What would be the necessary condition for a matrix of any $n \times n$ to have eigenvalue 1? I know that it must have a corresponding eigenvector - that is obvious - I want to know things like how values must be, or things like that.
What is the necessary condition for a matrix to have eigenvalue 1?
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linear-algebra
matrices
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0@Vilid: but $I-I = 0$ is most definitely singular. – 2012-12-11
1 Answers
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The necessary and sufficient condition for $A$ to have eigenvalue $1$ is that $A-I$ is singular. This is equivalent to any of
- $\det(A-I) = 0$
- $\text{Ker}(A-I) \ne \{0\}$
- $\text{Ran}(A-I) \ne {\mathbb F}^n$ (where we're working over the field $\mathbb F$)
- $\text{Ker}(A^T - I) \ne \{0\}$
- $\text{Ran}(A^T - I) \ne {\mathbb F}^n$