The construction of irreducible normalized representations of the general linear group

A method is described for constructing in an explicit form an irreducible representation T of M_n(F), the set of all n × n matrices over the real or complex field F, satisfying the condition T(A^*)=T^*(A) for all A∈M_n(F).     

A characterization of linear transformations which leave the doubly stochastic matrices

A characterization of linear transformations which leave the n×n doubly stochastic matrices invariant is given as a linear combination of functions of the type T(X)=AXB with certain restrictions posed on the n×n matrices A and B.     

Linear mappings which are rank-k nonincreasing, II

We prove the following result. Let F be an infinite field of characteristic other than two. Let k be a positive integer. Let S_n(F) denote the space of all n × n symmetric matrices with entries in F, and let T:S_n(F)→S_n(F) be a linear operator. Suppose that T is rank-k nonincreasing and its image contains a matrix with rank higher than K. Then, there exist λεF and PεF^n,n such that T(A)=λPAP^t for all AεS_n(F). λ can be chosen to be 1 if F is algebraically closed and ±1 if F=R, the real field.     

Linear operators preserving unitarily invariant norms of matrices

Let F_{m × n} be the set of all m × n matrices over the field F = C or R Denote by U_n(F) the group of all n × n unitary or orthogonal matrices according as F = C or F-R. A norm N() on Fm ×n, is unitarily invariant if N(UAV) = N(A): for all A ∈ F _m×n U ∈ U_m(F). and V ∈ U_n(F). We characterize those linear operators TF_{m × n} → F_{m × n}which satisfy N (T(A)) = N(A)for all A ∈ F_{m × n}

for a given unitarily invariant norm N(). It is shown that the problem is equivalent to characterizing those operators which preserve certain subsets in F_{m × n} To develop the theory we prove some results concerning unitary operators on F_{m × n} which are of independent interest.     

Quadratic expansions of spectral functions

A function, F, on the space of n×n real symmetric matrices is called spectral if it depends only on the eigenvalues of its argument, that is F(A)=F(UAU^T) for every orthogonal U and symmetric A in its domain. Spectral functions are in one-to-one correspondence with the symmetric functions on : those that are invariant under arbitrary swapping of their arguments. In this paper we show that a spectral function has a quadratic expansion around a point A if and only if its corresponding symmetric function has quadratic expansion around λ(A) (the vector of eigenvalues). We also give a concise and easy to use formula for the ‘Hessian' of the spectral function. In the case of convex functions we show that a positive definite ‘Hessian' of f implies positive definiteness of the ‘Hessian' of F.     

An edge version of the matrix-tree theorem and the wiener index

Let T be a tree on n vertices. The Laplacian matrix is L(T)=D(T)-A(T), where D(T) is the diagonal matrix of vertex degrees and A(T) is the adjacency matrix. A special case of the Matrix-Tree Theorem is that the adjugate of L(T) is the n-by-n matrix of l's. The (n-l)-square "edge version" of L(T)is K(T). The main result is a graph-theoretic interpretation of the entries of the adjugate of K(T). As an application, it is shown that the Wiener Index from chemistry is the trace of this adjugate.     

Decomposably non-negative matrices

Let A be an mn- by - mn symmetric matrix. Partition A into m²n - by - n blocks and suppose that each of these blocks is also symmetric. Suppose that for every decomposable (rank one) tensor ν ⊗ w, we have (ν ⊗ w)^t A(ν otimes; w) ≥ 0. Here, ν is a column m-tuple and w is a column n-tuple. We study the maximum number of negative eigenvalues such a matrix can have, as well as obtaining alternative characterizations of such matrices.     

Some generators for the algebra of n×n matrices

In this note, we show how the algebra of n×n matrices over a field can be generated by a pair of matrices AB, where A is an arbitrary nonscalar matrix and B can be chosen so that there is the maximum degree of linear independence between the higher commutators of B with A.     

The rank of the difference of matrices with prescribed similarity classes

Let F be a field and let A,B be n × n matrices over I. We study the rank of A' - B' when A and B run over the set of matrices similar to A and B, respectively.     

