span { { rand the eigenspace corresponding to is span { the algebraic and geometric multiplicities are in this case any pair of vectors taken from and forms basis for figure shows how and are transformed to av and av respectively - - - - figure the dashed arrows are the unit eigenvectors (blueand (redof matrix their transformed values av and av are indicated by solid arrows semi-simple diagonalizable matrix for which the algebraic and geometric multiplicities of all its eigenvalues are the same is called semi-simple this is equivalent to the matrix being diagonalizablemeaning that there is matrix and diagonal matrix such that vdv- eigendecomposition to see that this so-called eigen-decomposition holdssuppose is semi-simple matrix with eigenvalues } lr { } | { dr let be the diagonal matrix whose diagonal elements are the eigenvalues of aand let be matrix whose columns are linearly independent eigenvectors corresponding to these eigenvalues thenfor each (eigenvalueeigenvectorpair (lv)we have av lv hencein matrix notationwe have av vdand so vdv- leftand right-eigenvectors the eigenvector as defined in the previous section is called right-eigenvectoras it lies on the right of in the equation av lv if is complex matrix with an eigenvalue lthen the eigenvalue' complex conjugate is an eigenvalue of ato see thisdefine :li and :li asince is an eigenvaluewe have det( applying the identity det(bdet( )we see that