A note on idempotent matrices

Let H be an n × n matrix, and let the trace, the rank, the conjugate transpose, the Moore-Penrose inverse, and a g-inverse (or an inner inverse) of H be respectively denoted by trH, ρ(H), H¹, H², and H^?. This note develops two results: (i) the class of idempotent g-inverse of an idempotent matrix, and (ii) if H is an n × n matrix and ρ(H) = trH, then $\text{tr}\text{(H}{}^2\text{H}{}^2\text{H}{}^1\text{) ? ρ(H)}$, and the equality holds iff H is idempotent. This result is compared with the previous result of Khatri (1983), and some consequences of (i) and (ii) are given.     

Levy's stochastic area formula in higher dimensions

Let (W_t) = (W¹_t,W²_t,…,W^d_t), d ? 2, be a d-dimensional standard Brownian motion and let A(t) be a bounded measurable function from $\text{R}$⁺ into the space of d × d skew-symmetric matrices and x(t) such a function into $\text{R}$^d. A class of stochastic processes (L_t^A,x), a particular example of which is Levy's “stochastic area” $\text{L}{}_{\mathrm{t}}\text{=}\text{1}\text{2}\text{∝}{}_0{}^{\mathrm{?t}}\text{(W}{}^1{}_{\mathrm{s}}\text{,dW}{}^2{}_{\mathrm{s}}\text{? W}{}^2{}_{\mathrm{s}}\text{,dW}{}^1{}_{\mathrm{s}}\text{)}$, is dealt with.The joint characteristic function of W_t and L₁^A,x is calculated and based on this result a formula for fundamental solutions for the hypoelliptic operators which generate the diffusions (W_t,L_t^A,x) is given.     

Norm properties of C-numerical radii

Given n×n complex matrices A, C, the C-numerical radius of A is the nonnegative quantity

$\text{r}{}_{\mathrm{c}}\text{(A)≡ma{|tr(CU}{}^1\text{AU)|:U unitary}}$

. For C=diag(1,0,…,0) it reduces to the classical numerical radius $\text{r(A)= max{|x}{}^1\text{Ax|:x}{}^1\text{x=1}}$. We show that r_c is a generalized matrix norm if and only if C is nonscalar and trC≠0. Next, we consider an arbitrary generalized matrix norm and characterize all constants v?0 for which vN is multiplicative. A technique to obtain such v is then applied to C-numerical radii with Hermitian C. In particular we find that vr is a matrix norm if and only if v?4.     

Concrete model theory for a class of operators with unitary part

Let m and v_t, 0 ? t ? 2π be measures on T = [0, 2π] with m smooth. Consider the direct integral $\text{H}$ = ⊕L²(v^t) dm(t) and the operator $\text{(L?)(t, λ) = e}{}^{\mathrm{?i\lambda }}\text{?(t, λ) ? 2e}{}^{\mathrm{?i\lambda }}\text{∫}{}_{\mathrm{t}}{}^{\mathrm{2\pi }}\text{∫}{}_{\mathrm{T}}\text{?(s, x) e(s, t) dv}{}_{\mathrm{s}}\text{(x) dm(s)}$ on $\text{H}$, where e(s, t) = exp ∫_s^t ∫_Tdv_λ(θ) dm(λ). Let μ_t be the measure defined by $\text{∫}{}_{\mathrm{T}}\text{?(x) dμ}{}_{\mathrm{t}}\text{(x) = ∫}{}_0{}^{\mathrm{t}}\text{∫}{}_{\mathrm{T}}\text{?(x) dv}{}_{\mathrm{s}}\text{dm(s)}$ for all continuous ?, and let ?_t(z) = exp[?∫ (e^iθ + z)(e^iθ ? z)^?1dμ_t(gq)]. Call {v_t} regular iff for all $\text{t, ¦?}{}_{\mathrm{t}}\text{(e}{}^{\mathrm{i\theta }}\text{)¦ = ¦?}{}_{\mathrm{2\pi }}\text{(e}{}^{\mathrm{i\theta }}\text{)¦}$ for 1 a.e.     

