Practice Diagonalization Of Symmetric Matrices (33.8)

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8 questions

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6 sample questions

Question 1.
Define a symmetric matrix and provide an example.
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Think about the definition of the matrix being equal to its transpose.
Question 2.
What is the significance of real eigenvalues in symmetric matrices?
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Consider how this affects calculations.
3Question 3.
What characterizes a symmetric matrix?
- A = A^T
- A = -A
- A is always diagonal
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Recall the definition of symmetric matrices.
4Question 4.
True or False: All eigenvectors of a symmetric matrix are not necessarily orthogonal.
- True
- False
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Consider eigenvector properties we just discussed.
Question 5.
Consider the symmetric matrix A = [2 1; 1 2]. Diagonalize the matrix and explain the significance of each eigenvalue.
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Calculate eigenvalues and eigenvectors using the characteristic polynomial.
Question 6.
Show that the matrix A = [1 2; 2 1] can be expressed as A = QDQ^T. Determine Q and D.
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Work through the steps of finding characteristic polynomial and eigenvalues first.

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