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Two square matrices A,B of the same order n are similar if there exists an invertible matrix P of order
n such that
B= P-1AP.
Example 11.5.1
- 1.
- The identity matrix I is similar only to itself.
- 2.
- The matrices
and
are similar. (Check it with
).
Proposition 11.5.2
- 1.
- If A and B are similar, then |A|=|B|.
- 2.
- If A and B are similar, then
.
- 3.
- Similar matrices have the same characteristic polynomial, thus the same eigenvalues (with the same
multiplicities).
Noah Dana-Picard
2001-02-26