More bounds for elgenvalues using traces

Let the n × n complex matrix A have complex eigenvalues λ₁,λ₂,…λ_n. Upper and lower bounds for Σ(Reλ_i)² are obtained, extending similar bounds for Σ|λ_i|² obtained by Eberlein (1965), Henrici (1962), and Kress, de Vries, and Wegmann (1974). These bounds involve the traces of A¹A, B², C², and D², where $\text{B}\text{=}\text{1}\text{2}\text{(}\text{A}\text{+}\text{A}{}^1\text{)}$, $\text{C}\text{=}\text{1}\text{2}\text{(}\text{A}\text{?}\text{A}{}^1\text{) /i}$, and $\text{D}\text{=}\text{AA}{}^1\text{?}\text{A}{}^1\text{A}$, and strengthen some of the results in our earlier paper “Bounds for eigenvalues using traces” in Linear Algebra and Appl. [12].     

On the convergence of the Neumann series in interval analysis

Let $\text{A}$ be a real or complex n × n interval matrix. Then it is shown that the Neumann series $\text{Σ}{}^{\mathrm{\infty }}{}_{\mathrm{k=0}}\text{A}{}^{\mathrm{k}}$ is convergent iff the sequence {$\text{A}$^k} converges to the null matrix $\text{O}$, i.e., iff the spectral radius of the real comparison matrix $\text{B}$ constructed in [2] is less than one.     

Common solutions to the Lyapunov equations

Let A be an n×n matrix with complex entries. A necessary and sufficient condition is established for the existence of a Hermitian solution H to the equations

$\text{AH+HA}{}^1\text{=HA+A}{}^1\text{H=I}$

.     

Coding theory in white Gaussian channel with feedback

The message m = {m(t)} is a Gaussian process that is to be transmitted through the white Gaussian channel with feedback: $\text{Y(t) = ∫}{}_0{}^{\mathrm{t}}\text{F(s, Y}{}_0{}^{\mathrm{s}}\text{, m)ds + W(t)}$. Under the average power constraint, $\text{E[F}{}^2\text{(s, Y}{}_0{}^{\mathrm{s}}\text{, m)] ≤ P}{}_0$, we construct causally the optimal coding, in the sense that the mutual information I_t(m, Y) between the message m and the channel output Y (up to t) is maximized. The optimal coding is presented by $\text{Y(t) = ∫}{}_0{}^{\mathrm{t}}\text{A(s)[m(s) ?}\text{m}\text{?}\text{(s)] ds + W(t)}$, where $\text{m}\text{?}\text{(s) = E[m(s) ¦ Y(u), 0 ≤ u ≤ s]}$ and A(s) is a positive function such that $\text{A}{}^2\text{(s) E |m(s) ?}\text{m}\text{?}\text{(s)|}{}^2\text{= P}{}_0$.     

On solutions of “equations in symmetric groups”

Let x?S_n, the symmetric group on n symbols. Let θ? Aut(S_n) and let the automorphim order of x with respect to θ be defined by

$\text{γθ(x)=}\text{min}\text{{k:x xθ xθ}{}^2\text{? xθ}{}^{\mathrm{k?1}}\text{=1}}$

where xθ is the image of x under θ. Let α_g? Aut(S_n) denote conjugation by the element g?S_n. Let where s and k are positive integers and $\text{a}\text{b}$ denotes a divides b. Further h(s, k : n) ≡ b(1; s, k : n), where 1 denotes the identity automorphim. If g?S_n let c = f(g, s) denote the number of symbols in g which are in cycles of length not dividing the integer s, and let g_s denote the product of all cycles in g whose lengths do not divide s. Then g_s moves c symbols. The main results proved are: (1) recursion: if n ? c + 1 and t = n ? c ? 1 then $\text{b(g; s, 1:n)=∑}{}_{\mathrm{<mtext>i</mtext><mtext>s</mtext>}}\text{b(g; s, 1:n?1)(}{}^{\mathrm{t}}{}_{\mathrm{i?1}}\text{(i?1)!}$ (2) reduction: b(g; s, 1 : c)h(s, 1 : i) = b(g; s, 1 : i + c); (3) distribution: let D(θ, n) ≡ {(k, b) : k?Z⁺ and b = b(θ; 1, k : n) ≠ 0}; then D(θ, m) = D(φ, m) ∨ m ? N = N(θ, φ) iff θ is conjugate to φ; (4) evaluation: the number of cycles in g_s^s of any given length is smaller than the smallest prime dividing s iff b(g_s; s, 1 : c) = 1. If g = (12 … p^m)^t and $\text{sk}\text{p}{}^{\mathrm{m}}$ then $\text{b(g;s,k:p}{}^{\mathrm{m}}\text{) {}{}^0{}_{\mathrm{\pm 1}}\text{(}\text{mod}\text{p)}$.     

