Unitary similarity invariant function preservers of Hermitian matrix products

We show that the spectral theorem – which we understand to be a statement that every self-adjoint matrix admits a certain type of canonical form under unitary similarity – admits analogues over other ∗ -algebras distinct from the complex numbers. If these ∗ -algebras contain nilpotents, then it is shown that there is a consistent way in which many classic matrix decompositions – such as the Singular Value Decomposition, the Takagi decomposition, the skew-Takagi decomposition, and the Jordan decomposition, among others – are immediate consequences of these. If producing the relevant canonical form of a self-adjoint matrix were a subroutine in some programming language, then the corresponding classic matrix decomposition would be a 1-line invocation with no additional steps. We also suggest that by employing operator overloading in a programming language, a numerical algorithm for computing a unitary diagonalization of a complex self-adjoint matrix would generalize immediately to solving problems like SVD or Takagi. While algebras without nilpotents (like the quaternions) allow for similar unifying behaviour, the classic matrix decompositions which they unify are never obtained as easily. In the process of doing this, we develop some spectral theory over Clifford algebras of the form C l p , q , 0 ( R ) and C l p , q , 1 ( R ) where the former is admittedly quite easy. We propose a broad conjecture about spectral theorems.

Let the set of all n× n complex matrices and let be the set of all n× n hermitian matrices. We study the norms on that are invariant under unitary similarities (abbreviate to u.s.i. norms), i.e., the norms N(⋅) that satisfy N (A) = N (UAU∗ )for all unitary U. An important subclass of the u.s.i. norms on is the collection of all unitarily invariant (abbreviate to u.i.) norms, i.e., the norms N(.) that satisfy N (A) = S (UA) = N (AV) for all unitary U.In this paper we extend a fundamental result of u.i. norms on to u.s.i. norms on . It turns out that the C-numerical radii play an important role in the theory. We also show that on the collection of all the C-numerical radii which are norms and the collection of all the u.i. norms are two disjoint subclasses ot u.s.i. norms. A characterization of u.s.i. norms on in terms of Schur-convex norm functions is given. Then we identify those u.s.i. norms on , which are induced by inner products. Finally, using the results obtained, we prove some inequalities related ...

The structure of a unital linear map on hermitian matrices with the property that it preserves the set of invertible hermitian matrices with fixed indefinite inertia is examined. It turns out that such a map is either a unitary similarity or a unitary similarity followed by a transposition (the case when the fixed inertia has equal number of positive and negative eigenvalues is excluded).

Let M be the complex linear space of all n×n complex matrices or the real linear space of all n×n hermitian matrices. A norm N on M is invariant under unitary similarities if for any AeM and for any unitary matrix U. For N′ equal to the numerical radius or the spectral norm, we study the best constants α and β, i.e., the largest α and the smallest β, such that The results are then applied to study the multiplicativity factors ν for N with respect to different products ○ on M, i.e., constants ν>0 satisfying The particular case when N is a C-numerical radius and ○ is the usual product is studied in detail. Some other multiplicativity properties of N are also considered.

On a condensed form for normal matrices under finite sequences of elementary unitary similarities

Commuting Hermitian matrices may be simultaneously diagonalized by a common unitary matrix. However, the numerical aspects are delicate. We revisit a previously rejected numerical approach in a new algorithm called ‘do-one-then-do-the-other’. One of two input matrices is diagonalized by a unitary similarity, and then the computed eigenvectors are applied to the other input matrix. Additional passes are applied as necessary to resolve invariant subspaces associated with repeated eigenvalues and eigenvalue clusters. The algorithm is derived by first developing a spectral divide-and-conquer method and then allowing the method to break the spectrum into, not just two invariant subspaces, but as many as safely possible. Most computational work is delegated to a black-box eigenvalue solver, which can be tailored to specific computer architectures. The overall running time is a small multiple of a single eigenvalue-eigenvector computation, even on difficult problems with tightly clustered eigenvalues. The article concludes with applications to a structured eigenvalue problem and a highly sensitive eigenvector computation.

Analogs of some classical theorems on commuting matrices are proved. The new theorems deal with unitary congruences rather than unitary similarities; commutation is replaced by concommutation, defined in the paper, whereas normal and Hermitian matrices are replaced by conjugate-normal and symmetric matrices, respectively. Bibliography: 5 titles.

We show that a matrix is a Hermitian positive semidefinite matrix whose nonzero entries have modulus 1 if and only if it is similar to a direct sum of all I's matrices and a 0 matrix via a unitary monomial similarity. In particular, the only such nonsingular matrix is the identity matrix and the only such irreducible matrix is similar to an all l's matrix by means of a unitary diagonal similarity. Our results extend earlier results of Jain and Snyder for the case in which the nonzero entries (actually) equal 1. Our methods of proof, which reiy on the so called principal submatrix rank property, differ from the approach used by Jain and Snyder.

On normal and structured matrices under unitary structure-preserving transformations

Let be the real linear space of n × n complex Hermitian matrices. The unitary (similarity) orbit of C ∈ is the collection of all matrices unitarily similar to C. We characterize those C ∈ such that every matrix in the convex hull of can be written as the average of two matrices in . The result is used to study spectral properties of submatrices of matrices in , the convexity of images of under linear transformations, and some related questions concerning the joint C-numerical range of Hermitian matrices. Analogous results on real symmetric matrices are also discussed.

Descriptions are given of multiplicative maps on complex and real matrices that leave invariant a certain function, property, or set of matrices: norms, spectrum, spectral radius, elementary symmetric functions of eigenvalues, certain functions of singular values, (p, q )n umerical ranges and radii, sets of unitary, normal, or Hermitian matrices, as well as sets of Hermitian matrices with fixed inertia. The treatment of all these cases is unified, and is based on general group theoretic results concerning multiplicative maps of general and special linear groups, which in turn are based on classical results by Borel - Tits. Multiplicative maps that leave invariant elementary symmetric functions of eigenvalues and spectra are described also for matrices over a general commutative field.

The Jordan cononical form of a product of a Hermitian and a positive semidefinite matrix

A simple generalisation of elementary hermitian matrices is given and its application considered in connexion with the decomposition of a complex matrix into the product of a unitary matrix and an upper triangular matrix with real diagonal elements, and in connexion with the unitary transformation of a hermitian matrix to symmetric tridiagonal form.

Let $H_n$, $n\ge3$, be the space of all $n\times n$ Hermitian matrices. Assume that a map $\phi:H_n\to H_n$ preserves commutativity in both directions (no linearity or bijectivity of $\phi$ is assumed). Then $\phi$ is a unitary similarity transformation composed with a locally polynomial map possibly composed by the transposition. The same result holds for injective continuous maps on $H_n$ preserving commutativity in one direction only. We give counter-examples showing that these two theorems cannot be improved or extended to the infinite-dimensional case.

Certain interesting classes of functions on a real inner product space are invariant under an associated group of orthogonal linear transformations. This invariance can be made explicit via a simple decomposition. For example, rotationally invariant functions on ${\bf R}^2 $ are just even functions of the Euclidean norm, and functions on the Hermitian matrices (with trace inner product) which are invariant under unitary similarity transformations are just symmetric functions of the eigenvalues. We develop a framework for answering geometric and analytic (both classical and nonsmooth) questions about such a function by answering the corresponding question for the (much simpler) function appearing in the decomposition. The aim is to understand and extend the foundations of eigenvalue optimization, matrix approximation, and semidefinite programming.