Let-A-R-N-N-be-a-symmetric-positive-definite-matrix-and-b-R-N-a-vector-If-x-R-N-evaluate-the-integral-Z-A-b-e-1-2-x-T-Ax-b-T-x-dx-as-a-function-of-A-and-b-

Tinku Tara February 1, 2024 Vector Calculus 0 Comments

Question Number 203900 by necx122 last updated on 01/Feb/24

Let A ∈ R^(N×N) be a symmetric positive definite matrix and b ∈ R^N a vector. If x ∈ R^N , evaluate the integral Z(A,b) = ∫e^(−(1/2)x^T Ax + b^T x) dx as a function of A and b.

$${Let}\:{A}\:\in\:{R}^{{N}×{N}} \:{be}\:{a}\:{symmetric}\:{positive} \\ $$$${definite}\:{matrix}\:{and}\:{b}\:\in\:{R}^{{N}} \:{a}\:{vector}. \\ $$$${If}\:{x}\:\in\:{R}^{{N}} ,\:{evaluate}\:{the}\:{integral} \\ $$$${Z}\left({A},{b}\right)\:=\:\int{e}^{−\frac{\mathrm{1}}{\mathrm{2}}{x}^{{T}} {Ax}\:+\:{b}^{{T}} {x}} {dx}\:{as}\:{a}\:{function} \\ $$$${of}\:{A}\:{and}\:{b}. \\ $$

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Let-A-R-N-N-be-a-symmetric-positive-definite-matrix-and-b-R-N-a-vector-If-x-R-N-evaluate-the-integral-Z-A-b-e-1-2-x-T-Ax-b-T-x-dx-as-a-function-of-A-and-b-

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