Transcription of Down with Determinants! Sheldon Axler
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Down with Determinants! Sheldon Axler
down with Determinants! Sheldon Axler21 December 19941. IntroductionAsk anyone why a square matrix of complex numbers has an eigenvalue, and you llprobably get the wrong answer, which goes something like this: The characteristicpolynomial of the matrix which is defined via determinants has a root (by thefundamental theorem of algebra); this root is an eigenvalue of the s wrong with that answer? It depends upon determinants , that s are difficult, non-intuitive, and often defined without motivation. Aswe ll see, there is a better proof one that is simpler, clearer, provides more insight,and avoids paper will show how linear algebra can be done better without using determinants , we will define the multiplicity of an eigenvalue andprove that the number of eigenvalues, counting multiplicities, equals the dimensionof the underlying space.
Down with Determinants! Sheldon Axler 21 December 1994 1. Introduction Askanyonewhyasquarematrixofcomplexnumbershasaneigenvalue, andyou’ll probably get the wrong ...
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