Liner eigenvalue.docx - EIGENVALUES Eigenvalues are a special set of scalars associated with a linear system of equations that are sometimes also known

Liner eigenvalue.docx - EIGENVALUES Eigenvalues are a...

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EIGENVALUES Eigenvalues are a special set of scalars associated with a linear system of equations that are sometimes also known as characteristic roots, characteristic values proper values or latent roots . The determination of the eigenvalues and eigenvectors of a system is extremely important in physics and engineering, where it is equivalent to matrix diagonalization and arises in such common applications as stability analysis, the physics of rotating bodies, and small oscillations of vibrating systems, to name only a few. Each eigenvalue is paired with a corresponding so-called eigenvector . The decomposition of a square matrix into eigenvalues and eigenvectors is known in this work as eigen decomposition, and the fact that this decomposition is always possible as long as the matrix consisting of the eigenvectors of A is square is known as the eigen decomposition theorem. The Lanczos algorithm is an algorithm for computing the eigenvalues and eigenvectors for large symmetric sparse matrices. Let A be a linear transformation represented by a matrix A. If there is a vector such that for some scalar , then is called the eigenvalue of A with corresponding (right) eigenvector . Letting A be a square matrix with eigenvalue , then the corresponding eigenvectors satisfy which is equivalent to the homogeneous system
Equation ( 4 ) can be written compactly as where I is the identity matrix . As shown in Cramer's rule , a linear system of equations has nontrivial solutions if the determinant

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