2] into being a sum of a diagonal and an antisymmetric matrix did not cause any loss of generality, because any real-valued tensor can be rotated into such a form (its symmetric part is

diagonalizable by an orthogonal transformation), and rotation does not influence the FTLEs.

Section 3 gives expressions for the entries of the matrix M = K * K, where K is the Krylov matrix, as a function of the eigenvalues and eigenvectors of A for diagonalizable matrices.

In this section we characterize the entries of M = U* U = K*K as functions of the eigenvalues and eigenvectors of A and of the starting vector v for diagonalizable matrices A.

1] is assumed to be

diagonalizable and hence diagonal due to (1.

1]A constructed in this way is

diagonalizable but [P.

The matrices and are then

diagonalizable by the discrete cosine transform (DCT), which we will denote by C, making the computation of [x.

By constructing the matrices in the mean as described above with the same matrix Q, we obtain a set of simultaneously

diagonalizable, commuting matrices.

We first discuss the case of a

diagonalizable matrix.

Assume A is

diagonalizable and let A = X[LAMBDA][X.

Assuming that P is

diagonalizable, the invariant subspace is spanned by a set of right eigenvectors of P, and due to the 0 or 1 structure of n the elements in these eigenvectors must be constant over the aggregates.

z] are

diagonalizable for z [not equal to] 0 with eigenvalues 0 and z, hence [B.

In this paper we limit ourselves to the case where A is

diagonalizable, with A = [XDX.

More precisely, for residual minimizing methods (like GMRES), a sufficient condition for fast convergence is that the preconditioned matrix is

diagonalizable with well-conditioned eigenvector matrix and with all of its eigenvalues clustered away from zero; see, e.

m]B is symmetric, and hence unitarily

diagonalizable, [I.

Hence, the Bernstein transformation matrix is not

diagonalizable.