N matrix, when does the set of eigenvectors form a complete basis such that any v eC can be expressed as a hnear combination of eigenvectors  

Can we express a matrix as a decomposition involving matrices that comprise eigenvalues and eigenvectors  

Are there special classes of matrices for which the eigenvalues can be proved to be real, or for which the set of eigenvectors is not only linearly independent, but also orthogonal  

Such questions may seem abstract, but they are in fact very important in practice. Above, our proof of convergence of Jacobi s method is based upon the assumed existence of a complete eigenvector basis for — A). 

Because we caimot assume that the eigenvectors and eigenvalues are real, even for a real matrix, we consider the general case where A, e C, e C[, and is an TV x complex matrix. 

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