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 Mnm (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 Mnm or Mn as appropriate. Each square matrix AϵMnm 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 AϵMn with eigenvalues λ, and then by definition the eigenvectors of A satisfy the equation, AV = λV, where v={v1, v2, v3…………vn}. That is, AV=λV is equivalent to the homogeneous system of linear equation (A-λI) v=0. This homogeneous system can be written compactly as (A-λI) 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-λI) =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.
Highlights
IntroductionThe vibrational frequencies are determined by the eigenvalues of a symmetric 3x3 matrix
Each nxn square matrix A has a value associated with its determinant denoted by det (A), which is useful to solve a system of linear equation and if there exist a nonzero vector V and scalar λ such that AV = λ V, λ is called eigenvalue of matrix A with corresponding eigenvector V
This process will be stopped when the entries below the main diagonal of current matrix Am are sufficiently small, or if it appears that convergence will not happen. This implies that, QR factorization sequence process can fail to converge or the convergence can be extremely slow and expensive. If it converges to certain matrix, the diagonal entries of this current matrix are tends to be eigenvalues of matrix A, if not, it can be modified in order to speed up convergence or to accelerate the rate of convergence of the given real square matrix dramatically we use methods like shifting of origin
Summary
The vibrational frequencies are determined by the eigenvalues of a symmetric 3x3 matrix. Since computing the eigenvalue λ is equivalent to finding the roots of matrix’s characteristic polynomial, we can see that task is quickly too difficult for larger dimensional matrices/especially for the case of matrices which have more than three dimensions/ even if we know characteristic polynomial and algorithms such as Newton’s method for finding zeros cannot be depended upon produce all the zeros with reasonable speed and accuracy. Numerical analysts have found an entirely different ways to calculate eigenvalues of a given square matrix. Among those methods QR method is the most widely used, important, accurate and speedy one. Eyaya Fekadie Anley: The QR Method for Determining All Eigenvalues of Real Square Matrices
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