Stage 3 has built five separate pieces of machinery — the characteristic equation, diagonalisation, the diagonalisability criterion, two flavours of dynamical application, and the symmetric-matrix specialisation. This capstone note does not introduce anything new; it consolidates all five into a single decision procedure a reader can run against any eigenstructure problem, then works four exam-style problems end-to-end using that procedure, each drawing on a different piece of the toolkit.
Learning objectives. By the end of this note you should be able to:
- run the full eigenstructure decision procedure on an unfamiliar matrix without prompting;
- recognise which of Stage 3's four notes a given exam question is really testing;
- avoid the topic's most common computational slips;
- solve a multi-part eigenstructure problem end-to-end under time pressure.
In this lesson
- The decision procedure
- Worked exam-style problem 1: full toolkit on a matrix
- Worked exam-style problem 2: Markov chain stationary distribution
- Worked exam-style problem 3: symmetric matrix, all shortcuts
- Worked exam-style problem 4: coupled ODE stability
Before you start
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