Eigenvalues and Eigenvectors, Diagonalizable Matrices

For a given square matrix Ann our aim is to find and explore bases U of Rn or Cn in which the matrix representation Au = L 1 AglJ reveals much of the properties of the underlying linear transformation / that the generally dense standard matrix representation A = Ag 

Note that when solving linear systems Ax = b with real coefficient matrices A and real right-hand-side vectors b that are solvable, the solution vector x is real according to the row reduction process of subsection (B). 

This is not so for the matrix eigenvalue problem the eigenvalues (and eigenvectors) of real matrices n generally can only be found in the complex plane C (and in C ). The 

Some matrices are diagonalizable over C in the sense that they have a basis of - possibly complex - eigenvectors and these matrices can therefore be diagonalized by matrix similarity. For example, this is so for all n by n matrices with n distinct eigenvalues since eigenvectors for different eigenvalues are linearly independent. Other matrices such 

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