Problem 1E: The top matrix on the menu is the diagonal matrix A=( 5 4 00 3 4 ) Initially, when you select this... Problem 2E Problem 3E Problem 4E Problem 5E Problem 6E Problem 7E Problem 8E: The last item on the eigshow menu will generate a random 22 matrix each time that it is invoked. Try... Problem 9E Problem 10E Problem 11E Problem 12E: Consider the matrices A=(5 33 5) and B=(5 335) Note that the two matrices are the same except for... Problem 13E Problem 14E Problem 15E Problem 16E Problem 17E Problem 18E Problem 19E Problem 20E: Let A be a nonsingular 22 matrix with singular value decomposition A=USVT and singular values s1=s11... Problem 21E: Set A=[1,1;0.5,0.5] and use MATLAB to verify each of statements (a)-(d) in Exercise 20. Use the... Problem 22E Problem 23E Problem 24E Problem 25E Problem 1CTA: If A is an nn matrix whose eigenvalues are all nonzero, then A is nonsingular. Problem 2CTA: If A is nn matrix, then A and AT have the same eigenvectors. Problem 3CTA: If A and B are similar matrices, then they have the same eigenvalues. Problem 4CTA: If A and B are nn matrices with the same eigenvalues, then they are similar. Problem 5CTA: If A has eigenvalues of multiplicity greater than 1, then A must be defective. Problem 6CTA: If A is a 44 matrix of rank 3 and =0 is an eigenvalue of multiplicity 3, then A is diagonalizable. Problem 7CTA: If A is a 44 matrix of rank 1 and =0 is an eigenvalue of multiplicity 3, then A is defective. Problem 8CTA: The rank of an nn matrix A is equal to the number of nonzero eigenvalues of A, where eigenvalues are... Problem 9CTA: The rank of an mn matrix A is equal to the number of nonzero singular values of A, where singular... Problem 10CTA: If A is Hermitian and c is a complex scalar, then cA is Hermitian. Problem 11CTA: If an nn matrix A has Schur decomposition A=UTUH, then the eigenvalues of A are t11,t22,...,tnn. Problem 12CTA: If A is normal, but not Hermitian, then A must have at least one complex eigenvalue. Problem 13CTA Problem 14CTA Problem 15CTA: If A is symmetric, then eA is symmetric positive definite. Problem 1CTB: Let A=(10011 112 2) Find the eigenvalues of A. For each eigenvalue, find a basis for the... Problem 2CTB: Let A be a 44 matrix with real entries that has all 1's on the main diagonal (i.e.,... Problem 3CTB: Let A be a nonsingular nn matrix and let be an eigenvalue of A. Show that 0. Show that 1 is an... Problem 4CTB: Show that if A is a matrix of the form A=(a000a100a) then A must be defective. Problem 5CTB: Let A=(4222 10 102 10 14) Without computing the eigenvalues of A, show that A is positive definite.... Problem 6CTB Problem 7CTB Problem 8CTB: Let A be a 44 real symmetric matrix with eigenvalues 1=1,2=3=4=0 Explain why the multiple eigenvalue... Problem 9CTB: Let {u1,u2} be an orthonormal basis for 2 and suppose that a vector z can be written as a linear... Problem 10CTB: Let A be a 55 nonsymmetric matrix with rank equal to 3, let B=ATA, and let C=eB. What, if anything,... Problem 11CTB: Let A and B be nn matrices. If A is real and nonsymmetric with Schur decomposition UTUH, then what... Problem 12CTB: Let A be a matrix whose singular value decomposition is given by ( 2 5 2 5 2 5 2 5 3 5 2 5 2 5 ... format_list_bulleted