Asymptotic results for partitions (II) and a conjecture of Bateman and Erdös

Asymptotic results are obtained for p_A^(k)(n), the kth difference of the function p_A(n) which is the number of partitions of n into integers from A. Under certain restrictions on A it is shown that

$\text{P}{}_{\mathrm{A}}{}^{\mathrm{(k+1)}}\text{(n)}\text{P}{}_{\mathrm{A}}{}^{\mathrm{(k)}}\text{(n)}\text{= O(n}{}^{\mathrm{?1/2}}\text{) (n→ ∫)}$

thereby verifying for these A a conjecture of Bateman and Erdös.     

Determinants of Galois automorphisms of maximal commutative rings of 2×2 matrices

The determinants of solutions X to any of the 2×2 matrix equations: (1) XAX>^?1=A^t, t denoting transpose; (2) A=XY, X, Y symmetric; (3) X=AB?BA; and (4) $\text{XAX}{}^{\mathrm{?1}}\text{=A}{}^1$, the matrix of cofactors of A transposed, are characterized over a commutative ring R as the negative of norms from the quadratic extension R[A] whenever certain elements in the Picard group Pic (R[A]) are trivial. This characterization is realized by the utilization of galois cohomology and a generalized Latimer–MacDuffee correspondence between similarity classes and the elements in Pic (R[A]). The results represent a ramification of several articles by O. Taussky on the specializations of equations (1)–(3) to either the field of rational numbers or the ring of rational integers.     

The one-dimensional diffusion process and unitary representations of R1

Given a cocycle a(t) of a unitary group {U¹}, ?∞ < t < ∞, on a Hilbert space $\text{H}$, such that a(t) is of bounded variation on [O, T] for every T > O, a(t) is decomposed as a(t) = f;^t₀U^sxds + β(t) for a unique x ? $\text{H}$, β(t) yielding a vector measure singular with respect to Lebesgue measure. The variance is defined as $\text{σ}{}^2\text{({rmU}{}^{\mathrm{t}}\text{}, a(t)) =}\text{lim}{}_{\mathrm{T\to \infty }}\text{(}\text{1}\text{T}\text{)∥∝}{}^{\mathrm{t}}{}_0\text{U}{}^{\mathrm{s}}\text{x ds∥}{}^2$ if existing. For a stationary diffusion process on $\text{R}$¹, with Ω₁, the space of paths which are natural extensions backwards in time, of paths confined to one nonsingular interval J of positive recurrent type, an information function I(ω) is defined on $\text{Ω}{}_1$, based on the paths restricted to the time interval [0, 1]. It is shown that $\text{I(Ω)}$ is continuous and bounded on $\text{Ω}{}_1$. The shift τ_t, defines a unitary representation {U^t}. Assuming , dm being the stationary measure defined by the transition probabilities and the invariant measure on J, $\text{I(Ω)}$ has a C^∞ spectral density function f;. It is then shown that σ²({U^t}, I) = f;(O).     

Sharp Sobolev type inequalities for higher fractional derivativesInégalités optimales de type Sobolev pour les dérivées fractionelles d'ordre supérieur

On $\text{R}{}^{\mathrm{n}}$, n?1 and n≠2, we prove the existence of a sharp constant for Sobolev inequalities with higher fractional derivatives. Let s be a positive real number. For n>2s and $\text{q=}\text{2n}\text{n?2s}$ any function $\text{f∈}\text{H}{}^{\mathrm{s}}\text{(}\text{R}{}^{\mathrm{n}}\text{)}$ satisfies

$\text{6f6}{}^2{}_{\mathrm{q}}\text{?S}{}_{\mathrm{n,s}}\text{(?Δ)}{}^{\mathrm{s/2}}\text{f}{}^2{}_2\text{,}$

where the operator (?Δ)^s in Fourier spaces is defined by $\text{(?Δ)}{}^{\mathrm{s}}\text{f}\text{(k):=(2π|k|)}{}^{\mathrm{2s}}\text{f}\text{(k)}$. To cite this article: A. Cotsiolis, N.C. Tavoularis, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 801–804.     

