The QR Method for Determining All Eigenvalues of Real Square Matrices

Abstract

Eigenvalues are special sets of scalars associated with a given matrix. In other words for a given matrix A, if there exist a non-zero vector V such that, AV= λV for some scalar λ, then λ is called the eigenvalue of matrix A with corresponding eigenvector V. The set of all nxm matrices over a field F is denoted by M nm (F). If m = n, then the matrices are square, and which is denoted by Mn (F). We omit the field F = C and in this case we simply write M nm or M n as appropriate. Each square matrix AϵM nm has a value in R associated with it and it is called its determinant which is use full for solving a system of linear equation and it is denoted by det (A). Consider a square matrix AϵM n with eigenvalues λ, and then by definition the eigenvectors of A satisfy the equation, AV = λV, where v={v 1, v 2, v 3…………v n }. That is, AV= λ V is equivalent to the homogeneous system of linear equation (A- λI) v=0. This homogeneous system can be written compactly as (A- λ I) V = 0 and from Cramer’s rule, we know that a linear system of equation has a non-trivial solution if and only if its determinant is zero, so the solution λ is given by det (A- λ I) =0. This is called the characteristic equation of matrix A and the left hand side of the characteristic equation is the characteristic polynomial whose roots are equals to λ.