scholarly journals A necessary and sufficient condition for simultaneous diagonalization of two hermitian matrices and its application

1970 ◽  
Vol 11 (1) ◽  
pp. 81-83 ◽  
Author(s):  
Yik-Hoi Au-Yeung
Keyword(s):  
Complex Numbers ◽  
Single Element ◽  
Singular Matrix ◽  
Necessary And Sufficient Condition ◽  
Simultaneous Diagonalization ◽  
Hermitian Matrices ◽  
Skew Field ◽  
Conjugate Transpose ◽  
Necessary And Sufficient ◽  
Real Quaternions

We denote by F the field R of real numbers, the field C of complex numbers, or the skew field H of real quaternions, and by Fn an n dimensional left vector space over F. If A is a matrix with elements in F, we denote by A* its conjugate transpose. In all three cases of F, an n × n matrix A is said to be hermitian if A = A*, and we say that two n × n hermitian matrices A and B with elements in F can be diagonalized simultaneously if there exists a non singular matrix U with elements in F such that UAU* and UBU* are diagonal matrices. We shall regard a vector u ∈ Fn as a l × n matrix and identify a 1 × 1 matrix with its single element, and we shall denote by diag {A1, …, Am} a diagonal block matrix with the square matrices A1, …, Am lying on its diagonal.

A note on some theorems on simultaneous diagonalization of two Hermitian matrices

1971 ◽  
Vol 70 (3) ◽  
pp. 383-386 ◽  
Author(s):  
Yik-Hoi Au-Yeung
Keyword(s):  
Hermitian Matrix ◽  
Complex Numbers ◽  
Single Element ◽  
Singular Matrix ◽  
Simultaneous Diagonalization ◽  
Hermitian Matrices ◽  
Real Numbers ◽  
Skew Field ◽  
Conjugate Transpose ◽  
Real Quaternions

We denote by F the field R of real numbers, the field C of complex numbers, or the skew-field H of real quaternions, and by Fn an n-dimensional left vector space over F. If A is a matrix with elements in F, we denote by A* its conjugate transpose. In all three cases of F, an n × n matrix A is said to be Hermitian if A = A* and unitary if AA* = In, where In is the n × n identity matrix. An n × n Hermitian matrix A is said to be positive definite (postive semi-definite resp.) if uAu* > 0(uAu* ≥ 0 resp.) for all u (╪ 0) in Fn. Here and in what follows we regard u as a 1 × n matrix and identify a 1 × 1 matrix with its single element. In the following we shall always use A and B to denote two n×n Hermitian matrices with elements in F, and we say that A and B can be diagonalized simultaneously if there exists an n×n non-singular matrix V with elements in F such that VAV* and VBV* are diagonal matrices. We shall use diag {A1, A2} to denote a diagonal block matrix with the square matrices A1 and A2 lying on its diagonal.

scholarly journals A characterization of the definiteness of a Hermitian matrix

1974 ◽  
Vol 15 (1) ◽  
pp. 1-4
Author(s):  
Yik-Hoi Au-Yeung ◽  
Tai-Kwok Yuen
Keyword(s):  
Hermitian Matrix ◽  
Nonsingular Matrix ◽  
Complex Numbers ◽  
Identity Matrix ◽  
Hermitian Matrices ◽  
Real Numbers ◽  
Skew Field ◽  
Conjugate Transpose ◽  
Real Quaternions

We denote by F the field R of real numbers, the field C of complex numbers or the skew-field H of real quaternions, and by Fn an n-dimensional left vector space over F. If A is a matrix with elements in F, we denote by A* its conjugate transpose. In all three cases of F, an n × n matrix A is said to be hermitian (unitary resp.) if A = A* (AA*= identity matrix resp.). An n ×x n hermitian matrix A is said to be definite (semidefinite resp.) if uAu*vAv* ≥ 0 (uAu*vAv* ≧ 0 resp.) for all nonzero u and v in Fn. If A and B are n × n hermitian matrices, then we say that A and B can be diagonalized simultaneously into blocks of size less than or equal to m (abbreviated to d. s. ≧ m) if there exists a nonsingular matrix U with elements in F such that UAU* = diag{A1,…, Ak} and UBU* = diag{B1…, Bk}, where, for each i = 1, …, k, Ai and Bk are of the same size and the size is ≧ m. In particular, if m = 1, then we say A and B can be diagonalized simultaneously (abbreviated to d. s.).