Matrix identities of the fast fourier transform

Let A_n(ω) be the nxn matrix A_n(ω)=(a_ij with a_ij=ω^ij, 0?i,j?n?1, ωⁿ=1. For n=rs we show

$\text{A}{}_{\mathrm{n}}\text{(w)P}{}_{\mathrm{s}}{}^{\mathrm{r}}\text{P}{}_{\mathrm{r}}{}^{\mathrm{s}}\text{∵}\text{0}\text{s?1}\text{A}{}_{\mathrm{r}}\text{(w}{}^{\mathrm{s}}\text{)}\text{P}{}_{\mathrm{s}}{}^{\mathrm{r}}\text{{T}{}_{\mathrm{r}}{}^{\mathrm{s}}\text{(w)}}\text{∵}\text{0}\text{r?1}\text{A}{}_{\mathrm{s}}\text{(w}{}^{\mathrm{r}}\text{)}$

=(A_r?I_s)T^s_r(I_r?A_s). When r and s are relatively prime this identity implies a wide class of identities of the form PA_n(ω)Q^T=A_r(ω^αs)?A_s(ω^βr). The matrices P^s_r, P^r_s, P, and Q are permutation matrices corresponding to the “data shuffling” required in a computer implementation of the FFT, and T^s_r is a diagonal matrix whose nonzeros are called “twiddle factors.” We establish these identities and discuss their algorithmic significance.     

General logarithmic Sobolev inequalities and Orlicz imbeddings

We obtain, for a large class of measures μ, general inequalities of the form $\text{∫R}{}^{\mathrm{n}}\text{|u|}{}^{\mathrm{p}}\text{A(}\text{log}{}^1\text{|u|) dμ ? K(6u : W}{}^{\mathrm{m,p}}\text{(R}{}^{\mathrm{n}}\text{,dμ)6}{}^{\mathrm{p}}\text{+ 6 u 6}{}^{\mathrm{p}}\text{A(}\text{log}{}^1\text{6 u 6))}$, where $\text{6u6 = 6 u: L}{}^{\mathrm{p}}\text{(R}{}^{\mathrm{n}}\text{,dμ)6}{}^{\mathrm{p}}\text{,}\text{log}{}^1\text{t =}\text{max}\text{{1,}\text{log}\text{t}}$, and the function A depends in an appropriate way on μ. Our results extend similar results obtained by Rosen for the case p = 2, A(t) = t^s. We also investigate some implications of these inequalities for the imbedding of Sobolev spaces into Orlicz spaces.     

Families of finite sets in which no set is covered by the union of two others

Let f^k(n) denote the maximum of k-subsets of an n-set satisfying the condition in the title. It is proven that f^{2t ? 1}(n) ? f^2t(n + 1) ? $\text{(}{}_{\mathrm{t}}{}^{\mathrm{n}}\text{)}\text{(}{}_{\mathrm{t}}{}^{\mathrm{2t?1}}\text{)}$ with equalities holding iff there exists a Steiner-system S(t, 2t ? 1, n). The bounds are approximately best possile for k ? 6 and of correct order of magnitude for k >/ 7, as well, even if the corresponding Steiner-systems do not exist.Exponential lower and upper bounds are obtained for the case if we do not put size restrictions on the members of the family (i.e., the nonuniform case).     

