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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.
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.