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

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EIGENVALUESEigenvalues are a special set of scalars associated with alinear system of equationsthat aresometimes also known as characteristic roots, characteristic valuesproper values or latentroots.The determination of the eigenvalues and eigenvectors of a system is extremelyimportant in physics and engineering, where it is equivalent tomatrix diagonalizationand arisesin such common applications as stability analysis, the physics of rotating bodies, and smalloscillations of vibrating systems, to name only a few. Each eigenvalue is paired with acorresponding so-calledeigenvector. The decomposition of a square matrixinto eigenvaluesand eigenvectors is known in this work as eigen decomposition, and the fact that thisdecomposition is always possible as long as the matrix consisting of the eigenvectorsof A is square is known as the eigen decomposition theorem.The Lanczos algorithm is an algorithm for computing the eigenvalues and eigenvectors forlarge symmetric sparse matrices.Let A be a linear transformation represented by a matrix A. If there is a vectorsuchthatfor somescalar, thenis called the eigenvalue of A with corresponding (right)eigenvector.Letting A be asquare matrixwith eigenvalue, then the correspondingeigenvectorssatisfywhich is equivalent to the homogeneous system
Equation (4) can be written compactly aswhere I is theidentity matrix. As shown inCramer's rule, alinear system of equationshasnontrivial solutionsifthedeterminant

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