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The Similarity of Square Matrices 

Let A and B be n×n complex matrices. A and B are said to be similar if there exists an n×n invertible matrix P such that
It is not just a single matrix P that exists; it is almost a vector space. The invertibility requirement precludes the inclusion of a matrix of zeroes Vhich is required for a vector space. For the moment let the invertibility requirement be ignored. Let V be the set of all P such that AP=PB. If P and Q belong to V then (P+Q) belongs to V. Since AP=PB and AQ=QB then
Likewise if P belongs to V then for any scalar k, kP belongs to V since
Also the n×n matrix of zeroes, 0, belongs to V since A0=0=0B. And lastly, if P belongs to V then there exists a Q such that (P+Q)=0; namely (1)P. So V is a vector space.
Without the invertibility requirement any two n×n matrices A and B would be similar because A0=0B. Let V_{A,B} the vector space of n×n matrices P such that AP=PB and let W_{A,B} be the set of invertible n×n matrices P such that AP=PB. Then
Consider now the process of attempting to find for A and B a matrix P such that AP=PB. This is the same as seeking a solution to the linear homogeneous set of equations
Note that if A and B are similar and X is an eigenvector of B and λ is its eigenvalue then
and hence λ is also an eigenvalue of A and PX is an eigenvector. Thus in order for two matrices to be similar they must have the same eigenvalues.
(To be continued.)
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