Chapter4 - Chapter IV Eigenvalues and Eigenvectors 145 IV.1 Eigenvalues and Eigenvectors Prerequisites and Learning Goals After completing this section

Chapter4 - Chapter IV Eigenvalues and Eigenvectors 145 IV.1...

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Chapter IV Eigenvalues and Eigenvectors 145
IV.1. Eigenvalues and Eigenvectors Prerequisites and Learning Goals After completing this section, you should be able to Write down the definition of eigenvalues and eigenvectors and compute them using the stan- dard procedure involving finding the roots of the characteristic polynomial. You should be able to perform relevant calculations by hand or using specific MATLAB/Octave commands such as poly , roots , and eig . Define algebraic and geometric multiplicities of eigenvalues and eigenvectors; discuss when it is possible to find a set of eigenvectors that form a basis. Determine when a matrix is diagonalizable and use eigenvalues and eigenvectors to perform matrix diagonalization. Recognize the form of the Jordan Canonical Form for non-diagonalizable matrices. Explain the relationship between eigenvalues and the determinant and trace of a matrix. Use eigenvalues to compute powers of a diagonalizable matrix. IV.1.1. Definition Let A be an n × n matrix. A number λ and non-zero vector v are an eigenvalue eigenvector pair for A if A v = λ v Although v is required to be nonzero, λ = 0 is possible. If v is an eigenvector, so is s v for any number s 6 146
IV.1.2. Standard procedureThis leads to the standard textbook method of finding eigenvalues. The function ofλdefined byp(λ) = det(λI-A) is a polynomial of degreen, called the characteristic polynomial, whose zerosare the eigenvalues. So the standard procedure is:Compute the characteristic polynomialp(λ)Find all the zeros (roots) ofp(λ). This is equivalent to completely factoring

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