scholarly journals Real dimension of the Lie algebra of S-skew-Hermitian matrices

2022 ◽  
pp. 49-62
Author(s):  
Jonathan Caalim ◽  
Yu-ichi Tanaka
Keyword(s):  
Lie Algebra ◽  
Jordan Block ◽  
Complex Numbers ◽  
Block Decomposition ◽  
Hermitian Matrices ◽  
Real Dimension ◽  
Dimension Formula ◽  
Conjugate Transpose

Let $M_n(\mathbb{C})$ be the set of $n\times n$ matrices over the complex numbers. Let $S \in M_n(\mathbb{C})$. A matrix $A\in M_n(\mathbb{C})$ is said to be $S$-skew-Hermitian if $SA^*=-AS$ where $A^*$ is the conjugate transpose of $A$. The set $\mathfrak{u}_S$ of all $S$-skew-Hermitian matrices is a Lie algebra. In this paper, we give a real dimension formula for $\mathfrak{u}_S$ using the Jordan block decomposition of the cosquare $S(S^*)^{-1}$ of $S$ when $S$ is nonsingular.

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.).

When is a Matrix Positive

1974 ◽  
Vol 17 (3) ◽  
pp. 409-410
Author(s):  
Daniel Brand
Keyword(s):  
Complex Numbers ◽  
Identity Matrix ◽  
Conjugate Transpose

Throughout this note we shall use the following conventions and notations: All matrices have entries in the field of complex numbers. I denotes the identity matrix with compatible dimensions. A* is the conjugate transpose of a matrix A. A being self adjoint means A = A*.

scholarly journals Minimal representations for 6-dimensional nilpotent Lie algebra

2016 ◽  
Vol 15 (10) ◽  
pp. 1650191 ◽  
Author(s):  
Nadina Rojas
Keyword(s):  
Lie Algebra ◽  
Characteristic Zero ◽  
Faithful Representation ◽  
Nilpotent Lie Algebra ◽  
Dimension Formula ◽  
Minimal Dimension ◽  
Minimal Representations

Given a Lie algebra [Formula: see text], let [Formula: see text] and [Formula: see text] be the minimal dimension of a faithful representation and nilrepresentation of [Formula: see text], respectively. In this paper, we give [Formula: see text] and [Formula: see text] for each nilpotent Lie algebra [Formula: see text] of dimension [Formula: see text] over a field [Formula: see text] of characteristic zero. We also give a minimal faithful representation and nilrepresentation in each case.

scholarly journals Certain functors of nilpotent Lie algebras with the derived subalgebra of dimension two

2019 ◽  
Vol 19 (01) ◽  
pp. 2050012
Author(s):  
Farangis Johari ◽  
Peyman Niroomand
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Schur Multiplier ◽  
Nilpotent Lie Algebra ◽  
Nilpotent Lie Algebras ◽  
Dimension Formula ◽  
Tensor Square ◽  
Exterior Square

By considering the nilpotent Lie algebra with the derived subalgebra of dimension [Formula: see text], we compute some functors including the Schur multiplier, the exterior square and the tensor square of these Lie algebras. We also give the corank of such Lie algebras.

On Pearl's Paper "A Decomposition Theorem for Matrices"*

1969 ◽  
Vol 12 (6) ◽  
pp. 805-808 ◽  
Author(s):  
R. C. Thompson
Keyword(s):  
Decomposition Theorem ◽  
Real Case ◽  
Complex Numbers ◽  
Orthogonal Matrices ◽  
The Real ◽  
Conjugate Transpose ◽  
Published Paper

Let A be an m × n matrix of complex numbers. Let Aτ and A* denote the transpose and conjugate transpose, respectively, of A. We say A is diagonal if A contains only zeros in all positions (i, j) with i ≠ j. In a recently published paper [4], M.H. Pearl established the following fact: There exist real orthogonal matrices O1 and O2 (O1 m-square, O2 n-square) such that O1AO2 is diagonal, if and only if both AA* and A*A are real. It is the purpose of this paper to show that a theorem substantially stronger than this result of Pearl's is included in the real case of a theorem of N.A. Wiegmann [2]. (For other papers related to Wiegmann's, see [l; 3].)

scholarly journals Post-Lie algebra structures for nilpotent Lie algebras

2018 ◽  
Vol 28 (05) ◽  
pp. 915-933
Author(s):  
Dietrich Burde ◽  
Christof Ender ◽  
Wolfgang Alexander Moens
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Nilpotent Lie Algebras ◽  
Dimension Formula ◽  
Necessary And Sufficient

We study post-Lie algebra structures on [Formula: see text] for nilpotent Lie algebras. First, we show that if [Formula: see text] is nilpotent such that [Formula: see text], then also [Formula: see text] must be nilpotent, of bounded class. For post-Lie algebra structures [Formula: see text] on pairs of [Formula: see text]-step nilpotent Lie algebras [Formula: see text] we give necessary and sufficient conditions such that [Formula: see text] defines a CPA-structure on [Formula: see text], or on [Formula: see text]. As a corollary, we obtain that every LR-structure on a Heisenberg Lie algebra of dimension [Formula: see text] is complete. Finally, we classify all post-Lie algebra structures on [Formula: see text] for [Formula: see text], where [Formula: see text] is the three-dimensional Heisenberg Lie algebra.

scholarly journals Linear Transformations on Algebras of Matrices

1959 ◽  
Vol 11 ◽  
pp. 61-66 ◽  
Author(s):  
Marvin Marcus ◽  
B. N. Moyls
Keyword(s):  
Linear Transformation ◽  
Complex Numbers ◽  
List Type ◽  
Hermitian Matrices ◽  
Linear Transformations ◽  
Unimodular Group

Let Mn denote the algebra of n-square matrices over the complex numbers; and let Un, Hn, and Rk denote respectively the unimodular group, the set of Hermitian matrices, and the set of matrices of rank k, in Mn. Let ev(A) be the set of n eigenvalues of A counting multiplicities. We consider the problem of determining the structure of any linear transformation (l.t.) T of Mn into Mn having one or more of the following properties:(a)T(Rk) ⊆ for k = 1, …, n.(b)T(Un) ⊆ Un(c)det T(A) = det A for all A ∈ Hn.(d)ev(T(A)) = ev(A) for all A ∈ Hn.We remark that we are not in general assuming that T is a multiplicative homomorphism; more precisely, T is a mapping of Mn into itself, satisfyingT(aA + bB) = aT(A) + bT(B)for all A, B in Mn and all complex numbers a, b.

A note on positive definite matrices

1958 ◽  
Vol 3 (4) ◽  
pp. 173-175 ◽  
Author(s):  
W. N. Everitt
Keyword(s):  
Positive Definite ◽  
Vector Spaces ◽  
Hermitian Matrices ◽  
Positive Definite Matrices ◽  
Even Order ◽  
Conjugate Transpose

This note is concerned with an inequality for even order positive definite hermitian matrices together with an application to vector spaces.The abbreviations p.d. and p.s-d. are used for positive definite and positive semi-definite respectively. An asterisk denotes the conjugate transpose of a matrix.

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