Engineering Analysis ENG 3420 Fall 2009.ppt
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1、Engineering Analysis ENG 3420 Fall 2009,Dan C. Marinescu Office: HEC 439 B Office hours: Tu-Th 11:00-12:00,2,2,Lecture 17,Lecture 17,Reading assignment Chapters 10 and 11, Linear Algebra ClassNotes Last time: Symmetric matrices; Hermitian matrices. Matrix multiplication Today: Linear algebra functio
2、ns in Matlab The inverse of a matrix Vector products Tensor algebra Characteristic equation, eigenvectors, eigenvalues Norm Matrix condition number Next Time More on LU Factorization Cholesky decomposition,Matrix analysis in MATLAB,Norm Matrix or vector norm normest Estimate the matrix 2-normrank Ma
3、trix rankdet Determinanttrace Sum of diagonal elementsnull Null spaceorth Orthogonalizationrref Reduced row echelon formsubspace Angle between two subspaces,Eigenvalues and singular values,eig Eigenvalues and eigenvectorssvd Singular value decompositioneigs A few eigenvaluessvds A few singular value
4、spoly Characteristic polynomialpolyeig Polynomial eigenvalue problemcondeig Condition number for eigenvalueshess Hessenberg formqz QZ factorizationschur Schur decomposition,Matrix functions,Expm Matrix exponential Logm Matrix logarithm Sqrtm Matrix square root Funm Evaluate general matrix function,L
5、inear systems of equations, and / Linear equation solutioninv Matrix inversecond Condition number for inversioncondest 1-norm condition number estimatechol Cholesky factorizationcholinc Incomplete Cholesky factorizationlinsolve Solve a system of linear equationslu LU factorizationilu Incomplete LU f
6、actorizationluinc Incomplete LU factorizationqr Orthogonal-triangular decompositionlsqnonneg Nonnegative least-squarespinv Pseudoinverselscov Least squares with known covariance,The inverse of a square,If A is a square matrix, there is another matrix A-1, called the inverse of A, for which AA-1=A-1A
7、=I The inverse can be computed in a column by column fashion by generating solutions with unit vectors as the right-hand-side constants:,Canonical base of an n-dimensional vector space,100000010000001000.000.100000.010000.001,Matrix Inverse (cont),LU factorization can be used to efficiently evaluate
8、 a system for multiple right-hand-side vectors - thus, it is ideal for evaluating the multiple unit vectors needed to compute the inverse.,The response of a linear system,The response of a linear system to some stimuli can be found using the matrix inverse.,Distance and norms,Metric space a set wher
9、e the ”distance” between elements of the set is defined, e.g., the 3-dimensional Euclidean space. The Euclidean metric defines the distance between two points as the length of the straight line connecting them.A norm real-valued function that provides a measure of the size or “length” of an element
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