Diagonalization and matrix powers

6. The symmetric matrix \(\mathbf { M }\) has eigenvectors \(\left( \begin{array} { l } 2 \\ 2 \\ 1 \end{array} \right) , \left( \begin{array} { r } - 2 \\ 1 \\ 2 \end{array} \right)\) and \(\left( \begin{array} { r } 1 \\ - 2 \\ 2 \end{array} \right)\) with eigenvalues 5, 2 and - 1 respectively.

1. Find an orthogonal matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { P } ^ { \mathrm { T } } \mathbf { M } \mathbf { P } = \mathbf { D }$$ Given that \(\mathbf { P } ^ { - 1 } = \mathbf { P } ^ { \mathrm { T } }\)
2. show that $$\mathbf { M } = \mathbf { P D P } ^ { - 1 }$$
3. Hence find the matrix \(\mathbf { M }\).

3. $$\mathbf { A } = \left( \begin{array} { l l l } 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2 \end{array} \right)$$

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

3

1. Given the matrix \(\mathbf { Q } = \left( \begin{array} { r r r } 2 & - 1 & k \\ 1 & 0 & 1 \\ 3 & 1 & 2 \end{array} \right)\) (where \(k \neq 3\) ), find \(\mathbf { Q } ^ { - 1 }\) in terms of \(k\).
   Show that, when \(k = 4 , \mathbf { Q } ^ { - 1 } = \left( \begin{array} { r r r } - 1 & 6 & - 1 \\ 1 & - 8 & 2 \\ 1 & - 5 & 1 \end{array} \right)\). The matrix \(\mathbf { M }\) has eigenvectors \(\left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right) , \left( \begin{array} { r } - 1 \\ 0 \\ 1 \end{array} \right)\) and \(\left( \begin{array} { l } 4 \\ 1 \\ 2 \end{array} \right)\), with corresponding eigenvalues \(1 , - 1\) and 3 respectively.
2. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { M P } = \mathbf { D }\), and hence find the matrix \(\mathbf { M }\).
3. Write down the characteristic equation for \(\mathbf { M }\), and use the Cayley-Hamilton theorem to find integers \(a , b\) and \(c\) such that \(\mathbf { M } ^ { 4 } = a \mathbf { M } ^ { 2 } + b \mathbf { M } + c \mathbf { I }\). Section B (18 marks)

3 Let \(\mathbf { P } = \left( \begin{array} { r r r } 4 & 2 & k \\ 1 & 1 & 3 \\ 1 & 0 & - 1 \end{array} \right) (\) where \(k \neq 4 )\) and \(\mathbf { M } = \left( \begin{array} { r r r } 2 & - 2 & - 6 \\ - 1 & 3 & 1 \\ 1 & - 2 & - 2 \end{array} \right)\).

1. Find \(\mathbf { P } ^ { - 1 }\) in terms of \(k\), and show that, when \(k = 2 , \mathbf { P } ^ { - 1 } = \frac { 1 } { 2 } \left( \begin{array} { r r r } - 1 & 2 & 4 \\ 4 & - 6 & - 10 \\ - 1 & 2 & 2 \end{array} \right)\).
2. Verify that \(\left( \begin{array} { l } 4 \\ 1 \\ 1 \end{array} \right) , \left( \begin{array} { l } 2 \\ 1 \\ 0 \end{array} \right)\) and \(\left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right)\) are eigenvectors of \(\mathbf { M }\), and find the corresponding eigenvalues.
3. Show that \(\mathbf { M } ^ { n } = \left( \begin{array} { r r r } 4 & - 6 & - 10 \\ 2 & - 3 & - 5 \\ 0 & 0 & 0 \end{array} \right) + 2 ^ { n - 1 } \left( \begin{array} { r r r } - 2 & 4 & 4 \\ - 3 & 6 & 6 \\ 1 & - 2 & - 2 \end{array} \right)\). Section B (18 marks)

9 The matrix $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 1 & 4 \\ 1 & 5 & - 1 \\ 2 & 1 & 5 \end{array} \right)$$ has eigenvalues \(1,5,7\). Find a set of corresponding eigenvectors. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { n } = \mathbf { P D P } ^ { - 1 }\).
[0pt] [The evaluation of \(\mathbf { P } ^ { - 1 }\) is not required.]
Determine the set of values of the real constant \(k\) such that \(k ^ { n } \mathbf { A } ^ { n }\) tends to the zero matrix as \(n \rightarrow \infty\).

8 Find the eigenvalues and corresponding eigenvectors of the matrix \(\mathbf { A } = \left( \begin{array} { r r r } 4 & - 1 & 1 \\ - 1 & 0 & - 3 \\ 1 & - 3 & 0 \end{array} \right)\). Find a non-singular matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).

