Nine Eigen
Explore how matrices can fix vectors and vector directions.
Age
16 to 18
Challenge level
Problem
This problem involves the action of matrices on vectors in three dimensions. The first few questions look at fixed vector directions; the latter questions look at fixed vectors. As you consider each point, make use of geometric or algebraic arguments as appropriate. Draw diagrams and construct particular examples of matrices and vectors if needed. If there is no definitive answer to a given part, try to give examples of when the question posed is or is not true.
In the questions below: $R, S$ are rotation matrices; $P, Q$ are reflection matrices; $M$ is neither a rotation nor a reflection.
- Which of the different types of matrices can leave no vector directions fixed?
- Which of the different types of matrices can leave exactly one vector direction fixed?
- Which of the different types of matrices can leave more than one vector direction fixed?
- Is it ever the case that $RS$ can leave a vector invariant?
- Is it ever the case that $PQ$ can leave a vector invariant?
- Is it ever the case that $M$ will leave the direction of a vector invariant?
- Can a matrix with determinant zero leave a vector fixed?
- Can a matrix with determinant greater than $1$ leave a vector fixed?
- Can a matrix leave exactly two vectors fixed?
Getting Started
A vector ${\bf v}$ is fixed by a matrix if
$$
M{\bf v} = {\bf v}
$$
The vector direction of ${\bf v}$ is fixed if there is some number $\lambda\neq 0$ such that
$$ M{\bf v} = \lambda{\bf v} $$
This resource introduces concepts which appear in the later Further Pure Mathematics A Level modules.
Teachers' Resources
Why do this problem?
This problem asks a series of questions designed to provoke students' thinking about matrices which leave vectors fixed, and the properties that such matrices and vectors would have.Possible approach
It may be worthwhile to start with some preliminary work about
matrices in three dimensions. Students could find some examples of
$3 \times 3$ matrices which represent simple rotations and
reflections, which could be used in answering the problem.
The questions divide neatly into three sections - questions
1-3, 4-6 and 7-9. Students could tackle these questions in those
three sections, perhaps working with a partner, and feed back ideas
to the rest of the class after each section is answered.
For each section of questions, ask students to think about what they are being asked to do, use their intuition to make any initial comments, then think about the geometry of the situation and finally use some examples to support their thoughts algebraically.
Key questions
What can you say about a rotation that leaves the direction of
a vector unchanged?
What can you say about a reflection that leaves the direction
of a vector unchanged?
Possible extension
Fix Me or
Crush Me investigates matrices which fix certain
vectors and vectors which are fixed by certain
matrices.