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Let A = {Aij}i,j∈I, where I is an index set, be a doubly indexed family of matrices, where Aij is ni × nj. For each i∈I, let Vi be an ni-dimensional vector space. We say A is reducible in the coupled sense if there exist subspaces, Ui ⊆ Vi, with Ui ≠{0} for at least one i∈I, and Ui ≠ Vi for at least one i, such that Aij(Uj) ⊆ Ui for all i, j. Let B = {Bij}i,j∈I also be a doubly indexed family of matrices, where Bij is mi times mj. For each i∈I, let Xi be a matrix of size ni times mi. Suppose Aij} Xj = Xi Bij for all i, j. We prove versions of Schur's lemma for A, mathcal B satisfying coupled irreducibility conditions. We also consider a refinement of Schur's lemma for sets of normal matrices and prove corresponding versions for A, mathcal B satisfying coupled normality and coupled irreducibility conditions. [ABSTRACT FROM AUTHOR]

In this paper we derive closed-form expressions for the coefficients of the residual and characteristic polynomials in the full orthogonalization method and GMRES iterative Krylov method for solving linear systems with diagonalizable matrices. The coefficients are given as functions of the eigenvalues and eigenvectors of the matrix A and of the right-hand side b. These results yield the residual vectors and the explicit solution of the optimization problem minp∈πk ||p(A)b||, where πk is the set of polynomials of degree k with a value 1 at the origin. In addition, the Ritz values and harmonic Ritz values can be written explicitly for the first four iterations of the Arnoldi algorithm. Moreover, from the coefficients of the characteristic polynomials, we obtain lower bounds for the distances of the eigenvalues of A to the Ritz and harmonic Ritz values. [ABSTRACT FROM AUTHOR]

The higher rank numerical range is closely connected to the construction of quantum error correction code for a noisy quantum channel. It is known that if a normal matrix A ∈ Mn has eigenvalue a1,…, an, then its higher rank numerical range Λk(A) is the intersection of convex polygons with vertices aji,…,ajn-k+1, where 1 ≤ j1 < ⋯ < jn-k+1 ≤ n. In this paper, it is shown that the higher rank numerical range of a normal matrix with m distinct eigenvalues can be written as the intersection of no more than max{m,4} closed half planes. In addition, given a convex polygon P, a construction is given for a normal matrix A ∈ Mn with minimum n such that Λk(A) = P. In particular, if P has p vertices, with p ≥ 3, there is a normal matrix A ∈ Mn with n ≤ max{p + k - 1, 2k + 2} such that Λk(A) = P. [ABSTRACT FROM AUTHOR]

A short recurrence for orthogonalizing Krylov subspace bases for a matrix A exists if and only if the adjoint of A is a low-degree polynomial in A (i.e., A is normal of low degree). In the area of iterative methods, this result is known as the Faber-Manteuffel theorem [V. Faber and T. Manteuffel, SIAM J. Numer. Anal., 21 (1984), pp. 352-362]. Motivated by the description by J. Liesen and Z. Strako?s, we formulate here this theorem in terms of linear operators on finite dimensional Hilbert spaces and give two new proofs of the necessity part. We have chosen the linear operator rather than the matrix formulation because we found that a matrix-free proof is less technical. Of course, the linear operator result contains the Faber-Manteuffel theorem for matrices. [ABSTRACT FROM AUTHOR]

We show that the adjoint A+ of a matrix A with respect to a given inner product is a rational function in A, if and only if A is normal with respect to the inner product. We consider such matrices and analyze the McMillan degrees of the rational functions r such that A+ = r(A). We introduce the McMillan degree of A as the smallest among these degrees, characterize this degree in terms of the number and distribution of the eigenvalues of A, and compare the McMillan degree with the normal degree of A, which is defined as the smallest degree of a polynomial p for which A+ = p(A). We show that unless the eigenvalues of A lie on a single circle in the complex plane, the ratio of the normal degree and the McMillan degree of A is bounded by a small constant that depends neither on the number nor on the distribution of the eigenvalues of A. Our analysis is motivated by applications in the area of short recurrence Krylov subspace methods. [ABSTRACT FROM AUTHOR]

We study necessary and sufficient conditions that a nonsingular matrix A can be B-orthogonally reduced to upper Hessenberg form with small bandwidth. By this we mean the existence of a decomposition AV=VH, where H is upper Hessenberg with few nonzero bands, and the columns of V are orthogonal in an inner product generated by a hermitian positive definite matrix B. The classical example for such a decomposition is the matrix tridiagonalization performed by the hermitian Lanczos algorithm, also called the orthogonal reduction to tridiagonal form. Does there exist such a decomposition when A is nonhermitian? In this paper we completely answer this question. The related (but not equivalent) question of necessary and sufficient conditions on A for the existence of short-term recurrences for computing B-orthogonal Krylov subspace bases was completely answered by the fundamental theorem of Faber and Manteuffel [SIAM J. Numer. Anal.}, 21 (1984), pp. 352--362]. We give a detailed analysis of B-normality, the central condition in both the Faber--Manteuffel theorem and our main theorem, and show how the two theorems are related. Our approach uses only elementary linear algebra tools. We thereby provide new insights into the principles behind Krylov subspace methods, that are not provided when more sophisticated tools are employed. [ABSTRACT FROM AUTHOR]