Criterion of unitary similarity for upper triangular matrices in general position

Each square complex matrix is unitarily similar to an upper triangular matrix with diagonal entries in any prescribed order. Let A and B be upper triangular n-by-n matrices that (i) are not similar to direct sums of matrices of smaller sizes, or (ii) are in general position and have the same main diagonal. We prove that A and B are unitarily similar if and only if ||h(A_k)||=||h(B_k)|| for all complex polynomials h(x) and k=1, 2, . . , n, where A_k and B_k are the principal k-by-k submatrices of A and B, and ||M|| is the Frobenius norm of M.Comment: 16 page

Open problems on GKK tau-matrices

We propose several open problems on GKK tau-matrices raised by examples showing that some such matrices are unstableComment: To appear in Linear Algebra and its Applications (LAA

A complete unitary similarity invariant for unicellular matrices

We present necessary and sufficient conditions for an n\times n complex matrix B to be unitarily similar to a fixed unicellular (i.e., indecomposable by similarity) n\times n complex matrix AComment: 16 page

Sensitivity and Dynamic Distance Oracles via Generic Matrices and Frobenius Form

Algebraic techniques have had an important impact on graph algorithms so far. Porting them, e.g., the matrix inverse, into the dynamic regime improved best-known bounds for various dynamic graph problems. In this paper, we develop new algorithms for another cornerstone algebraic primitive, the Frobenius normal form (FNF). We apply our developments to dynamic and fault-tolerant exact distance oracle problems on directed graphs. For generic matrices $A$ over a finite field accompanied by an FNF, we show (1) an efficient data structure for querying submatrices of the first $k\geq 1$ powers of $A$, and (2) a near-optimal algorithm updating the FNF explicitly under rank-1 updates. By representing an unweighted digraph using a generic matrix over a sufficiently large field (obtained by random sampling) and leveraging the developed FNF toolbox, we obtain: (a) a conditionally optimal distance sensitivity oracle (DSO) in the case of single-edge or single-vertex failures, providing a partial answer to the open question of Gu and Ren [ICALP'21], (b) a multiple-failures DSO improving upon the state of the art (vd. Brand and Saranurak [FOCS'19]) wrt. both preprocessing and query time, (c) improved dynamic distance oracles in the case of single-edge updates, and (d) a dynamic distance oracle supporting vertex updates, i.e., changing all edges incident to a single vertex, in $\tilde{O}(n^2)$ worst-case time and distance queries in $\tilde{O}(n)$ time.Comment: To appear at FOCS 202

On the number of invariant polynomials of matrix commutators

We study the possible numbers of nonconstant invariant polynomials of the matrix commutator XA - AX, when X varies.Praxis ProgramCAU

Polynomial traces and elementary symmetric functions in the latent roots of a non-central Wishart matrix

Hypergeometric functions and zonal polynomials are the tools usually addressed in the literature to deal with the expected value of the elementary symmetric functions in non-central Wishart latent roots. The method here proposed recovers the expected value of these symmetric functions by using the umbral operator applied to the trace of suitable polynomial matrices and their cumulants. The employment of a suitable linear operator in place of hypergeometric functions and zonal polynomials was conjectured by de Waal in 1972. Here we show how the umbral operator accomplishes this task and consequently represents an alternative tool to deal with these symmetric functions. When special formal variables are plugged in the variables, the evaluation through the umbral operator deletes all the monomials in the latent roots except those contributing in the elementary symmetric functions. Cumulants further simplify the computations taking advantage of the convolution structure of the polynomial trace. Open problems are addressed at the end of the paper

On the Non-Symmetric Spectra of Certain Graphs

The study of eigenvalue list multiplicities of matrices with certain graphs has appeared in volumes for symmetric real matrices. Very interesting properties, such as interlacing, equivalent geometric and algebraic multiplicities of eigenvalues, and Parter-Weiner-Etc. Theory drive the study of symmetric real matrices. We diverge from this and analyze non-symmetric real matrices and ask if we can attain more possible eigenvalue list multiplicities. We fully describe the possible algebraic list multiplicities for matrices with graphs $P_n, S_n, K_n$ and $K_n-K_m$