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35 Functions of Matrix ArgumentProperties

§35.2 Laplace Transform


For any complex symmetric matrix Z,

35.2.1 g(Z)=Ωetr(-ZX)f(X)dX,

where the integration variable X ranges over the space Ω.

Suppose there exists a constant X0Ω such that |f(X)|<etr(-X0X) for all XΩ. Then (35.2.1) converges absolutely on the region (Z)>X0, and g(Z) is a complex analytic function of all elements zj,k of Z.

Inversion Formula

Assume that 𝒮|g(U+iV)|dV converges, and also that its limit as U is 0. Then

35.2.2 f(X)=1(2πi)m(m+1)/2etr(ZX)g(Z)dZ,

where the integral is taken over all Z=U+iV such that U>X0 and V ranges over 𝒮.

Convolution Theorem

If gj is the Laplace transform of fj, j=1,2, then g1g2 is the Laplace transform of the convolution f1*f2, where

35.2.3 f1*f2(T)=0<X<Tf1(T-X)f2(X)dX.