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b) 3 -2 Given matrix A = -2 6 2. Suppose the eigenvalues of A and the bases for the 4 2 corresponding eigenspaces are as follows A₁ = -2, Ex=-2 2-{(3)}) and 2 i. Explain why A is orthogonally diagonalizable. ii. Give the relation between Ex=-2 and Ex-7. Hence, find an orthogonal matrix Q that diagonalizes matrix A and the corresponding diagonal matrix D such that D = Q-¹AQ. iii. Hence, calculate A². ---(Q)-()} 2 and X₂ = 7, Ex=7

b) 3 -2 Given matrix A = -2 6 2. Suppose the eigenvalues of A and the bases for the 4 2 corresponding eigenspaces are as follows A₁ = -2, Ex=-2 2-{(3)}) and 2 i. Explain why A is orthogonally diagonalizable. ii. Give the relation between Ex=-2 and Ex-7. Hence, find an orthogonal matrix Q that diagonalizes matrix A and the corresponding diagonal matrix D such that D = Q-¹AQ. iii. Hence, calculate A². ---(Q)-()} 2 and X₂ = 7, Ex=7

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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b) 3 -2 Given matrix A = -2 6 2. Suppose the eigenvalues of A and the bases for the A=1-/²2 2 corresponding A₁ = -2, Ex=-2 4 eigenspaces are as follows 2-((-)) and 2 ---(Q)-()} 2 and X₂ = 7, Ex=7 i. Explain why A is orthogonally diagonalizable. ii. Give the relation between Ex=-2 and Ex-7. Hence, find an orthogonal matrix Q that diagonalizes matrix A and the corresponding diagonal matrix D such that D = Q-¹AQ. iii. Hence, calculate A².
Transcribed Image Text:b) 3 -2 Given matrix A = -2 6 2. Suppose the eigenvalues of A and the bases for the A=1-/²2 2 corresponding A₁ = -2, Ex=-2 4 eigenspaces are as follows 2-((-)) and 2 ---(Q)-()} 2 and X₂ = 7, Ex=7 i. Explain why A is orthogonally diagonalizable. ii. Give the relation between Ex=-2 and Ex-7. Hence, find an orthogonal matrix Q that diagonalizes matrix A and the corresponding diagonal matrix D such that D = Q-¹AQ. iii. Hence, calculate A².
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