A \(3 \times 3\) matrix \(\mathbf { A }\) has eigenvalues \(- 1,1,2\), with corresponding eigenvectors $$\left( \begin{array} { r } 0 \\ 1 \\ - 1 \end{array} \right) , \quad \left( \begin{array} { r } - 1 \\ 0 \\ 1 \end{array} \right) , \quad \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) ,$$ respectively. Find

1. the matrix \(\mathbf { A }\),
2. \(\mathbf { A } ^ { 2 n }\), where \(n\) is a positive integer.

5 A matrix \(\mathbf { A }\) has eigenvalues \(- 1,1\) and 2 , with corresponding eigenvectors $$\left( \begin{array} { r } 0 \\ 1 \\ - 2 \end{array} \right) , \quad \left( \begin{array} { r } - 1 \\ - 1 \\ 3 \end{array} \right) \quad \text { and } \quad \left( \begin{array} { r } 2 \\ - 3 \\ 5 \end{array} \right) ,$$ respectively. Find \(\mathbf { A }\).

A \(3 \times 3\) matrix \(\mathbf { A }\) has distinct eigenvalues 2, 1, 3, with corresponding eigenvectors $$\left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) , \quad \left( \begin{array} { r } - 1 \\ 0 \\ b \end{array} \right) , \quad \left( \begin{array} { r } 0 \\ 1 \\ - 1 \end{array} \right)$$ respectively, where \(b\) is a positive constant.

1. Find \(\mathbf { A }\) in terms of \(b\).
2. Find \(\mathbf { A } ^ { - 1 } \left( \begin{array} { r } 0 \\ 2 \\ - 2 \end{array} \right)\).
3. It is given that $$\mathbf { A } ^ { n } \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) = \left( \begin{array} { l } 4 \\ 4 \\ 0 \end{array} \right) \quad \text { and } \quad \mathbf { A } ^ { n } \left( \begin{array} { r } - 1 \\ 0 \\ b \end{array} \right) = \left( \begin{array} { c } - 1 \\ 0 \\ b ^ { - 1 } \end{array} \right) .$$ Find the values of \(n\) and \(b\).

10 Find the eigenvalues and corresponding eigenvectors of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 6 & 4 & 1 \\ - 6 & - 1 & 3 \\ 8 & 8 & 4 \end{array} \right)$$ Hence find a non-singular matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } + \mathbf { A } ^ { 2 } + \mathbf { A } ^ { 3 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\).

10 It is given that the eigenvalues of the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r } 4 & 1 & - 1 \\ - 4 & - 1 & 4 \\ 0 & - 1 & 5 \end{array} \right)$$ are \(1,3,4\). Find a set of corresponding eigenvectors. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { M } ^ { n } = \mathbf { P } \mathbf { D } \mathbf { P } ^ { - 1 }$$ where \(n\) is a positive integer. Find \(\mathbf { P } ^ { - 1 }\) and deduce that $$\lim _ { n \rightarrow \infty } 4 ^ { - n } \mathbf { M } ^ { n } = \left( \begin{array} { r r r } - \frac { 1 } { 3 } & 0 & - \frac { 1 } { 3 } \\ \frac { 4 } { 3 } & 0 & \frac { 4 } { 3 } \\ \frac { 4 } { 3 } & 0 & \frac { 4 } { 3 } \end{array} \right)$$

9 Find the eigenvalues and corresponding eigenvectors of the matrix $$\mathbf { A } = \left( \begin{array} { r r r } 3 & - 1 & 0 \\ - 1 & 2 & - 1 \\ 0 & - 1 & 3 \end{array} \right)$$ Find a non-singular matrix \(\mathbf { M }\) and a diagonal matrix \(\mathbf { D }\) such that \(( \mathbf { A } - 2 \mathbf { I } ) ^ { 3 } = \mathbf { M D M } ^ { - 1 }\), where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix.

6 The matrix A, where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 0 & 0 \\ 10 & - 7 & 10 \\ 7 & - 5 & 8 \end{array} \right)$$ has eigenvalues 1 and 3 .

1. Find corresponding eigenvectors.
   It is given that \(\left( \begin{array} { l } 0 \\ 2 \\ 1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\).
2. Find the corresponding eigenvalue.
3. Find a diagonal matrix \(\mathbf { D }\) and matrices \(\mathbf { P }\) and \(\mathbf { P } ^ { - 1 }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }\).

6 The matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 0 & 0 \\ 10 & - 7 & 10 \\ 7 & - 5 & 8 \end{array} \right)$$ has eigenvalues 1 and 3. Find corresponding eigenvectors. It is given that \(\left( \begin{array} { l } 0 \\ 2 \\ 1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\). Find the corresponding eigenvalue. Find a diagonal matrix \(\mathbf { D }\) and matrices \(\mathbf { P }\) and \(\mathbf { P } ^ { - 1 }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }\).