Inversion lemma

The above equations are developed using the theory of least squares and making use of the matrix inversion lemma... 

To solve the least squares problem for the estimate of the measurement errors we need to invert the covariance matrix <. It is possible to relate to through a simple recursive formula. Let us recall the following matrix inversion lemma (Noble, 1969) ... 

The application of the matrix inversion lemma leads to recursive formulas similar to those previously obtained in the conventional approach set out in Section 6.3. 

It should be noted that the solution of the minimization problem simplifies to the updating step of a Kalman filter. In fact, if instead of applying the matrix inversion lemma to Eq. (8.19) to produce Eq. (8.20), the inversion is performed on the estimate equation (8.18), the well-known form of the Kaman filter equations is obtained. 

Now, since the purpose of an experimental program is to gain information, in attempting to design the experiment we will try to plan the measurements in such a way that the final error estimate covariance is minimal. In our case, this can be achieved in a sequential manner by applying the matrix inversion lemma to Eq. (9.17), as we have shown in previous chapters. 

The likelihood function in Eq. 3.46 is used during the sparsiiication procedure in order to optimise the hyperparameters and the sparse points. At first sight, it seems that the inverse of an A x A matrix has to be calculated, the computational cost of which would scale as N. However, by using the matrix inversion lemma, also known as the Woodbury matrix identity, the computational cost scales only with NM if iV M. If we want to find the inverse of a matrix, which can be written in the form Z + UWV, the Woodbury matrix identity states that... 

This follows from the rewriting XjX,= Xt.i Xt.i+ x, x,, X,V = XnV-i + and the application of the matrix inversion lemma. This leads to the Kalman filter equations... 

It was around this time that Gian-Carlo Rota was developing his elegant theory of Mobius Inversion, a theory with far-reaching consequences in many branches of combinatorial mathematics. The basic paper on this topic is [RotG64J. Mobius inversion can be used to derive a proof of P ya s Theorem which is not only different from the usual proof via Burnside s Lemma, but also has potential for generalization and application in new directions. 

We note in passing that Lemma 1 and Theorem 1 guarantee the existence of an inverse operator defined only on TZ A), the range of A, which is not obliged to coincide with H. If the range of an operator A happens to be the entire space H, TZ(A) = H, then the conditions of Lemma 1 or Theorem 1 ensure the existence of an operator A with T>[A ) = H. In particular, a positive operator A with the range TZ A) = H possesses an inverse with V[A ) = H, since the condition Ax, x) > 0 for all x Q implies that Ax yf 0 for x yf 0 and Lemma 1 applies equally well to such a setting. 

Proof. Inserting the expression for - given in Lemma 3 into the left-hand side of (34) and multiplying the equation thus obtained by the inverse of the nonsingular matrix H we arrive at the system of matrix equations, whose left-hand sides are the linear combinations of the linearly independent matrices E, Syield system of Eqs. (39). The assertion is proved. 

A similar statement is also valid for the product b (u)b(Hkp). Let us designate the contour from Lemma 1, through Cfa/i Then the inversely directed contour will be designated as Qe/fa ... 

Proof, (i) Note that the inverse map of an isomorphism is an isomorphism. Thus, the claim is an immediate consequence of Lemma 5.2.1. 

A homomorphism of affine group schemes is a natural map G -+ H for which each G(R) - H(R) is a homomorphism. We have already seen the example det GL - Gm. The Yoneda lemma shows as expected that such maps correspond to Hopf algebra homomorphisms. But since any map between groups preserving multiplication also preserves units and inverses, we need to check only that A is preserved. An algebra homomorphism between Hopf algebras which preserves A must automatically preserve S and e. 

Lemma 8.3 The gTSL are closed under inverse mgsm mapping. 

Lemma 8.2.2. The J are regular formal schemes and the inverse images i 1 (resjp.j p. if 1 are reKular divisors... 

Lemma-Definition 3.3.5. // / X Y, X Y are functors with respective left adjoints f Y X., g Y X, then with Horn denoting functorial morphisms, the following natural compositions are inverse isomorphisms ... 

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