Several aspects of the classification problem in linear algebra are considered: classification of pairs of mutually annihilating operators, classification of matrices that are self-congruent only via matrices of determinant one, criterion of unitary similarity for upper triangular matrices in general position and normal matrices, simultaneous unitary equivalences, and reduction of a pair of skew-symmetric matrices to its canonical form under congruence.
Pairs (A,B) of mutually annihilating operators AB = BA = 0 on a finite dimensional vector space over an algebraically closed field were classified by I.Gelfand and V.Ponomarev by method of linear relations. The classification of (A,B) over any field was derived by L.Nazarova, A.Roiter, V.Sergeichuk, and V.Bondarenko from the classification of finitely generated modules over a dyad of two local Dedekind rings. It is given canonical matrices of (A,B) over any field in an explicit form and our proof is constructive: the matrices of (A,B) are sequentially reduced to their canonical form by similarity transformations (A, B) ↦ (S^(-1)AS,S^(-1)BS).
D.Docovi ́c and F. Szechtman considered a vector space V endowed with a bilinear form. They proved that all isometries of V over a field F of characteristic not 2 have determinant 1 if and only if V has no orthogonal summands of odd dimension (the case of characteristic 2 was also considered). Their proof is based on Riehm’s classification of bilinear forms. E. Coakley, F. Dopico, and R. Johnson gave another proof of this criterion over R and C using Thompson’s canonical pairs of symmetric and skew-symmetric matrices for congruence. Let M be the matrix of the bilinear form on V. It is given another proof of this criterion over F using our canonical matrices for congruence and obtain necessary and sufficient conditions involving canonical forms of M for congruence, of (M^T, M) for equivalence, and of M^(-T) M (if M is nonsingular) for similarity.
Each square complex matrix is unitarily similar to an upper triangular matrix with diagonal entries in any prescribed order. Let A=[a_ij] and B=[b_ij] be upper triangular n × n matrices that
• are not similar to direct sums of matrices of smaller sizes, or
• are in general position and have the same main diagonal.
It is proved that A and B are unitarily similar if and only if
∥h(A_k)∥ = ∥h(B_k)∥ for all h ∈ C[x] and k = 1,...,n,
where A_k∶=[a_ij]_(i,j=1)^k and B_k∶=[b_ij]_(i,j=1)^k are the leading principal k×k submatrices of A and B, and ∥ ⋅ ∥ is the Frobenius norm.
It is given several criteria of unitary similarity of a normal matrix A and any matrix B in terms of the Frobenius and spectral norms, characteristic polynomi- als, and traces of matrices.
Let S_1, S_2, S_3, S_4 be given finite sets of pairs of n-by-n complex matrices. It is described an algorithm to determine, with finitely many computations, whether there is a single unitary matrix U such that each pair of matrices in S_1 is unitarily similar via U, each pair of matrices in S_2 is unitarily congruent via U, each pair of matrices in S_3 is unitarily similar via U ̅, and each pair of matrices in S_4 is unitarily congruent via U ̅.
Let (A, B) be a pair of skew-symmetric matrices over a field of characteristic not 2. Its regularization decomposition is a direct sum
(▁▁A, ▁▁B)⊕(A_1,B_1)⊕⋅⋅⋅⊕(A_t,B_t)
that is congruent to (A, B), in which (▁▁A, ▁▁B) is a pair of nonsingular matrices and (A_1,B_1)⊕⋅⋅⋅⊕(A_t,B_t ) are singular indecomposable canonical pairs of skew- symmetric matrices under congruence. It is given an algorithm that constructs a regularization decomposition. We also give a constructive proof of the known canonical form of (A,B) under congruence over an algebraically closed field of characteristic not 2.
