In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems...

mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization...

an orthogonal matrix, it admits an exponential form. Correspondingly, the matrix S writes as exponential of a skew-symmetric block matrix Σ {\displaystyle...

In probability theory, the matrix-exponential distribution is an absolutely continuous distribution with rational Laplace–Stieltjes transform. They were...

to exponentiation, including: Exponential function, also: Matrix exponential, the matrix analogue to the above Exponential decay, decrease at a rate proportional...

Pascal's triangle in terms of the matrix exponential can be given: Pascal's triangle is the exponential of the matrix which has the sequence 1, 2, 3, 4...

eigenvalues), one can derive a relation between the trace function, the matrix exponential function, and the determinant: det ( exp ⁡ ( A ) ) = exp ⁡ ( tr ⁡...

determine the product of exponentials for a kinematic chain, with the goal of parameterizing an affine transformation matrix between the base and tool...

written as U = eiH, where e indicates the matrix exponential, i is the imaginary unit, and H is a Hermitian matrix. For any nonnegative integer n, the set...

is the identity matrix. Thus, in the setting of matrix Lie groups, the exponential map is the restriction of the matrix exponential to the Lie algebra...

rotation matrix or differential rotation matrix is a matrix representing an infinitely small rotation. While a rotation matrix is an orthogonal matrix R T...

computer design. The exponential function can also be defined as a power series, which is readily applied to real, complex, and even matrix arguments. The complex...

exponential map is a map from the Lie algebra g of a Lie group G into G. In case G is a matrix Lie group, the exponential map reduces to the matrix exponential...

formal matrix exponential solution may be reduced to a simple form. Below, this solution is displayed in terms of Putzer's algorithm. The matrix equation...

orthogonal matrix group consists of skew-symmetric matrices. Going the other direction, the matrix exponential of any skew-symmetric matrix is an orthogonal...

Lie group is the exponential map, which is defined using the standard matrix exponential series for eA For any skew-symmetric matrix A, exp(A) is always...

the exponential itself is just 1, which makes it the generic group element of SU(2). A more abstract version of formula (2) for a general 2 × 2 matrix can...

then the exponential map takes the Lie algebra of G {\displaystyle G} into G {\displaystyle G} ; thus, we have an exponential map for all matrix groups...

can be used to compute the matrix exponential eA, a need frequently arising in solving linear differential equations, matrix logarithms and square roots...

\end{aligned}}} This approach can be generalized to matrix exponential and other matrix functions that can be defined as power series. For example...

\mathbf {y} } . If A {\displaystyle A} is skew-Hermitian, then the matrix exponential e A {\displaystyle e^{A}} is unitary. The space of skew-Hermitian...

s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} ). In terms of the matrix exponential, R = exp ⁡ ( θ K ) . {\displaystyle \mathbf {R} =\exp(\theta \mathbf...

sequence Exponential smoothing Exponential stability Exponential sum Exponential time Sub-exponential time Exponential tree Exponential type Exponentially equivalent...

satisfy ATJA = J. Thus, the matrix exponential of a Hamiltonian matrix is symplectic. However the logarithm of a symplectic matrix is not necessarily Hamiltonian...

trickier due to the matrix exponential integral. It can, however, be computed by first constructing a matrix, and computing the exponential of it F = [ − A...

case, we can define Φ {\displaystyle \mathbf {\Phi } } , using the matrix exponential, as Φ ( t , t 0 ) = e A ( t − t 0 ) {\displaystyle \mathbf {\Phi }...

idempotent. The matrix exponential of J is exp ⁡ ( μ J ) = I + e μ n − 1 n J {\displaystyle \exp(\mu J)=I+{\frac {e^{\mu n}-1}{n}}J} The all-ones matrix arises...

when inverting the exponential map, that is, when finding a rotation vector that corresponds to a given rotation matrix. The exponential map is onto but...

x and y are replaced with matrices A and B, and the exponential replaced with a matrix exponential, it is usually necessary for A and B to commute for...

integral in the commutative algebras. In practice the ordered exponential is used in matrix and operator algebras. It is a kind of product integral, or...