This procedure is based on the definition of the inverse matrix: any matrix multiplied by its inverse is equal to the identity (or unit) matrix. So matrix X is the unknown of the matrix equation. 3, we found that v Tx p l f p where l solves the. In our nonlinear-equation setting f(x p)0 of Sec. Now, to simplify the following steps, we will name A the square matrix that has the coefficients of the unknowns, X the column matrix with the unknowns, and B the column matrix with the independent terms: Another way of thinking about adjoint methods is that they correspond to the observation that the vector Jacobian product vT x p (a vJp), for any given vector v 2RM, is much cheaper to compute than the M P Ja-cobian matrix x p itself. We can verify that these matrices correspond to the system of equations by multiplying the matrices, since we would obtain the two equations of the system. Let’s see an example of how to do it:Ī system of equations can be expressed with matrices: Well, one of the applications of the inverse matrix is the resolution of systems of linear equations. Now you may be wondering… what is the inverse matrix for? Is it really used for something? Solving a system of equations with the inverse matrix
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