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Exercise: Let A be an invertible n x n matrix. Suppose that u ER" is an eigenvector of A with corresponding eigenvalue λ = 2 and v E R" is an eigenvector of A with corresponding eigenvalue λ = -7. Fill in the following blanks. (Enteras a/b, where applicable. Work out any powers before inputting your answer.) 1. Au = 3. Asu = 5. A-6u = u u u 2. Av= 4. A³v = 6. A 2y = V

Exercise: Let A be an invertible n x n matrix. Suppose that u ER" is an eigenvector of A with corresponding eigenvalue λ = 2 and v E R" is an eigenvector of A with corresponding eigenvalue λ = -7. Fill in the following blanks. (Enteras a/b, where applicable. Work out any powers before inputting your answer.) 1. Au = 3. Asu = 5. A-6u = u u u 2. Av= 4. A³v = 6. A 2y = V

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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Exercise: Let A be an invertible n x n matrix. Suppose that u ER" is an eigenvector of A with corresponding eigenvalue λ = 2 and v ER" is an eigenvector of A with corresponding eigenvalue = -7. Fill in the following blanks. (Enteras a/b, where applicable. Work out any powers before inputting your answer.) 1. Au = 3. Asu = 5. A u = u u u 2. Av= 4. A³v = 6. A-²y = V Example: Suppose that A is an invertible n x n matrix and that x ER" is an eigenvector of A with corresponding eigenvalue λ = -3. Since x is an eigenvector of A with corresponding eigenvalue λ = -3 we have that x = 0 and Ax = X. (a) Show that x is also an eigenvector of B = 2A²-6A-¹ + 51 and find the corresponding eigenvalue (of B).
Transcribed Image Text:Exercise: Let A be an invertible n x n matrix. Suppose that u ER" is an eigenvector of A with corresponding eigenvalue λ = 2 and v ER" is an eigenvector of A with corresponding eigenvalue = -7. Fill in the following blanks. (Enteras a/b, where applicable. Work out any powers before inputting your answer.) 1. Au = 3. Asu = 5. A u = u u u 2. Av= 4. A³v = 6. A-²y = V Example: Suppose that A is an invertible n x n matrix and that x ER" is an eigenvector of A with corresponding eigenvalue λ = -3. Since x is an eigenvector of A with corresponding eigenvalue λ = -3 we have that x = 0 and Ax = X. (a) Show that x is also an eigenvector of B = 2A²-6A-¹ + 51 and find the corresponding eigenvalue (of B).
Since x is an eigenvector of A with corresponding eigenvalue = -3 we have that x = 0 and Ax= (a) Show that x is also an eigenvector of B = 2A²-6A-¹ + 51 and find the corresponding eigenvalue (of B). Bx (2A²6A-¹ +51)x = 2A²x - 6A-¹x + 5x = 2(−3)2x − 6(_)x+5x - = = 18x + 2x + 5x Therefore, the nonzero vector x is an eigenvector of B with corresponding eigenvalue λ = (b) Verify (to yourself) that x is also an eigenvector of C = -81 + 18A-2 + A³ - 4A and find the corresponding eigenvalue (of C). The vector x is an eigenvector of C with corresponding eigenvalue λ =
Transcribed Image Text:Since x is an eigenvector of A with corresponding eigenvalue = -3 we have that x = 0 and Ax= (a) Show that x is also an eigenvector of B = 2A²-6A-¹ + 51 and find the corresponding eigenvalue (of B). Bx (2A²6A-¹ +51)x = 2A²x - 6A-¹x + 5x = 2(−3)2x − 6(_)x+5x - = = 18x + 2x + 5x Therefore, the nonzero vector x is an eigenvector of B with corresponding eigenvalue λ = (b) Verify (to yourself) that x is also an eigenvector of C = -81 + 18A-2 + A³ - 4A and find the corresponding eigenvalue (of C). The vector x is an eigenvector of C with corresponding eigenvalue λ =
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