A note on the convexity of the indefinite joint numerical range

2014 ◽  
Vol 27 ◽  
Author(s):  
Hiroshi Nakazato ◽  
Natalia Bebiano ◽  
Joao Da Providencia
Keyword(s):  
Numerical Range ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Hermitian Matrices ◽  
Krein Spaces ◽  
Necessary And Sufficient ◽  
Joint Numerical Range

This note investigates the convexity of the indefinite joint numerical range of a tuple of Hermitian matrices in the setting of Krein spaces. Its main result is a necessary and sufficient condition for convexity of this set. A new notion of “quasi-convexity” is introduced as a refinement of pseudo-convexity.

ON THE GROUP INVERSE FOR THE SUM OF MATRICES

2013 ◽  
Vol 96 (1) ◽  
pp. 36-43
Author(s):  
CHANGJIANG BU ◽  
XIUQING ZHOU ◽  
LIANG MA ◽  
JIANG ZHOU
Keyword(s):  
Sufficient Condition ◽  
Group Inverse ◽  
Necessary And Sufficient Condition ◽  
Skew Field ◽  
Block Matrices ◽  
Necessary And Sufficient

AbstractLet${ \mathbb{K} }^{m\times n} $denote the set of all$m\times n$matrices over a skew field$ \mathbb{K} $. In this paper, we give a necessary and sufficient condition for the existence of the group inverse of$P+ Q$and its representation under the condition$PQ= 0$, where$P, Q\in { \mathbb{K} }^{n\times n} $. In addition, in view of the natural characters of block matrices, we give the existence and representation for the group inverse of$P+ Q$and$P+ Q+ R$under some conditions, where$P, Q, R\in { \mathbb{K} }^{n\times n} $.

The Structure and Distribution of Roots of 2×2 Matrices

2013 ◽  
Vol 765-767 ◽  
pp. 667-669
Author(s):  
Yuan Yuan Li
Keyword(s):  
Infinitely Many Solutions ◽  
Sufficient Condition ◽  
Singular Matrix ◽  
Necessary And Sufficient Condition ◽  
Number Of Solutions ◽  
Jordan Canonical Form ◽  
The Matrix ◽  
Explicit Formulae ◽  
Necessary And Sufficient ◽  
Distribution Of Roots

This paper is concerned with Jordan canonical form theorem of algebraic formulae giving all the solutions of the matrix equation Xm= A where n is a positive integer greater than 2 and A is a 2 × 2 matrix with real or complex elements. If A is a 2 × 2 non-singular matrix, the equation Xm = A has infinitely many solutions and we obtain explicit formulae giving all the solutions. If A is a 2 × 2 singular matrix, and we obtained necessary and sufficient condition of square root . This leads to very simple formulae for all the solutions when A is either a singular matrix or a non-singular matrix with two coincident eigenvalues. We also determine the precise number of solutions in various cases.

Lp Behavior of the Eigenfunctions of the Invariant Laplacian

1993 ◽  
Vol 36 (4) ◽  
pp. 458-465
Author(s):  
E. G. Kwon
Keyword(s):  
Harmonic Functions ◽  
Maximum Modulus ◽  
Open Unit ◽  
Complex Numbers ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Maximum Modulus Principle ◽  
Open Unit Ball ◽  
Image Position ◽  
Necessary And Sufficient

AbstractLet be the invariant Laplacian on the open unit ball B of Cn and let Xλ denote the set of those f € C2(B) such that counterparts of some known results on X0, i.e. on M-harmonic functions, are investigated here. We distinguish those complex numbers λ for which the real parts of functions in Xλ belongs to Xλ. We distinguish those λ for which the Maximum Modulus Priniple remains true. A kind of weighted Maximum Modulus Principle is presented. As an application, setting α ≥ ½ and λ = 4n2α(α — 1), we obtain a necessary and sufficient condition for a function f in Xλ to be represented asfor some F ∊ LP(∂B).

