Symmetric matrices and variance

Remember that the Sage command A.right_eigenmatrix() attempts to find a basis for $\real^m$ consisting of eigenvectors of $A$. In particular, the assignment D, P = A.right_eigenmatrix() provides a diagonal matrix $D$ constructed from the eigenvalues of $A$ with the columns of $P$ containing the associated eigenvectors.

1. For each of the following matrices, determine whether there is a basis for $\real^2$ consisting of eigen…

Each of the following matrices is symmetric so the Spectral Theorem tells us that each is orthogonally diagonalizable. The point of this activity is to find an orthogonal diagonalization for each matrix.

To begin, find a basis for each eigenspace. Use this basis to find an orthogonal basis for each eigenspace and put these bases together to find an orthogonal basis for $\real^m$ consisting of eigenvectors.…

For each of the following matrices, find the eigenvalues and a basis for each eigenspace. Determine whether the matrix is diagonalizable and, if so, find a diagonalization. Determine whether the matrix is orthogonally diagonalizable and, if so, find an orthogonal diagonalization.

1. $\begin{bmatrix} 5 & 1 \\ -1 & 3 \\ \end{bmatrix}$

2. $\begin{bmatrix} 0 & 1 \\ 1 & 0 \\ \end{bmatrix}$

3. $\begin{bmatrix} 1 & 0 & 0 \\ 2 & -2 & 0 \\ 0 & 1 & 4 \\ \end{bmatrix}$

4. $\begin{bmatrix} 2 & 5 & -4\\ 5 & -7 & 5 \\ -4 & 5 & 2 \\ \end{bmatrix}$

Consider the matrix $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \\ \end{bmatrix}$ whose eigenvalues are $\lambda_1=5$, $\lambda_2=-1$, and $\lambda_3 = -1$.

1. Explain why $A$ is orthogonally diagonalizable.

2. Find an orthonormal basis for the eigenspace $E_5$.

3. Find a basis for the eigenspace $E_{-1}$.

4. Now find an orthonormal basis for $E_{-1}$.

5. Find matrices $D$ and $Q$ such that $A=QDQ^T$.

Find an orthogonal diagonalization, if one exists, for the following matrices.

1. $\begin{bmatrix} 11 & 4 & 12 \\ 4 & -3 & -16 \\ 12 & -16 & 1 \end{bmatrix}$.

2. $\begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 2 \\ -2 & -2 & 1 \\ \end{bmatrix}$.

3. $\begin{bmatrix} 9 & 3 & 3 & 3\\ 3 & 9 & 3 & 3\\ 3 & 3 & 9 & 3\\ 3 & 3 & 3 & 9\\ \end{bmatrix}$.

Suppose that $A$ is an $m\times n$ matrix and that $B=A^TA$.

1. Explain why $B$ is orthogonally diagonalizable.

2. Explain why $\vvec\cdot(B\vvec) = \len{A\vvec}^2$.

3. Suppose that $\uvec$ is an eigenvector of $B$ with associated eigenvalue $\lambda$ and that $\uvec$ has unit length. Explain why $\lambda = \len{A\uvec}^2$.

4. Explain why the eigenvalues of $B$ are nonnegative.

5. If $C$ is the covariance matrix associated to a demeaned dataset, explain why the eigenvalues of $C$ are nonnegative.

Suppose that you have the data points

1. Find the demeaned data points.

2. Find the total variance $V$ of the dataset.

3. Find the variance in the direction $\evec_1 = \twovec10$ and the variance in the direction $\evec_2=\twovec01$.

4. Project the demeaned data points onto the line defined by $\vvec_1=\twovec21$ and find the variance of these projected points.

5. Project the demeaned data points onto the line defined by $\vvec_2=\twovec1{-2}$ and find the variance of these projected points.

6. Ho…

Suppose that $C$ is the covariance matrix of a demeaned dataset.

1. Suppose that $\uvec$ is an eigenvector of $C$ with associated eigenvalue $\lambda$ and that $\uvec$ has unit length. Explain why $V_{\uvec} = \lambda$.

2. Suppose that the covariance matrix of a demeaned dataset can be written as $C=QDQ^T$ where What is $V_{\uvec_2}$? What does this tell you about the demeaned data?

3. Explain why the total variance of a dataset equals the sum of the eigenvalues of the c…

Determine whether the following statements are true or false and explain your thinking.

1. If $A$ is an invertible, orthogonally diagonalizable matrix, then so is $A^{-1}$.

2. If $\lambda=2+i$ is an eigenvalue of $A$, then $A$ cannot be orthogonally diagonalizable.

3. If there is a basis for $\real^m$ consisting of eigenvectors of $A$, then $A$ is orthogonally diagonalizable.

4. If $\uvec$ and $\vvec$ are eigenvectors of a symmetric matrix associated to eigenva…

Suppose that $A$ is a noninvertible, symmetric $3\times3$ matrix having eigenvectors and associated eigenvalues $\lambda_1 = 20$ and $\lambda_2 = -4$. Find matrices $Q$ and $D$ such that $A = QDQ^T$.

Suppose that $W$ is a plane in $\real^3$ and that $P$ is the $3\times3$ matrix that projects vectors orthogonally onto $W$.

1. Explain why $P$ is orthogonally diagonalizable.

2. What are the eigenvalues of $P$?

3. Explain the relationship between the eigenvectors of $P$ and the plane $W$.