Exponential Map on complex matrices (original) (raw)

Multiplicative maps on invertible matrices that preserve matricial properties

The Electronic Journal of Linear Algebra, 2003

Descriptions are given of multiplicative maps on complex and real matrices that leave invariant a certain function, property, or set of matrices: norms, spectrum, spectral radius, elementary symmetric functions of eigenvalues, certain functions of singular values, (p, q) numerical ranges and radii, sets of unitary, normal, or Hermitian matrices, as well as sets of Hermitian matrices with fixed inertia. The treatment of all these cases is unified, and is based on general group theoretic results concerning multiplicative maps of general and special linear groups, which in turn are based on classical results by Borel-Tits. Multiplicative maps that leave invariant elementary symmetric functions of eigenvalues and spectra are described also for matrices over a general commutative field.

Property L and Commuting Exponentials in Dimension at Most Three

Bulletin of the Australian Mathematical Society, 2013

Let A,BA, BA,B be two square complex matrices of the same dimension nleq3n\leq 3nleq3. We show that the following conditions are equivalent. (i) There exists a finite subset UsubsetmathbbNgeq2U\subset { \mathbb{N} }_{\geq 2} UsubsetmathbbNgeq2 such that for every tinmathbbNsetminusUt\in \mathbb{N} \setminus UtinmathbbNsetminusU, exp(tA+B)=exp(tA)exp(B)=exp(B)exp(tA)\exp (tA+ B)= \exp (tA)\exp (B)= \exp (B)\exp (tA)exp(tA+B)=exp(tA)exp(B)=exp(B)exp(tA). (ii) The pair (A,B)(A, B)(A,B) has property L of Motzkin and Taussky and exp(A+B)=exp(A)exp(B)=exp(B)exp(A)\exp (A+ B)= \exp (A)\exp (B)= \exp (B)\exp (A)exp(A+B)=exp(A)exp(B)=exp(B)exp(A). We also characterise the pairs of real matrices (A,B)(A, B)(A,B) of dimension three, that satisfy the previous conditions.

Some properties of the matrix exponential

IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 2001

We give a simple condition on a matrix for which if the exponential matrix is diagonal, lower or upper triangular, then so is . It is also shown that for diagonalizable and any matrix , and commute if and only if and commute. These results are useful in problems in which knowledge about has to be extracted from structural information about its exponential, such as in large-scale sampled-data systems. They also find application in the area of blind system identification.

On additive transformations preserving a multiplicative matrix function

Applied Mathematics and Computation, 2007

Let M n ðKÞ be the ring of all n Â n matrices over a division ring K, and f be a multiplicative matrix function from M n ðKÞ to a multiplicative Abelian group with zero G [ f0g ðf ðABÞ ¼ f ðAÞf ðBÞ; 8A; B 2 M n ðKÞÞ. We call an additive transformation / on M n ðKÞ preserves a multiplicative matrix function f, if f ð/ðAÞÞ ¼ f ðAÞ; 8A 2 M n ðKÞ. In this paper, we characterize all additive surjective transformations on M n ðKÞ over any division ring K ðchK 6 ¼ 2Þ that leave a non-trivial multiplicative matrix function invariant. Applications to several related preservers are considered.

A matrix generalization of Euler identity e jϕ = cosϕ + j sinϕ

2007

In this work we present a matrix generalization of the Euler identity about exponential representation of a complex number. The concept of matrix exponential is used in a fundamental way. We define a notion of matrix imaginary unit which generalizes the usual complex imaginary unit. The Euler-like identity so obtained is compatible with the classical one. Also, we derive some exponential representation for matrix real and imaginary unit, and for the first Pauli matrix.

On the product of matrix exponentials

Linear Algebra and its Applications, 1994

We study the family of positive definite Hermitian matrices of the form (ets/zerAets/z)l/t for t > 0, where A and B are Hermitian. In particular, we show that the above matrix family converges to a limit when t + ~0.

A matrix generalization of Euler identity e^(ix) = cosx + i sinx

A Memoir on the Theory of Matrices

The term matrix might be used in a more general sense, but in the present memoir I consider only sqnare and rectangular matrices, and the term matrix used without qualification is to be understood as meaning a square matrix; in this restricted sense, a set of quantities arranged in the form of a sqnare, . ,.

Notes on linear combinations of two tripotent, idempotent, and involutive matrices that commute

2008

The aim of this paper is to provide alternate proofs of all the results of our previous paper [2] in the particular case when the given two matrices A1 and A2 in the linear combination A = c1A1 + c2A2 commute. 1 Introduction and Preliminaries Let C and C m,n denote the sets of complex numbers and m × n complex matrices. Moreover, C * will mean C \ {0}. Now, consider a linear combination of the form A = c 1 A 1 + c 2 A 2 , (1.1) where A 1 , A 2 ∈ C n,n are nonzero matrices and c 1 , c 2 ∈ C *. The aim of this paper is to provide alternate proofs of all the results of our previous paper [2] in the particular case that A 1 and A 2 in (1.1) are commuting matrices, i.e. A 1 A 2 = A 2 A 1. Recall that a matrix B ∈ C n,n is said to be similar to a matrix A ∈ C n,n if there exists a nonsingular matrix P ∈ C n,n such that B = P −1 AP.

The Real Matrices forms of the Bicomplex Numbers and Homothetic Exponential motions

2014

In this paper, a bicomplex number is described in four- dimensional space and its a variety of algebraic properties is presented. In addition, Pauli-spin matrix elements corresponding to base the real matrices forms of the bicomplex numbers are obtained and its the algebraic properties are given. Like i and j in two different spaces are defined terms of Euler's formula. In the last section velocities become higher order by giving an exponential homothetic motion for the bicomplex numbers. And then, Due to the way in which the matter is presented, the paper gives some formula and facts about exponential homothetic motions which are not generally known

Exponential Map on complex matrices (original) (raw)

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