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Reflections 25.8 Reflect on your understanding
In addition to the reflection activities below, re-read
Section 25.2 Terminology and notation . Be sure you understand each of the new definitions introduced in this chapter, and spend some time committing them to memory.
1. Necessary conditions for transition matrix that diagonalizes.
What
two conditions must the columns of transition matrix
\(P \) meet in order for
\(\inv{P} A P \) to be diagonal?
2. Diagonal form entries.
In the case that
\(\inv{P} A P \) is diagonal, what relationship will the diagonal entries of that diagonal matrix have to the original matrix
\(A \text{?}\)
3. Necessary condition to be diagonalizable.
State the condition that
\(A \) must meet in order to be diagonalizable.
4. Diagonalization procedure.
Summarize the full procedure for creating a transition matrix
\(P \) so that
\(\inv{P} A P \) is diagonal (assuming
\(A \) is diagonalizable).
5. Non-uniqueness of diagonal forms.
Explain why a diagonalizable matrix that has at least two different eigenvalues will be similar to at least two different diagonal matrices.
6. Non-uniqueness of transition matrices.
Explain why, in the case of a diagonalizable matrix, there are an infinite number of transition matrices that realize the similarity between that matrix and a specific diagonal form matrix to which it is similar.
