Unit 7 test: Symmetric Matrices

Which statement about an $n \times n$ matrix $A$ is equivalent to "$A$ is orthogonally diagonalizable"?

Find the eigenvalues of $A = \begin{bmatrix} 4 & -2 \\ -2 & 1 \end{bmatrix}$.

Let $A = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix}$. Its eigenvalues are $2$ and $-1$. What is the dimension of the eigenspace for $\lambda = -1$?

A symmetric $2 \times 2$ matrix $A$ has eigenvalues $2$ and $-3$, and $(3, 4)$ is an eigenvector for $\lambda = 2$. Find the $(1, 2)$ entry of $A$.

The eigenspace of a symmetric matrix $A$ for $\lambda = 1$ has basis $(1, 1, 0)$, $(1, 0, 1)$. To build the orthogonal matrix $P$ in $A = PDP^T$, which pair of vectors (after normalizing) should represent this eigenspace?

Let $A$ be the matrix of the quadratic form $Q(\mathbf{x}) = 4x_1^2 - 10x_1x_2 + x_2^2$. Find the $(2, 1)$ entry of $A$.

Let $A = \begin{bmatrix} 1 & 2 & 0 \\ 2 & -1 & 3 \\ 0 & 3 & 2 \end{bmatrix}$ and $\mathbf{x} = (1, 2, 1)$. Compute $\mathbf{x}^TA\mathbf{x}$.

Classify $Q(\mathbf{x}) = -x_1^2 + 4x_1x_2 - 4x_2^2$.

An orthogonal change of variable $\mathbf{x} = P\mathbf{y}$ turns $Q(\mathbf{x}) = 2x_1^2 + 4x_1x_2 + 5x_2^2$ into $\lambda_1y_1^2 + \lambda_2y_2^2$, with no cross term. Find $\lambda_1$ and $\lambda_2$.

Find the maximum value of $Q(\mathbf{x}) = 3x_1^2 + 8x_1x_2 - 3x_2^2$ subject to $x_1^2 + x_2^2 = 1$.

Find the largest singular value of $A = \begin{bmatrix} 1 & -1 \\ -2 & 2 \\ 2 & -2 \end{bmatrix}$.

Let $A = \begin{bmatrix} 2 & 1 \\ -1 & 2 \end{bmatrix}$. Find the maximum of $\|A\mathbf{x}\|$ over all unit vectors $\mathbf{x}$.

A $5 \times 4$ matrix $A$ has singular values $9, 4, 1, 0$. What is the rank of $A$?

A $3 \times 3$ matrix $A$ has singular values $4$, $3$ and $\tfrac{1}{2}$. Find $|\det A|$.

Which statement about the singular value decomposition $A = U\Sigma V^T$ of an $m \times n$ matrix $A$ is false?