Topics In Linear Algebra and Its Applications - STEM2

26 The Spectral TheoremVarious versions of the spectral theorem in linear algebra show how a lineartransformation may be characterized by its eigenvalues and eigenvectors. Inthe most simple form of a symmetric matrix it is shown that the matrix maybe diagonalized by introducing a basis if eigenvectors. The eigenvectors areorthogonal to each other and the eigenvalues are real. A very good surveyof the spectral theorem, in both finite dimensional and infinite dimensionalspaces isLorch Edward R, The Spectral Theorem, In Studies in Modern AnalysisVolume I, R. C. Buck editor, MAA Studies in Mathematics, The MathematicalAssociation of America, 1962.27 TensorsA tensor is a multilinear functional. That is, it is a function defined ona cartesian product of vector spaces. It is a linear transformation in eachindividual vector space v i . Thusf(v 1 , v 2 , ..., v i + u i , ...v n ) = f(v 1 , v 2 , ..., v i , ...v n ) + f(v 1 , v 2 , ..., u i , ...v n )and so on.28 Application of Linear Algebra to VibrationTheory, Normal CoordinatesSee the document Vibration, (vibra.tex).28.1 A Simple Spring ExampleSpring force:Spring Potential Energy:F = kx.V = k x22 .25