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
10、of a vector space.,Vector Norms,The p-norm of a vector X is: Important examples of vector p-norms include:,Matrix Norms,Common matrix norms for a matrix A include: Note - max is the largest eigenvalue of ATA.,Matrix Condition Number,The matrix condition number CondA is obtained by calculating CondA=
11、|A|A-1| In can be shown that: The relative error of the norm of the computed solution can be as large as the relative error of the norm of the coefficients of A multiplied by the condition number. If the coefficients of A are known to t digit precision, the solution X may be valid to only t-log10(Co
12、ndA) digits.,Built-in functions to compute norms and condition numbers,norm(X,p) Compute the p norm of vector X, where p can be any number, inf, or fro (for the Euclidean norm) norm(A,p) Compute a norm of matrix A, where p can be 1, 2, inf, or fro (for the Frobenius norm) cond(X,p) or cond(A,p) Calc
13、ulate the condition number of vector X or matrix A using the norm specified by p.,LU Factorization,LU factorization involves two steps: Decompose the A matrix into a product of:a lower triangular matrix L with 1 for each entry on the diagonal. and an upper triangular matrix U Substitution to solve f
14、or x Gauss elimination can be implemented using LU factorization The forward-elimination step of Gauss elimination comprises the bulk of the computational effort. LU factorization methods separate the time-consuming elimination of the matrix A from the manipulations of the right-hand-side b.,Gauss E
15、limination as LU Factorization,To solve Ax=b, first decompose A to get LUx=b MATLABs lu function can be used to generate the L and U matrices: L, U = lu(A) Step 1 solve Ly=b; y can be found using forward substitution. Step 2 solve Ux=y, x can be found using backward substitution. In MATLAB: L, U = l
16、u(A) d = Lb x = Ud LU factorization requires the same number of floating point operations (flops) as for Gauss elimination. Advantage once A is decomposed, the same L and U can be used for multiple b vectors.,Cholesky Factorization,A symmetric matrix a square matrix, A, that is equal to its transpos
17、e:A = AT (T stands for transpose). The Cholesky factorization based on the fact that a symmetric matrix can be decomposed as:A= UTU The rest of the process is similar to LU decomposition and Gauss elimination, except only one matrix, U, needs to be stored. Cholesky factorization with the built-in ch
18、ol command: U = chol(A) MATLABs left division operator examines the system to see which method will most efficiently solve the problem. This includes trying banded solvers, back and forward substitutions, Cholesky factorization for symmetric systems. If these do not work and the system is square, Gauss elimination with partial pivoting is used.,
