For a 2 X 2 matrix (except the identity matrix) having eigenvalues 1,1 is it necessary for the matrix to satisfy a two degree monic polynomial (X-1)(X-K) for some real K (K is not equal to 1) (the matrix clearly cannot satisfy a monomial).
For example, an identity matrix can satisfy X(X-1) (the identity matrix also has eigenvalues 1,1) but can we always find an n degree polynomial except the characteristic polynomial for all n x n matrices having a repeated eigenvalue.

  • 3
    $\begingroup$ If the minimal polynomial of the matrix is $X-1$, it can. But this means the matrix is the identity matrix. $\endgroup$
    – Bernard
    Jul 7 '20 at 14:54

The matrix must satisfy its characteristic polynomial, i.e. $(X-I)^2=0$. If also $(X-I)(X-kI) = 0$, then their difference $(X-I)^2 - (X-I)(X-kI) = (X-I)(k-1) = 0$ so if $k \ne 1$, $X-I = 0$ i.e. $X$ is the identity matrix.

  • $\begingroup$ So a 2 by 2 matrix with a single eigenvalue can only satisfy a single monic two degree polynomial i.e it's characteristic polynomial? $\endgroup$
    – Aspirant
    Jul 7 '20 at 16:06
  • 2
    $\begingroup$ Unless it's a scalar multiple of the identity matrix, yes. $\endgroup$ Jul 7 '20 at 16:24

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