23: Suppose matrix A is similar to B. What is the relationship between the characteristic polynomials of A and B? What does your answer tell you about the eigenvalues of A and B?
Characteristic polynomials of A and B are the same, therefore their eigenvalues are also same.
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled.
If A and B are similar matrices then, , for an invertible T. If det , then
This means that the characteristics polynomials of A and B are the same. From this also follows that their eigenvalues are exactly the same.
Hence, characteristic polynomials of A and B are the same, therefore their eigenvalues are also same.
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