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04-11 and det(A). mind the complex values. Calculate A²-24 +27. Calculate trace(4)

04-11 and det(A). mind the complex values. Calculate A²-24 +27. Calculate trace(4)

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter7: Eigenvalues And Eigenvectors
Section7.2: Diagonalization
Problem 45E
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part F G

3 Eigenvalues and Eigenvectors Represent the matrix A as A = V-¹AV, where A = diag(A... An) is the diagonal matrix with the eigenvalues X EC of A on its diagonal, and V = [...] is the matrix whose columns are the eigenvectors, such that , E C is an eigenvector associated with the eigen- value A. Whenever possible, normalize the eigenvector so that the first coordinate of the vector is 1 (if the first coordinate is 0, normalize it so that the second coordinate is 1). (a) A= -5 3 -18 10 (b) A = 1. Calculate A² and A²-3A +27. Calculate trace(A) and det(A). (c) A = (Can you conclude what is A-¹ without any additional calculation?) 3 -0.5 (d) A= (e) A = 10 -24 10 10] Calculate A² and A²-41. Calculate trace(A) and det(A). . (f) A = -41- ... mind the complex values. Calculate A²-2A +21. Calculate trace(4) and det(A). (8) 4 = 41¹]. (This matrix represents contra-clockwise rotation by 45 degrees.) If you like extra practice, you can find the decomposition using the following trick: First calculate A², this is the matrix representing the contra-clockwise rotation by 90 degrees. Represent A² as V-¹AV, and then A = V-¹AV, where A solves the equation A² = A (simply take square root of each entry on the diagonal, however, keep in mind that both √ and -√ solve the equation X² = Ã; remember that Vi= (1 + i) and i=(1-i)). You should then verify that indeed A = V-¹AV, because in fact there are 4 matrices A such that A² = Ã. Note that this trick is not universal, for example AS = I, but you won't be able to find the decomposition of A using the decomposition of I. - 69.
Transcribed Image Text:3 Eigenvalues and Eigenvectors Represent the matrix A as A = V-¹AV, where A = diag(A... An) is the diagonal matrix with the eigenvalues X EC of A on its diagonal, and V = [...] is the matrix whose columns are the eigenvectors, such that , E C is an eigenvector associated with the eigen- value A. Whenever possible, normalize the eigenvector so that the first coordinate of the vector is 1 (if the first coordinate is 0, normalize it so that the second coordinate is 1). (a) A= -5 3 -18 10 (b) A = 1. Calculate A² and A²-3A +27. Calculate trace(A) and det(A). (c) A = (Can you conclude what is A-¹ without any additional calculation?) 3 -0.5 (d) A= (e) A = 10 -24 10 10] Calculate A² and A²-41. Calculate trace(A) and det(A). . (f) A = -41- ... mind the complex values. Calculate A²-2A +21. Calculate trace(4) and det(A). (8) 4 = 41¹]. (This matrix represents contra-clockwise rotation by 45 degrees.) If you like extra practice, you can find the decomposition using the following trick: First calculate A², this is the matrix representing the contra-clockwise rotation by 90 degrees. Represent A² as V-¹AV, and then A = V-¹AV, where A solves the equation A² = A (simply take square root of each entry on the diagonal, however, keep in mind that both √ and -√ solve the equation X² = Ã; remember that Vi= (1 + i) and i=(1-i)). You should then verify that indeed A = V-¹AV, because in fact there are 4 matrices A such that A² = Ã. Note that this trick is not universal, for example AS = I, but you won't be able to find the decomposition of A using the decomposition of I. - 69.
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