Examples on Matrix Transformations

Comp.145  October 1996  Dr. A.J. Mestel

1. Suppose

A = ( \begin{matrix}1 & 2 & 0 \\ −1 & 0 & 1 \\ 0 & 3 & 1 \end{matrix} )  and  B = ( \begin{matrix}0 & −2 & 1 \\ 1 & 0 & 0 \\ 0 & −2 & 1 \end{matrix} ) .

Calculate the matrix products (A + B)(A − B) and A^{2} − B^{2}. Why are the two results not equal?

2. A graphical operation on the position vector (x, y, z)^{T} consists, in order, of projecting onto the x-y plane, rotating about the x-axis through 45^{∘}, stretching the z-coordinate by a factor of 3, and then rotating back about the x-axis through 45^{∘} in the opposite sense. Show that the entire operation can be represented by post-multiplication by the matrix

E = ( \begin{matrix}1 & 0 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 0 \end{matrix} ) .

3. Show that for a matrix A of dimensions m × n the product AA^{T} exists and is a symmetric square matrix.

4. Find a matrix A such that A\underline{R} = \underline{b} ∧ \underline{R} for every vector \underline{R}, where \underline{b} = (b_{1}, b_{2}, b_{3})^{T} is a given constant vector.

5. Describe, in geometrical terms, the effect on the position vector \underline{R} = (x, y, z)^{T} of the transformation matrix

P = ( \begin{matrix}h_{1}^{2} & h_{1}h_{2} & h_{1}h_{3} \\ h_{2}h_{1} & h_{2}^{2} & h_{2}h_{3} \\ h_{3}h_{1} & h_{3}h_{2} & h_{3}^{2} \end{matrix} )  where  h_{1}^{2} + h_{2}^{2} + h_{3}^{2} = 1 .

[Hint: First relate the transformation to (\underline{h} · \underline{R})\underline{h} where \underline{h} = (h_{1}, h_{2}, h_{3})^{T}.]