Mathematical Immaturity

2.21 Miscellaneous review exercises on matrices

• If a square matrix has a row of zeros or a column of zeros, prove that it is singular.
• For each of the following statements about $n \times n$ matrices, give a proof or exhibit a counter example.
  (a) $\quad$ If $AB + BA = O,$ then $A^2B^3 = B^3A^2.$
  (b) $\quad$ If $A$ and $B$ are nonsingular, then $A + B$ is nonsingular.
  (c) $\quad$ If $A$ and $B$ are nonsingular, then $AB$ is nonsingular.
  (d) $\quad$ If $A,$ $B,$ and $A + B$ are nonsingular, then $A - B$ is nonsingular.
  (e) $\quad$ If $A^3 = O,$ then $A - I$ is nonsingular.
  (f) $\quad$ If the product of $k$ matrices $A_1 \cdots A_k$ is nonsingular, then each matrix $A_i$ is nonsingular.
• If $A = \begin{bmatrix} 1 & 2 \\ 5 & 4 \end{bmatrix},$ find a nonsingular matrix $P$ such that $P^{-1}AP = \begin{bmatrix} 6 & 0 \\ 0 & -1 \end{bmatrix}.$
• The matrix $A = \begin{bmatrix} a & i \\ i & b \end{bmatrix},$ where $i^2 = -1,$ $a = \frac{1}{2}(1 + \sqrt{5}),$ and $b = \frac{1}{2}(1 - \sqrt{5}),$ has the property that $A^2 = A.$ Describe completely all $2 \times 2$ matrices $A$ with complex entries such that $A^2 = A.$
• If $A^2 = A,$ prove that $(A + I)^k = I + (2^k - 1)A.$
• The special theory of relativity makes use of a set of equations of the form $x' = a(x - vt),$ $y' = y,$ $z' = z,$ $t' = a(t - vx/c^2).$ Here $v$ represents the velocity of a moving object, $c$ the speed of light, and $a = c/\sqrt{c^2 - v^2},$ where $|v| \lt c.$ The linear transformation which maps the two-dimensional vector $(x, t)$ onto $(x', t')$ is called a Lorentz transformation. Its matrix relative to the usual bases is denoted by $L(v)$ and is given by $$L(v) = a \begin{bmatrix} 1 & -v \\ -vc^{-2} & 1 \end{bmatrix}.$$ Note that $L(v)$ is nonsingular and that $L(0) = I.$ Prove that $L(v)L(u) = L(w),$ where $w = (u + v)c^2/(uv + c^2).$ In other words, the product of two Lorentz transformations is another Lorentz transformation.
• If we interchange the rows and columns of a rectangular matrix $A,$ the new matrix so obtained is called the transpose of $A$ and is denoted by $A^t.$ For example, if we have $$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}, \quad \text{then} \quad A^t = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}.$$ Prove that transposes have the following properties:
  (a) $\quad$ $(A^t)^t = A.$
  (b) $\quad$ $(A + B)^t = A^t + B^t.$
  (c) $\quad$ $(cA)^t = cA^t.$
  (d) $\quad$ $(AB)^t = B^tA^t.$
  (e) $\quad$ $(A^{-1})^t = (A^t)^{-1}$ if $A$ is nonsingular.
• A square matrix $A$ is called an orthogonal matrix if $AA^t = I.$ Verify that the $2 \times 2$ matrix $\begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$ is orthogonal for each real $\theta.$ If $A$ is any $n \times n$ orthogonal matrix, prove that its rows, considered as vectors in $V_n,$ form an orthonormal set.
• For each of the following statements about $n \times n$ matrices, give a proof or else exhibit a counter example.
  (a) $\quad$ If $A$ and $B$ are orthogonal, then $A + B$ is orthogonal.
  (b) $\quad$ If $A$ and $B$ are orthogonal, then $AB$ is orthogonal.
  (c) $\quad$ If $A$ and $AB$ are orthogonal, then $B$ is orthogonal.
• Hadamard matrices, named for Jacques Hadamard (1865-1963), are those $n \times n$ matrices with the following properties:
  I. Each entry is 1 or $-1.$
  II. Each row, considered as a vector in $V_n,$ has length $\sqrt{n}.$
  III. The dot product of any two distinct rows is 0.
  Hadamard matrices arise in certain problems in geometry and the theory of numbers, and they have been applied recently to the construction of optimum code words in space communication. In spite of their apparent simplicity, they present many unsolved problems. The main unsolved problem at this time is to determine all $n$ for which an $n \times n$ Hadamard matrix exists. This exercise outlines a partial solution.
  (a) $\quad$ Determine all $2 \times 2$ Hadamard matrices (there are exactly 8).
  (b) $\quad$ This part of the exercise outlines a simple proof of the following theorem: If $A$ is an $n \times n$ Hadamard matrix, where $n > 2,$ then $n$ is a multiple of 4. The proof is based on two very simple lemmas concerning vectors in $n$-space. Prove each of these lemmas and apply them to the rows of Hadamard matrix to prove the theorem.

  $\quad$ Lemma 1. $\quad$ If $X,$ $Y,$ $Z$ are orthogonal vectors in $V_n,$ then we have $$(X + Y) \cdot (X + Z) = \|X\|^2.$$

  $\quad$ Lemma 2. $\quad$ Write $X = (x_1, \ldots, x_n),$ $Y = (y_1, \ldots, y_n),$ $Z = (z_1, \ldots, z_n).$ If each component $x_i,$ $y_i,$ $z_i$ is either 1 or $-1,$ then the product $(x_i + y_i)(x_i + z_i)$ is either 0 or 4.