Some remarks to supplement Friday’s lectures:

- We mentioned that has only 1 eigenvalue and its geometric multiplicity is 1 (strictly less than its algebraic multiplicity 2), so is not diagonalizable.
Indeed for this simple case, we may check that is not diagonalizable by

*brute-force*, as follows:Suppose for some nonsingular .

Let us write . Then,

Case 1) and , then .

Case 2) and , then .

Case 3) and , then .

Case 4) and , then .

All cases are impossible as .

- Cayley-Hamilton’s Theorem says that for an matrix , if its characteristic polynomial is , then
(zero matrix).

- Some classmates asked if the characteristic equation has no real solution, does it mean has no eigenvalue? The answer is NO: it only means has
eigenvalues! If we use complex numbers, then has (complex) eigenvalues and the theorem of Jordan Canonical Form holds for the case of complex eigenvalues.**no real**

After Class Exercises: Ex 9.3 Qn 7, 31; Ex 11.1 Qn 2; Ex 11.2 Qn 3 (b). (Solution — To be available.)

**Exercises**. Find a Jordan Canonical Form of the following matrices:

, .

Next week we shall turn to the new topic — inner product spaces. Hence let us have a round up with the following chart which indicates how/what are developed in our study of linear transformations.

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