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-15 8 -7 2 8 -5 -1 1 Determine the four eigenvalues of A. ● Determine all of the ordinary eigenvectors of A that you can. For each unique eigenvalue, what is its multiplicity k, and how many defects d does it have? Given the system x' = Ax = -16 41 -10 5 7 -1 -2 -2 x, find its eigenvalue(s) and eigenvectors.

-15 8 -7 2 8 -5 -1 1 Determine the four eigenvalues of A. ● Determine all of the ordinary eigenvectors of A that you can. For each unique eigenvalue, what is its multiplicity k, and how many defects d does it have? Given the system x' = Ax = -16 41 -10 5 7 -1 -2 -2 x, find its eigenvalue(s) and eigenvectors.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Given the system x' = Ax = -15 8 -16 4 -10 5 7 -1 -7 2 8 -5 -1 1 -2-2- x, find its eigenvalue(s) and eigenvectors. Determine the four eigenvalues of A. Determine all of the ordinary eigenvectors of A that you can. For each unique eigenvalue, what is its multiplicity k, and how many defects d does it have? Determine any missing eigenvectors. You should use the Excel "=MMULT(matrix, matrix)" (the form of the Excel function) for your work. Otherwise, simple arithmetic mistakes are too likely. Determine the four linearly independent solution vectors, designating them x₁(t), x₂(t), x3(t), and X4(t). Write the general solution vector x(t) as a linear combination of the four solution vectors from the question immediately above, using coefficients c₁ through c4, respectively, but leaving them as unknowns. You are encouraged to write this solution row-by-row, such as x₁(t) = ..., X₂(t) = ..., and so on, for rows 1, 2, and so on, respectively. Note that these x's are functions, not vectors. Form the matrix Q from the chain(s) of generalized eigenvectors you determined. Determine Q¹¹. You may use the Excel "-MINVERSE(matrix)" function. • Find the Jordan normal form matrix J = Q¹AQ and confirm that has the correct eigenvalues and block form. Use MMULT, first for two of the three matrices and then for this product times the third matrix. Make sure to preserve the order of the three matrices in the multiplications.
Transcribed Image Text:Given the system x' = Ax = -15 8 -16 4 -10 5 7 -1 -7 2 8 -5 -1 1 -2-2- x, find its eigenvalue(s) and eigenvectors. Determine the four eigenvalues of A. Determine all of the ordinary eigenvectors of A that you can. For each unique eigenvalue, what is its multiplicity k, and how many defects d does it have? Determine any missing eigenvectors. You should use the Excel "=MMULT(matrix, matrix)" (the form of the Excel function) for your work. Otherwise, simple arithmetic mistakes are too likely. Determine the four linearly independent solution vectors, designating them x₁(t), x₂(t), x3(t), and X4(t). Write the general solution vector x(t) as a linear combination of the four solution vectors from the question immediately above, using coefficients c₁ through c4, respectively, but leaving them as unknowns. You are encouraged to write this solution row-by-row, such as x₁(t) = ..., X₂(t) = ..., and so on, for rows 1, 2, and so on, respectively. Note that these x's are functions, not vectors. Form the matrix Q from the chain(s) of generalized eigenvectors you determined. Determine Q¹¹. You may use the Excel "-MINVERSE(matrix)" function. • Find the Jordan normal form matrix J = Q¹AQ and confirm that has the correct eigenvalues and block form. Use MMULT, first for two of the three matrices and then for this product times the third matrix. Make sure to preserve the order of the three matrices in the multiplications.
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