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hw7-solutions

hw7-solutions - Assignment 6 More on Eigenvalues and...

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Assignment 6: More on Eigenvalues and Eigenvectors Written by: Saber Mirzaei Problem 1. (a) ( ) (b) { } ( ) Problem 2. (a) ( ) Now after finding we can find { } and [ ] [ ] (b) Following the same approach we get: [ ] [ ]
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Problem 3. (a) Step1: Find the orthogonal projection of on to ( let’s call is ) Based on what explained in previous problem: Step1: Find the orthogonal projection of on to ( let’s call is ) Step 3: Find the orthogonal projection of on to { } (let’s call is ) Step 4: obviously our goal vector is any multiple of Following these steps the answer is any multiple of [ ] (b) Following steps of part (a) the answer is any multiple of [ ]
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Problem 4. Step 1. Find the set of eigenvalues of Step 2. For each eigenvalue find the basis of corresponding eigenspace Step 3. If the set of eigenvectors are not mutually orthogonal then using Gram-Schmidt process find a set of orthogonal vectors that span the subspace as the original one. Step 4. Normalize the vectors Step 5. Set up the matrix and (a) Step 1. Following step 1 we get Step 2. The eigenvectors for
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