Orthogonal matrix diagonalization

5. $$\mathbf { M } = \left( \begin{array} { r r r } 6 & - 2 & - 1 \\ - 2 & 6 & - 1 \\ - 1 & - 1 & 5 \end{array} \right)$$ Given that 8 is an eigenvalue of \(\mathbf { M }\)

1. determine an eigenvector corresponding to the eigenvalue 8
2. Determine the other two eigenvalues of \(\mathbf { M }\).
3. Hence find an orthogonal matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { T } \mathbf { M P } = \mathbf { D }\) 5.
   

4. $$\mathbf { M } = \left( \begin{array} { r r r } 0 & - 1 & 3 \\ - 1 & 4 & - 1 \\ 3 & - 1 & 0 \end{array} \right)$$ Given that \(\left( \begin{array} { r } 1 \\ - 2 \\ 1 \end{array} \right)\) is an eigenvector of \(\mathbf { M }\)

1. determine its corresponding eigenvalue. Given that - 3 is an eigenvalue of \(\mathbf { M }\)
2. determine a corresponding eigenvector. Hence, given that \(\left( \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right)\) is also an eigenvector of \(\mathbf { M }\)
3. determine a diagonal matrix \(\mathbf { D }\) and an orthogonal matrix \(\mathbf { P }\) such that \(\mathbf { D } = \mathbf { P } ^ { \mathrm { T } } \mathbf { M P }\)

2. $$\mathbf { A } = \left( \begin{array} { l l } 3 & 2 \\ 2 & 6 \end{array} \right)$$

1. Find the eigenvalues and corresponding normalised eigenvectors of the matrix \(\mathbf { A }\).
2. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { \mathrm { T } } \mathbf { A P } = \mathbf { D }\).

3. $$\mathbf { M } = \left( \begin{array} { r r r } - 2 & 5 & 0 \\ 5 & 1 & - 3 \\ 0 & - 3 & 6 \end{array} \right)$$ Given that \(\mathbf { i } + \mathbf { j } + \mathbf { k }\) is an eigenvector of \(\mathbf { M }\),

1. determine the corresponding eigenvalue. Given that 8 is an eigenvalue of \(\mathbf { M }\),
2. determine a corresponding eigenvector.
3. Determine a diagonal matrix \(\mathbf { D }\) and an orthogonal matrix \(\mathbf { P }\) such that $$\mathbf { D } = \mathbf { P } ^ { \mathrm { T } } \mathbf { M P }$$

6. $$\mathbf { M } = \left( \begin{array} { r r r } p & - 2 & 0 \\ - 2 & 6 & - 2 \\ 0 & - 2 & q \end{array} \right)$$ where \(p\) and \(q\) are constants.
Given that \(\left( \begin{array} { r } 2 \\ - 2 \\ 1 \end{array} \right)\) is an eigenvector of the matrix \(\mathbf { M }\),

1. find the eigenvalue corresponding to this eigenvector,
2. find the value of \(p\) and the value of \(q\). Given that 6 is another eigenvalue of \(\mathbf { M }\),
3. find a corresponding eigenvector. Given that \(\left( \begin{array} { l } 1 \\ 2 \\ 2 \end{array} \right)\) is a third eigenvector of \(\mathbf { M }\) with eigenvalue 3
4. find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { P } ^ { \mathrm { T } } \mathbf { M } \mathbf { P } = \mathbf { D }$$

8. $$\mathbf { A } = \left( \begin{array} { l l l } 1 & 0 & 4 \\ 0 & 5 & 4 \\ 4 & 4 & 3 \end{array} \right)$$

1. Verify that \(\left( \begin{array} { r } 2 \\ - 2 \\ 1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\) and find the corresponding eigenvalue.
2. Show that 9 is another eigenvalue of \(\mathbf { A }\) and find the corresponding eigenvector.
3. Given that the third eigenvector of \(\mathbf { A }\) is \(\left( \begin{array} { r } 2 \\ 1 \\ - 2 \end{array} \right)\), write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { P } ^ { \mathrm { T } } \mathbf { A } \mathbf { P } = \mathbf { D } .$$