6  Problems

Problem 1

Consider the matrix \[ \boldsymbol A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}. \]

  1. Determine \(\boldsymbol A'\). Is \(\boldsymbol A\) symmetric?
  2. Is \(\boldsymbol A\) idempotent?
  3. Compute the determinant and the rank. Is \(\boldsymbol A\) nonsingular?
  4. Compute the inverse.
  5. Compute the trace.

Problem 2

  1. Let \(\boldsymbol A \boldsymbol B = \boldsymbol C\), where \[ \boldsymbol A = \begin{pmatrix} 1 & 0 \\ 2 & 2 \end{pmatrix}, \quad \boldsymbol C = \begin{pmatrix} 1 & 3 \\ 2 & 5 \end{pmatrix}. \] Compute \(\boldsymbol B\).
  2. \(\boldsymbol \delta\) and \(\boldsymbol \gamma\) are \(c \times 1\) vectors, \(\boldsymbol X\) is a \(d \times c\) matrix, and \(\boldsymbol Y\) is a \(c \times d\) matrix. Determine the orders of \(\boldsymbol X \boldsymbol Y\), \(\boldsymbol Y \boldsymbol X\), \(\boldsymbol \gamma' \boldsymbol \gamma\), \(\boldsymbol \gamma \boldsymbol \gamma'\), and \(\boldsymbol \delta' \boldsymbol Y \boldsymbol X \boldsymbol \gamma\). Under which conditions do the expressions \(\boldsymbol Y^{-1}\) and \(\boldsymbol \delta' \boldsymbol Y \boldsymbol X + \boldsymbol \gamma' \boldsymbol \gamma\) exist?
  3. Compute \(\mathop{\mathrm{tr}}(\lambda \boldsymbol R' \boldsymbol R)\) for \(\lambda \in \mathbb R\) and \[ \boldsymbol R = \begin{pmatrix} \frac{1}{4} & \frac{\sqrt 3}{4} \\ \frac{\sqrt 3}{4} & \frac{3}{4} \end{pmatrix}. \]

Problem 3

Let \(\boldsymbol A\) be nonsingular. Simplify the expression \[ \bigg( \frac{1}{\sqrt 2} \boldsymbol A^{-1} \bigg( \frac{1}{\sqrt 2} \boldsymbol A'' + \frac{\sqrt 2}{2} \boldsymbol A \bigg) \bigg). \]

Problem 4

Consider the \(n \times k\) matrix \(\boldsymbol X\) with \(\mathop{\mathrm{rank}}(\boldsymbol X) = k\). Moreover, let \(\boldsymbol P = \boldsymbol X (\boldsymbol X' \boldsymbol X)^{-1} \boldsymbol X'\), and let \(\boldsymbol M = \boldsymbol I_n - \boldsymbol P\)

  1. Determine the order of the following matrices: \(\boldsymbol I_n\), \(\boldsymbol X' \boldsymbol X\), \(\boldsymbol P\), \(\boldsymbol M\)
  2. Which matrices from a) are symmetric?
  3. Which matrices from a) are idempotent?
  4. Compute the trace of \(\boldsymbol I_n\) and \(\boldsymbol P\).

Problem 5

Let \(\boldsymbol X\) be a \(n \times k\) matrix. Show that \(\boldsymbol X' \boldsymbol X\) is positive semi-definite. Under which condition is \(\boldsymbol X' \boldsymbol X\) positive definite?

Problem 6

Let \(\boldsymbol y \in \mathbb R^n\), \(\boldsymbol X\) be a \(n \times k\) matrix, and \(\boldsymbol \beta \in \mathbb R^k\). Compute the derivatives \[ \frac{\partial f(\boldsymbol \beta)}{\partial \boldsymbol \beta}, \quad \frac{\partial^2 f(\boldsymbol \beta)}{\partial \boldsymbol \beta \partial \boldsymbol \beta'}, \] for the function \(f(\boldsymbol \beta) = (\boldsymbol y - \boldsymbol X \boldsymbol \beta)'(\boldsymbol y - \boldsymbol X \boldsymbol \beta)\).

6.1 Solutions

Solutions to the problems are available here (unfortunately only in German so far)