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

$L e t A \in R^{N \times N} b e a s y m m e t r i c p o s i t i v e$ $d e f i n i t e m a t r i x a n d b \in R^{N} a v e c t o r .$ $I f x \in R^{N}, e v a l u a t e t h e i n t e g r a l$ $Z (A, b) = \int e^{- \frac{1}{2} x^{T} A x + b^{T} x} d x a s a f u n c t i o n$ $o f A a n d b .$
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