# Study and analyze the eigenvalues and eigenvectors of a square matrix and study their applications through mathematical linear effects

## Abstract

In this paper we will investigate what properties are intrinsic to a matrix, or its associated linear application. As we will see, the fact that there are many bases in a vector space makes the expression of matrices or linear applications relative: it depends on which reference base we take. However, there are elements associated with this matrix, which do not depend on the reference base or bases that we choose, for example: a null spaces and a column spaces of a matrix, and their respective    dimensions. The eigen values of matrix isa root of a character polynomial. Find a eigenvalue of matrix is equivalent to finding a rootof its polynomial. For matrices of size n ≥ 5, there does not generally existclosed expressions for the roots of the characteristic polynomial based on primary expressions (additions, subtractions, multiplications, divisions and roots). This result implies that the methods to find the valuesof a matrix must be iterativeOne way to calculate the eigen values would be to calculate the roots of thecharacteristic polynomial using a numerical method of calculating roots, like roots in Matlab. Roots in Python. But find the roots of a polynomial is usually a poorly conditioned problem. The conditioning of a problem has not been defined, but a wrong problemconditioned is a problem for which a small change in the data can induce an uncontrolled change in the results.

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