You are given that a 2 x 2 real matrix A has the following unit eigenvectors corresponding to a single eigenvalue X: 1 (1, -1), √2 1 X2 = (1,1) √2 1. Is A guaranteed to be symmetric, skew-symmetric, or neither? Is A guaranteed to be real, imginary, or neither? 2. Use the given information to write out a diagonalization of A that does not include any matrix inverses. 3. Write an expression for A as a summation of two projection matrices. 4. Is it guaranteed that Ax is a multiple of x for any 2-vector x? Explain why or why not. X1 =
You are given that a 2 x 2 real matrix A has the following unit eigenvectors corresponding to a single eigenvalue X: 1 (1, -1), √2 1 X2 = (1,1) √2 1. Is A guaranteed to be symmetric, skew-symmetric, or neither? Is A guaranteed to be real, imginary, or neither? 2. Use the given information to write out a diagonalization of A that does not include any matrix inverses. 3. Write an expression for A as a summation of two projection matrices. 4. Is it guaranteed that Ax is a multiple of x for any 2-vector x? Explain why or why not. X1 =
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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Transcribed Image Text:You are given that a 2 x 2 real matrix A has the following unit eigenvectors corresponding to a single eigenvalue X:
1
(1, -1),
√2
1
X2 = (1,1)
√2
1. Is A guaranteed to be symmetric, skew-symmetric, or neither? Is A guaranteed to be real, imginary, or neither?
2. Use the given information to write out a diagonalization of A that does not include any matrix inverses.
3. Write an expression for A as a summation of two projection matrices.
4. Is it guaranteed that Ax is a multiple of x for any 2-vector x? Explain why or why not.
5. Consider the ratio
||A²+1x||
and assume that λ 0. What is the smallest value of k for which this ratio is equal to X? Explain whether
||Akx||
this also holds true for larger values of k.
X1 =
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