# Analysing the properties of a probability distribution is a question of general interest. In this paper we describe the properties of the matrix-exponential class of distributions, developing some

The Exponential Map. Note that the exponential of a matrix is always an invertible matrix.The inverse matrix of eX is given by e−X.This is analogous to the fact that the exponential of a …

A3 + It is not difﬁcult to show that this sum converges for all complex matrices A of any ﬁnite dimension. But we will not prove this here. If A is a 1 t1 matrix [t], then eA = [e ], by the The sum of the infinite series is called the matrix exponential and denoted as This series is absolutely convergent. In the limiting case, when the matrix consists of a single number i.e. has a size of 1 Properties of the Matrix Exponential Let A be a real or complex n×n matrix.

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These properties are easily verifiable and left as … Notes on the matrix exponential Erik Wahlén erik.wahlen@math.lu.se ebruaryF 14, 2012 1 Introduction The purpose of these notes is to describe how one can compute the matrix exponential eA when A is not diagonalisable. This is done in escThl by transforming A into Jordan normal form. As we will see here, it is not necessary to go this far. where we have de ned the \matrix exponential" of a diagonalizable matrix as: eAt= Xe tX 1 Note that we have de ned the exponential e t of a diagonal matrix to be the diagonal matrix of the e tvalues. Equivalently, eAtis the matrix with the same eigenvectors as A but with eigenvalues replaced by e t.

1 Properties of the Matrix Exponential Let A be a real or complex n×n matrix. The exponential of A is deﬁned via its Taylor series, eA = I + X∞ n=1 An n!, (1) where I is the n×n identity matrix.

## exponential is not. Matrix-matrix exponentiation A 0 is defined for certain matrices A and B (Definition 3.1 ), and properties of A0 analogous to those of scalar-scalar exponentiation ab are described. Von Neumann's quantum-mechanical entropy can be written using matrix-matrix exponentiation in a form 76 0022-247X/94 S6.00

Example16.Let D= 2 0 0 2 ; N= 0 1 0 0 and A= D+ N= 2 1 0 2 : The matrix Ais not diagonalizable, since the only eigenvalue is 2 and Cx = 2 x hasthesolution x = z 1 0 ; z2C: SinceDisdiagonal,wehavethat etD= e2t 0 0 e2t : Moreover,N2 = 0 (conﬁrmthis!),so etN = I+ tN= 1 t 0 1 8 A is simply the matrix for this linear operator in the standard basis fe 1;:::;e ng, where e 1 = (1;0;0;:::;0), e 2 = (0;1;0;:::;0), etc. If we choose a new basis ff 1;:::;f n g, then the matrix for the operator in the new basis is B = T 1AT, where T is the matrix whose columns consist of the coordinates for the vectors f j in the old basis (the standard basis).

### A more con- ceptual explanation is that matrix exponential manipulations do not work as in the scalar case unless the matrices involved commute. Such is the

In mathematics, the matrix exponentialis a matrix functionon square matricesanalogous to the ordinary exponential function. It is used to solve systems of linear differential equations.

The radius of convergence of the above series is inﬁnite. Consequently, eq. (1) converges for all matrices A. In these notes, we discuss a
10.4 Matrix Exponential 505 10.4 Matrix Exponential The problem x′(t) = Ax(t), x(0) = x0 has a unique solution, according to the Picard-Lindel¨of theorem. Solve the problem n times, when x0 equals a column of the identity matrix,
where we have de ned the \matrix exponential" of a diagonalizable matrix as: eAt= Xe tX 1 Note that we have de ned the exponential e t of a diagonal matrix to be the diagonal matrix of the e tvalues. Equivalently, eAtis the matrix with the same eigenvectors as A but with eigenvalues replaced by e t. In mathematics, the matrix exponentialis a matrix functionon square matricesanalogous to the ordinary exponential function.

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The matrix P X is idempotent, and more generally, the trace of any idempotent matrix equals its own rank. Exponential trace. Expressions like tr(exp(A)), where A is a square matrix, occur so often in some fields (e.g. multivariate statistical theory), that a shorthand notation has become common: Analysing the properties of a probability distribution is a question of general interest.

In some cases, it is a simple matter to express
Moreover,. M(t) is an invertible matrix for every t. These two properties characterize fundamental matrix solutions.) (Remark 2: Given a linear system, fundamental
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential
Matrix exponentials provide a concise way of describing the solutions to systems of homoge- neous linear and have reasonable properties. Limits and infinite
However, from a theoretical point of view it is important to know properties of this matrix function.

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### ferential equations, hence the asymptotic properties of matrix exponential func- Here, as explained in Section 2.2, exp(tA)=etA stands for the matrix ft(A).

9/17/99. Transformations. Now that we have a representation of the solution of constant- coefficient. Depending on the properties of A different numerical alternatives might be used ( see e.g.

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### A is simply the matrix for this linear operator in the standard basis fe 1;:::;e ng, where e 1 = (1;0;0;:::;0), e 2 = (0;1;0;:::;0), etc. If we choose a new basis ff 1;:::;f n g, then the matrix for the operator in the new basis is B = T 1AT, where T is the matrix whose columns consist of the coordinates for the vectors f j in the old basis (the standard basis).

A more con- ceptual explanation is that matrix exponential manipulations do not work as in the scalar case unless the matrices involved commute. Such is the That's equvialent to an upper triangular matrix, with the main diagonal elements equal to 1.