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Matrix Methods and Applications

by: Dr. Filomena Hegmann

Matrix Methods and Applications APPM 3310

Marketplace > University of Colorado at Boulder > Applied Math > APPM 3310 > Matrix Methods and Applications
Dr. Filomena Hegmann

GPA 3.76


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Class Notes
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This 2 page Class Notes was uploaded by Dr. Filomena Hegmann on Thursday October 29, 2015. The Class Notes belongs to APPM 3310 at University of Colorado at Boulder taught by Staff in Fall. Since its upload, it has received 29 views. For similar materials see /class/231871/appm-3310-university-of-colorado-at-boulder in Applied Math at University of Colorado at Boulder.


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Date Created: 10/29/15
APPM 3310 new material for the nal 1 Gram Schmidt and A QR factorization 0 Be able to perform the Gram Schmidt procedure to transform a collection of vectors into an orthonormal set of vectors with the same span 0 Know how to write the original vectors in terms of the new or thonormal vectors o If the original collection of vectors are columns of a matrix A7 know how to deduce the A QR factorization from this information 0 Know that if A has independent columns7 then A can be factored 0 Know what an orthonormal matrix is Know what an orthogonal matrix is Be able to provide examples of each 0 Know what it means for orthonormal matrices to preserve length 0 Understand how the normal equations for the least squares solu tion to A b simplify to R QTb if A is factored A QR 0 Understand why this formulation could be advantageous for nu merical computations 2 Eigenvalues7 eigenvectors7 and diagonalization 0 Know how to compute the eigenvalues and eigenvectors of a square matrix 0 Know that the sum of eigenvalues is the trace of the matrix and the product of eigenvalues is the determinant 0 Know what the characteristic polynomial of a matrix is 0 Know the statement of the Cayley Hamilton theorem 0 Know the de nitions of algebraic and geometric multiplicity of an eigenvalue 0 Know that the algebraic multiplicity is always greater than or equal to the geometric multiplicity 0 Be able to give examples of eigenvalues with equal algebraic and geometric multiplicity7 and algebraic multiplicity strictly great than geometric multiplicity 0 Know that eigenvectors corresponding to di erent eigenvalues are independent 0 Know that if the algebraic multiplicity equals the geometric mul tiplicity for each eigenvalue7 then the matrix is diagonalizable In this case7 know that the matrix of eigenvectors S satis es A SDS l 0 Understand how the forumla for Ak simpli es upon diagonaliza tion of A 3 Linear di erence equations 0 Be able to write down an expression for the general solution of uk1 Auc7 where A is a square matrix Know how the expression simpli es if A is diagonalizable 0 Be able to compute limCHOO uh the steady state behavior of the system7 using the above expression 0 Understand why the only equilibrium solutions correspond to eigen vectors for the eigenvalue 1 4 Linear di erential equations 0 Be able to write down an expression for the general solution of u Au7 where A is a square matrix 0 Be able to write down a formula for the matrix exponential pltAtgt when A is diagonalizable 0 Understand why no 0 is the only equilibrium solution for the system Understand how its stability is determined by the real parts of the eigenvalues of A


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