scholarly journals On the Irreducibility of Artin's Group of Graphs

2015 ◽  
Vol 7 (2) ◽  
pp. 117
Author(s):  
Malak M. Dally ◽  
Mohammad N. Abdulrahim
Keyword(s):  
Complex Numbers ◽  
Sufficient Condition ◽  
Artin Group ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

We consider the graph $E_{3,1}$ with three generators $\sigma_1, \sigma_2, \delta$, where $\sigma_1$ has an edge with each of $\;\sigma_2$ and  $\;\delta$. We then define the Artin group of the graph $E_{3,1}$ and consider its  reduced  Perron representation of degree three. After we specialize the indeterminates used in defining the representation to  non-zero complex numbers, we obtain a necessary and sufficient condition that guarantees the irreducibility of the representation.<br />

Random iterations of polynomials of the form $z^2+c_n$: connectedness of Julia sets

1999 ◽  
Vol 19 (5) ◽  
pp. 1221-1231 ◽  
Author(s):  
RAINER BRÜCK ◽  
MATTHIAS BÜGER ◽  
STEFAN REITZ
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Complex Numbers ◽  
Sufficient Condition ◽  
Fatou Set ◽  
Necessary And Sufficient Condition ◽  
Quadratic Polynomials ◽  
Necessary And Sufficient

For a sequence $(c_n)$ of complex numbers we consider the quadratic polynomials $f_{c_n}(z):=z^2+c_n$ and the sequence $(F_n)$ of iterates $F_n:= f_{c_n} \circ \dotsb \circ f_{c_1}$. The Fatou set $\mathcal{F}_{(c_n)}$ is by definition the set of all $z \in \widehat{\mathbb{C}}$ such that $(F_n)$ is normal in some neighbourhood of $z$, while the complement of $\mathcal{F}_{(c_n)}$ is called the Julia set $\mathcal{J}_{(c_n)}$. The aim of this paper is to study the connectedness of the Julia set $\mathcal{J}_{(c_n)}$ provided that the sequence $(c_n)$ is bounded and randomly chosen. For example, we prove a necessary and sufficient condition for the connectedness of $\mathcal{J}_{(c_n)}$ which implies that $\mathcal{J}_{(c_n)}$ is connected if $|c_n| \le \frac{1}{4}$, while it is almost surely disconnected if $|c_n| \le \delta$ for some $\delta>\frac{1}{4}$.

scholarly journals Linear diophantine equations with cyclic coefficient matrices and its applications to Riemann surfaces

1985 ◽  
Vol 28 (3) ◽  
pp. 369-380
Author(s):  
Nobumasa Takigawa
Keyword(s):  
Riemann Surfaces ◽  
Diophantine Equations ◽  
Linear Equations ◽  
Integral Solution ◽  
Complex Numbers ◽  
Necessary And Sufficient Condition ◽  
Linear Diophantine Equations ◽  
Fixed Complex ◽  
Image Position ◽  
Necessary And Sufficient

Let co, c1, …, cn-1 be the nonzero complex numbers and let C = (cu+1,v+1) = (cn+u-v), O≦u,v≦n — 1, be a cyclic matrix, where n + u — v is taken modulo n. In this paper we shall give the solution of the linear equationswhere Lu (0≦u≦n —1) is a fixed complex number. In Theorem 1 weshall give a necessary and sufficient condition for (1) to have an integral solution.

scholarly journals On the Irreducibility of Fourth Dimensional Tuba's Representation of the Pure Braid Group on Three Strands

2018 ◽  
Vol 11 (3) ◽  
pp. 682-701
Author(s):  
Hasan A. Haidar ◽  
Mohammad N. Abdulrahim
Keyword(s):  
Braid Group ◽  
Sufficient Conditions ◽  
Complex Numbers ◽  
Closed Field ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Algebraically Closed Field ◽  
Pure Braid Group ◽  
Necessary And Sufficient

We consider Tuba's representation of the pure braid group, $%P_{3} $, given by the map $\phi :P_{3}\longrightarrow GL(4,F)$, where $F$ is an algebraically closed field. After, specializing the indeterminates used in defining the representation to non- zero complex numbers, we find sufficient conditions that guarantee the irreducibility of Tuba's representation of the pure braid group $P_{3}$ with dimension $d=4$. Under further restriction for the complex specialization of the indeterminates, we get a necessary and sufficient condition for the irreducibility of $\phi

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