Diagonalization of matrices, classification of quadratic forms, and linear programmingMain objectivesThe objective of this activity is to consolidate your learning about linear applications and diagonalization.In addition, a linear programming problem will be modeled using systems of linear inequalities and their associated matrix, and the solution will be proposed using the simplex algorithm.Description of the activity and instructionsThe activity consists of four exercises, with several sections each one that you will have to solve.The exercises must be solved showing the solving process. It will not be evaluated if the student only presents the solution of the exercise without explaining how he/she has reached the result.

Task 1. Diagonalization of matrices. Obtain, if possible, the diagonal matrix of the following matrices. Show the spectrum of the matrices, the multiplicities of the eigenvalues, the eigen subspaces and if possible, obtain the diagonal matrix and the invertible matrix. N to H r"'_-_""--. MN DON Oil5N DUO ~..____...r Task 2. Diagonalization and classification of quadratic forms. It is required to construct the matrix for these quadratic forms and applying the diagonalization method studied, obtain the diagonal matrix and classify the quadratic form. a) Classify the quadratic form by obtaining its diagonal matrix q(x,y,z) = 2x2 + y2 + 422 4x)! + 10312 b) Classify the quadratic form by obtaining its diagonal matrix QOCJAZ) = x2 + 4y2 + 222 430) + 6x2 + 10yz