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By Terence Tao

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R / be a Gaussian vector. 3) B Lt M 28 Chapter 1. Gaussian integral operators Proof. R0 / ! Rn / and we apply the previous theorem. 3 Categories. Rn / and morphisms are Gaussian operators. Recall the definition of abstract categories. RQ/P: Example 1. The objects are finite-dimensional linear spaces Cn , morphisms are linear operators. , etc. Example 2. Rn / ! Rm / are Gaussian operators BŒS with Re S > 0. Example 3. The objects are nonnegative integers n D 0, 1, 2,…, morphisms n ! m L such that S D S t and Re S > 0.

10) Example. 7. Linear relations. 12) of linear equations. They only differ in signs. Let us formulate our observation more precisely. Rm ˚ Rn / be a Gaussian distribution. Consider the subspace P . / V2mC2n defined in the previous subsection. Vm Consider the subspace L. 10). 3. v ˚ w / 2 P . w C ˚ w / ˚ .. v C / ˚ v / 2 L. /: Below we formulate this result in a more closed form. 4 A preliminary remark on products of two Gaussian operators. 1) for products of Gaussian operators. 4. Let v ˚ w 2 LŒS2  and w ˚ y 2 LŒS1 .

Emulation of basic definitions of matrix theory 39 Remark. Informally, indef P is the image of 0 under P . Remark. Obviously, P is the graph of an operator V ! W if and only if indef P D 0 and dom P D V . If P is the graph of an operator W ! V , then ker P D 0 and im P D W . 2) 5B . We also define the rank of P : rk P D dim P dim ker P dim indef P D dim dom P dim ker P D dim im P dim indef P: This definition extends the definition of the rank of a linear operator. rk Q; rk P /: 6B . The last definition of this series is a pseudoinverse linear relation P .

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