MATH221 December 2007
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Question 07 (c)
Consider the matrix . For each eigenvalue, find the dimension of the corresponding eigenspace.
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From part (b), we know that the eigenvalues of A are 1 and 2. We also saw in part (b) that the eigenvalue 2 is a repeated root of the characteristic polynomial for A, while the eigenvalue 1 is not.
By definition, the dimension of each respective eigenspace is at least 1. As well, recall that the dimension of each eigenspace is at most the multiplicity of the eigenvalue as a root of the characteristic polynomial.
- As the eigenvalue 1 is not a repeated root of the characteristic polynomial for A, the dimension of its eigenspace must be at least 1, and at most 1, i.e. it is equal to 1.
- The eigenvalue 2 is a twice repeated root of the characteristic polynomial for A; therefore the dimension of its eigenspace may be either 1 or 2. We will need to investigate further to determine which one it is.
To calculate the dimension of the eigenspace corresponding to the eigenvalue 2, we want to determine , that is, the dimension of the kernel of A - 2I. We have
All three rows of A - 2I are multiples of each other, so . Therefore, by the rank-nullity theorem,
- For the eigenvalue 1, the dimension of its eigenspace is 1.
- For the eigenvalue 2, the dimension of its eigenspace is 2.
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