We introduce the concept of change matrix. (See Lect18.pdf) Next lecture we shall see how it is used to answer the question: what is the relation between and ?

Some preparation: In the coming lecture we shall (re-)visit eigenvalues and eigenvectors that you are supposed to know their definitions and the method to find them. To refresh your memory, please read the following.

Given an matrix . If and satisfy , then is called the eigenvalue of and is said to be an eigenvector of corresponding to . Note that by definition is NOT an eigenvector of .

**Exercise.** Let be an eigenvalue of . Show that

is a subspace of . We call the eigenspace of corresponding to , thus is the set of all eigenvectors corresponding to and the zero vector.

**Method:**

- The eigenvalues of are found by solving the characteristic equation
which is in fact a polynomial in of degree . ( the identity matrix.)

- The eigenvectors corresponding to are found by solving the matrix equation in :
All nonzero solutions of are the eigenvectors for .

(See Textbook p.151-153 for examples.)

After Class Exercises: Ex 9.2 Qn 1(b), 4(b), 5(b), 7(b) (See textbook’s solution and L18-ace.pdf)

Revision: Ex 9.1 Qn 16, 22. (See L18-ace.pdf)

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