Eigenvalue inequalities associated with the cartesian decomposition

Let T=A+iB where AB are Hermitian matrices. We obtain several inequalities relating the l_p distance between the eigenvalues of A and those of iB with the Schatten p-norm of T. The majorization results which lead to these inequalities are also used to get simple proofs of some known lower and upper bounds for the determinant of T.     

The Best Lower Bound for the Permanent of a Correlation Matrix of Rank Two

If 1≤k≤n, then Cor(n,k) denotes the set of all n×n real correlation matrices of rank not exceeding k. Grone and Pierce have shown that if A∈Cor (n, n-1), then per(A)≥n/(n-1). We show that if A∈Cor(n,2), then , and that this inequality is the best possible.     

Flat portions on the boundary of the numerical ranges of certain Toeplitz matrices

We give criterions for a flat portion to exist on the boundary of the numerical range of a matrix. A special type of Teoplitz matrices with flat portions on the boundary of its numerical range are constructed. We show that there exist 2 × 2 nilpotent matrices A₁,A₂, an n  × n nilpotent Toeplitz matrix N_n, and an n  × n cyclic permutation matrix S_n(s) such that the numbers of flat portions on the boundaries of W(A₁⊕N_n) and W(A₂⊕S_n(s)) are, respectively, 2(n - 2) and 2n.     

An inequality for M-matrices

If AB are n × n M matrices with dominant principal diagonal, we show that 3[det(A + B)]^1/n ≥ (det A)^1/n + (det B)^1/n.     

On the fourth power moment of Fourier coefficients of cusp form

Let a(n)be the Fourier coefficients of a holomorphic cusp form of weightκ=2n≥12 for the full modular group and A(x)=∑_(n≤x)a(n).In this paper,we establish an asymptotic formula of the fourth power moment of A(x)and prove that ∫T1A~4(x)dx=3/(64κπ~4)s_4;2()T~(2κ)+O(T~(2κ-δ_4+ε))with δ_4=1/8,which improves the previous result.     

A formula for the number of Latin squares

We establish an explicit formula for the number of Latin squares of order n:

, where B_n is the set of n×n(0,1) matrices, σ₀(A is the number of zero elements of the matrix A and per A is the permanent of the matrix A.     

Linear preservers of matrices of rank-2

Let T be a linear operator on the space of all m×n matrices over any field. we prove that if T maps rank-2 matrices to rank-2 matrices then there exist nonsingular matrices U and V such that either T(X)=UXV for all matrices X, or m=n and T(X)=UX^tV for all matrices X where X^t denotes the transpose of X.     

Alternative proofs of a theorem of Moyls and Marcus on the numerical range of a square matrix

Moyls and Marcus [4] showed that for n≤4,n×n an complex matrix A is normal if and only if the numerical range of A is the convex hull of the eigenvalues of A. When n≥5, there exist matrices which are not normal, but such that the numerical range is still the convex hull of the eigenvalues. Two alternative proofs of this fact are given. One proof uses the known structure of the numerical range of a 2×2 matrix. The other relies on a theorem of Motzkin and Taussky stating that a pair of Hermitian matrices with property L must commute.     

On k-spaces of real matrices

It is shown that if W is a linear subspace of real n × n matrices, such that rank (A) = k for all 0 ≠ A ∈ W, then dim W≤ n. If dim W = n.5≤ n is prime, and 2 is primitive modulo n then k =1.     

A matrix problem associated with computing Cohen-Macaulay type

Suppose A∈M_n×m(F), B∈M_n×t(F) for some field F. Define Г(AB) to be the set of n×n diagonal matrices D such that the column space of DA is contained in the column space of B. In this paper we determine dim Г(AB). For matrices AB of the same rank we provide an algorithm for computing dim Г(AB).     

Multiplicative properties of generalized matrix functions

We consider scalar-valued matrix functions for n×n matrices A=(a_ij) defined by Where G is a subgroup of S_n the group of permutations on n letters, and χ is a linear character of G. Two such functions are the permanent and the determinant. A function (1) is multiplicative on a semigroup S of n×n matrices if d(AB)=d(A)d(B) AB∈S.

With mild restrictions on the underlying scalar ring we show that every element of a semigroup containing the diagonal matrices on which (1) is multiplicative can have at most one nonzero diagonal(i.e., diagonal with all nonzero entries)and conversely, provided that χ is the principal character(χ≡1).