# EigenVe2(M) = Eigenvectors of square matrix M by SVD

EigenVe2(M) = Eigenvectors of square matrix M by SVD:

### ```Ve = EigenVe2(M) // V = eigenvectors of M by SVD { P = CharPoly(M) // P = Characteristic polynomial of square matrix M R = PolRoots(P) // R = All of the roots of polynomial P f = 1 for (i = M.n − 1; i >= 1; i−−) // Delete roots that are duplicates if (r[i] == r[i + 1) Delete root i from r for (i = 1; i <= r.m; i++) { U = M for (j = 1; j <= M.n; j++) // Subtract i-th root from u[j, j] = u[j, j] − r[i] // all diagonal elements of U (U, W, V) = SVD(U) // Decompose U to (U, W, V) for (j = M.n; j >= 1; j−−) // Delete columns of V for W = 0 { if (w[j] != 0) Delete column j from matrix V } for (ci = 1; ci <= V.n; ci++)// Concat Columns of V to Ve { for (ri = 1; ri <= M.n; ri++) { Ve[ri, f] = v[ri, ci] } f++ } } for (ci = 1; ci <= M.n; ci++) // For all columns of Ve { di = 0 for (ri = n; ri >= 1; ri−−) // Find last non-zero element if (Ve[ri, ci] != 0) // in this column { di = Ve[ri, ci] break } if (di == MultiCD.cZero) // if column is all zeros { Delete column ci from matrix Ve } else { for (ri = 1; ri <= M.n; ri++) // Make last non-zero element Ve[ri, ci] = Ve[ri, ci] / di // equal to one } } } // EigenVe2```

See: Complex SVD -- Numerical Recipes Forum