Factorization of fredholm operators on analytic functions

Let Ω be a simply connected domain in the complex plane, and $\text{A(Ω}{}^{\mathrm{n}}\text{)}$, the space of functions which are defined and analytic on $\text{Ω}{}^{\mathrm{n}}$, if K is the operator on elements $\text{u(t, a}{}_1\text{, …, a}{}_{\mathrm{n}}\text{)}\text{of}\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ defined in terms of the kernels k_i(t, s, a₁, …, a_n) in $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$ by is the identity operator on $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$, then the operator I ? K may be factored in the form (I ? K)(M ? W) = (I ? ΠK)(M ? ΠW). Here, W is an operator on $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ defined in terms of a kernel w(t, s, a₁, …, a_n) in $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$ by Wu = ∝_an^tw(t, s, a₁, …, a_n) u(s, a₁, …, a_n) ds. ΠW is the operator; ΠWu = ∝_{an ? 1}w(t, s, a₁, …, a_n) u(s, a₁, …, a_n) ds. ΠK is the operator; ΠKu = ∑_{i = 1}^{n ? 1} ∝_ai^tk_i(t, s, a₁, …, a_n) ds + ∝_{an ? 1}^tk_n(t, s, a₁, …, a_n) u(s, a₁, …, a_n) ds. The operator M is of the form m(t, a₁, …, a_n)I, where $\text{m ? A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ and maps elements of $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ into itself by multiplication. The function m is uniquely derived from K in the following manner. The operator K defines an operator $\text{K}{}^1$ on functions u in $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$, by . A determinant $\text{δ(I ? K}{}^1\text{)}$ of the operator $\text{I ? K}{}^1$ is defined as an element $\text{m}{}^1\text{(t, a}{}_1\text{, …, a}{}_{\mathrm{n\; +\; 1}}\text{)}$ of $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$. This is mapped into $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ by setting a_{n + 1} = t to give m(t, a₁, …, a_n). The operator I ? ΠK may be factored in similar fashion, giving rise to a chain factorization of I ? K. In some cases all the matrix kernels k_i defining K are separable in the sense that k_i(t, s, a₁, …, a_n) = P_i(t, a₁, …, a_n) Q_i(s, a₁, …, a_n), where P_i is a 1 × p_i matrix and Q_i is a p_i × 1 matrix, each with elements in $\text{A(Ω}{}_{\mathrm{n\; +\; 1}}\text{)}$, explicit formulas are given for the kernels of the factors W. The various results are stated in a form allowing immediate extension to the vector-matrix case.     

A generalization of Lavoie's inequality concerning the sum of idempotent matrices

Let a complex pxn matrix A be partitioned as A′=(A′₁,A′₂,…,A′_k). Denote by ?(A), A′, and A^? respectively the rank of A, the transpose of A, and an inner inverse (or a g-inverse) of A. Let A⁽¹⁴⁾ be an inner inverse of A such that A⁽¹⁴⁾A is a Hermitian matrix. Let B=(A⁽¹⁴⁾₁,A⁽¹⁴⁾₂,…,A_k⁽¹⁴⁾) and $\text{ρ(A)=}\text{∑}\text{i=1}\text{k}\text{ρ(A}{}_{\mathrm{i}}\text{)}$.Then the product of nonzero eigenvalues of BA (or AB) cannot exceed one, and the product of nonzero eigenvalues of BA is equal to one if and only if either B=A⁽¹⁴⁾ or $\text{A}{}_{\mathrm{i>}}\text{A}{}_{\mathrm{<mtext>j</mtext>}}{}^1\text{=0}$ for all i ≠ j,i, j=1,2,…,k . The results of Lavoie (1980) and Styan (1981) are obtained as particular cases. A result is obtained for k=2 when the condition $\text{ρ(A)=}\text{∑}\text{i=1}\text{k}\text{ρ(A}{}_{\mathrm{i}}\text{)}$ is no longer true.     

Positive definite Hermitian matrices and reproducing kernels

Some quadratic identities associated with positive definite Hermitian matrices are derived by use of the theory of reproducing kernels. For example, the following identity is obtained: Let{A_j}^m_j=1 be N × N positive definite Hermitian matrices. Then, for any complex vector x ∈ $\text{C}$^N, we have the identity

$\text{x}{}^1\text{∑}\text{j=}\text{1}\text{m}\text{A}{}^{-1}{}_{\mathrm{j}}{}^{-1}\text{x = min}\text{∑}\text{j=}\text{1}\text{m}\text{x}{}^1{}_{\mathrm{j}}\text{A}{}_{\mathrm{j}}\text{x}{}_{\mathrm{j}}$

. The minimum is taken here over all the decompositions x =∑^m_j=1x_j. This identity gives, in a sense, a precise converse for an inequality which was derived by T. Ando. Moreover, this paper shows that the sum of two reproducing kernels is naturally related to the harmonic-arithmetic-mean inequality for matrices and also that the geometric-arithmetic-mean inequality for matrices can be naturally interpreted in terms of tensor-product spaces.