What do eigenvalues and eigenvectors of a Matrix A tell us about a transformation T (T: R^3 -> R^3)?

I want to know if I'm on the right page, close to, or not correct about any of these. Let's say I have a 3x3 matrix and its eigenvalues are 0,2, and 5.

1) Eigenvalues can tell us if A is an invertible matrix if and only if zero is not an eigenvalue of A, but because it is in this example we know that A is not an invertible matrix. That means that the transformation of A is not invertible as well.

2) If the vectors are eigenvectors that correspond with the eigenvalues of an nxn matrix A then the set of vectors are linearly independent. Since the eigenvalues of A and eigenvectors correspond to each other then the set of vectors is linearly independent. Meaning the vectors of the transformation of A are linearly independent.

3) If A has 'n' linearly independent eigenvectors then the nxn matrix A is diagonalizable. Because A has all 3 linearly independent eigenvectors we can conclude that A is diagonalizable and the Transformation of A is diagonalizable.

  • $\begingroup$ The second is only true when the eigenvalues are distinct. $\endgroup$
    – amd
    May 4 '20 at 17:21
  1. Yes, this is correct.
  2. As stated, it is not true. If $v$ is an eigenvector, then so is $2v$. But $\{v,2v\}$ is linearly dependent. However, if $\lambda_1,\ldots,\lambda_n$ are distinct eigenvalues and if, for each $k\in\{1,\ldots,n\}$, $v_k$ is an eigenvector corresponding to the eigenvaulue $\lambda_k$, then the set $\{v_1,\ldots,v_n\}$ is linearly independent.
  3. Yes, and actually that if is a “if and only if”.
  • $\begingroup$ If I was to restate number two as you presented it in your answer and all the conditions you mentioned are satisfied then it would become true? $\endgroup$ May 4 '20 at 17:34
  • $\begingroup$ Yes, it will.${}$ $\endgroup$ May 4 '20 at 17:35
  • $\begingroup$ Okay, thank you so much for the clarifications! $\endgroup$ May 4 '20 at 17:37
  • $\begingroup$ Is there anything more that can be added? Can they tell something about the null or column space? $\endgroup$ May 4 '20 at 18:09
  • $\begingroup$ The null space consists of the null vector together with the eigenvectors corresponding to the eigenvalue $0$. And I see nothing that I could add concerning the column space. $\endgroup$ May 4 '20 at 18:16

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