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# LINEAR ALGEBRA FOR APPL MATH 640

Texas A&M

GPA 3.6

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This 4 page Class Notes was uploaded by Vivien Bradtke V on Wednesday October 21, 2015. The Class Notes belongs to MATH 640 at Texas A&M University taught by Staff in Fall. Since its upload, it has received 10 views. For similar materials see /class/226058/math-640-texas-a-m-university in Mathematics (M) at Texas A&M University.

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Date Created: 10/21/15

Math 640 Final examination 12203 Review sheet For the nal exam you should be well acquainted with Vector spaces Rank and linear independent The Hermitian theory of matrices Jordan normal form Factorization theory Similarity theory unitary and general Positive matrices Positive irreducible matrices General matrix theory and norm theory If you are asked to prove a result from the lectures you must give the full proof If you are asked to prove a result that requires a result from the lectures you may cite the result The following list of problems contains some challenging examples and some fairly easy examples However the distribution of these problems is not necessarily representative ofthe nal exam 1 How do the LU and QR decompostions differ What are the LU and QR algorithms Not every matrix has a LU decomposition Give conditions when it will 2 Let B 6 MAR is nonnegative Suppose that bi 2 forz39 12 Show that B2 2 B ifz39 j 3 Characterize all nilpotent symmetric matrices 4 Let G I f be a blocked matrix with blocksA and B square Under what conditions on A B C and D is G Hermitian 5 LetA a b andB a f c d Study conditions when AB BA Are there general conditions that yield commutivity 6 LetA 6 MAR be symmetric Prove that the null spaces ofA and AT are equal Prove that this is not true in general 7 For symmetric matrices there is an effective way to find the eigenvalues Called the RayleighRitz method it establishes that Ax xgt and Lmin min ltAx xgt lax M24 9696 where Am and 1mm are the maximum and minimum eigenvalues ofA Am max Hxllz1 respectively LetA a Z Assume that a c and dare real Apply C the RayleighRitz theorem to reduce the problem of nding the maximum eigenvalue to maximization of the function axz 2ch1 x2 d dx2 over the interval 11 LetA 6 MAR be nonnegative in the sense that a 2 0 for 1 5 139 j 5 n Let A pA be a positive eigenvalue ofA and x x1 xquot be a nonzero nonnegative eigenvector ofA Let cmax and cm be the maximum and minimum column sums ofA respectively Show that cm 5 A 5 cm Prove the same result for the maximum and minimum row sums ofA LetP3 denote the cubic polynomials and let wx We now restrict x our attention to P3 on the interval 1 1 De ne the inner product of two polynomialsp and q in P3 by ltpqgt pxgtqxgtwxgtdx Show that pq is an inner product Apply the GramSchmidt procedure to determine an orthogonal basis for P3 You may begin with p1 pr 1 So you need to nd p2 p3 and p4 Hint You may use Maple or some other computer algebra system for this as Suppose that the eigenvalues ofA e M12C are 05 and 09 a Prove using norm theory that limHoAquot b Prove using the Jordan canonical form that that limnswAquot 0 Prove that a necessary and suf cient condition that a matrixA 6 MAC be an isometry is that the columns ofA be orthonormal vectors Prove or disprove all isometries on Cquot are similar Letx e Cquot with llxll2 1 Prove that1 2ww is unitary a f Study conditions when AB BA LetA a b and B c d Are there general conditions that yield commutivity LetA 6 MAC be normal Prove that the null spaces ofA and A are equal Prove that this is not true in general LetA 6 MAC What does the minimal polynomial tell us aboutA How does the minimal polynomial differ from the characteristic polynomial Given A 6 MAC is nonnegative Then we know that the spectral radius is bounded between the minimum and maximum column sums Show that the same is true for the minimum and maximum row sums Compute the CollatzWeilandt function for the matrix A A A N A w A p A 9 A q A 9 A 1 3 4 2 1 1 3 4 5 for the vectorsx 111102 and 213 Deduce a lower bound for the spectral radius A GivenA e MnR is nonnegative De ne the function Axl xi gAx 123 Prove that i gA is homogeneous of degree 0 and ii ifx is a nonnegative ntuple such that x gt 0 whenever Ax gt 0 and a is the least numberfor which ix Ax 2 0 then a gA LetA e MnltC be Hermitian Show that 2 rankA 2 tr A 2 Show that the nonnegative matrixA e Mquot is reducible if and only ifthere is a proper subset 611 eh ejk of the standard basis ofRquot such that the span Of A6119 Aejz quot399 Aejk C ejla ejz elk Show that the n x n permutation matrix with 1 s in the postions 12 23 n 1n and 171 is irreducible Prove that the product of irreducible matrices is irreducible 15 Every nonnegative nilpotent matrix is reducible The transition matrix of a Markov chain is a column stochastic matrix S Suppose that all the probabilities ie the entries are strictly positive Let x1 be some state Le a nonnegative vector with sum 1 and de ne let an an Show that limp an exists and is independent Ofxl Given B e MnR is nonnegative Suppose that x 6 A7 andy Bx Prove that ngincj S gyi S mjaxcj where the of are the column sums ofB Let B e MnR is nonnegative Suppose that 11 2 forz39 12 Show that B2 2 B ifz j Suppose that B e MnR is nonnegative Suppose that B has each row ofB has one entry greater than or equal to one and that these entries occur in distinct columns Prove that pB 2 1 N O N N N N w M 9 LetG LetA ab be be a blocked matrix with blocksA and B square Under what conditions on A B C and D is G Hermitian Normal Unitary Assume that a b and c are real Apply the RayleighRitz theorem to reduce the problem of nding the maximum eigenvalue to maximization of the function axz 2ch1 x2 d dx2 over the interval 11 Let xy e Rquot We de ne the antiinner product ofx and yto be ltxya 21xiym1 ls ltxya a true inner product Why or why not Show that there is a basis xh xn of Rquot that is antiorthogonal in the sense that xjxka 0 If k Show that given an antiorthogonal basis x1 happen that xjxja 0 for any 1 Suppose we are given an antiorthogonal basis x1xn ome and the vectors are arranged into the rows of a matrix A Find the inverse ofA xn ofRn then it cannot

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