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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.43 Jordan Normal Form

Suppose that you are presented with an n × n real or complex MATRIX [I.3 §4.2] A and would like to understand it. You might ask how it behaves as a LINEAR MAP [I.3 §4.2] on Imagen or Imagen, or you might wish to know what the powers of A are. In general, answering these questions is not particularly easy, but for some matrices it is very easy. For example, if A is a diagonal matrix (that is, one whose nonzero entries all lie on the diagonal), then both questions can be answered immediately: if x is a vector in n or n, then Ax will ...

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