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. Without determinants , we ll define the characteristic andminimal polynomials and then prove that they behave as expected.
det 4 3. Generalized eigenvectors Unfortunately, the eigenvectors of T need not span V.For example, the linear operator on C2 whose matrix is 01 00 has only one eigenvalue, namely 0, and its eigenvectors form a one-dimensional
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