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The definition of diagonalizable matrix and diagonalizable linear map is given here https://en.wikipedia.org/wiki/Diagonalizable_matrix and the poof that every idempotent matrix is diagonalizable is given in Proof 1 here Idempotent matrix is diagonalizable? , but I do not understand how the definition of diagonalizable linear map is used in Proof 1, could anyone clarify this for me please?

Thanks!

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He doesn't use that definition, but he showed that $A:V\longrightarrow V$ is a diagonalizable linear map. Indeed he found a basis such that the associated linear map is represented by a diagonal matrix.

InsideOut
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  • he used the definition of a diagonalizable linear map? –  Sep 16 '17 at 06:58
  • your answer is not clear for me.... please what is the definition of a diagonalizable linear map that he used? –  Sep 16 '17 at 07:00
  • Yes, a map is diagonalizable if there exists a basis such that the associated matrix is diagonal. He found a basis of eigenvectors for $A$ and then respect to it the matrix id diagonal – InsideOut Sep 16 '17 at 07:01
  • The same given in Wikipedia that you cited above – InsideOut Sep 16 '17 at 07:01
  • a linear map T : V → V is called diagonalizable if there exists an ordered basis of V with respect to which T is represented by a diagonal matrix..... where is the ordered basis that he found? and how he showed that with respect to this basisT is represented by a diagonal matrix? –  Sep 16 '17 at 07:04
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    He found $r$ linear indipendent eigenvectors of eigenvalue $1$. Precisely he took a basis for the image and then a preimage of such basis, note that these are linearly indipendent. After that he chose a basis of the kernel, they are linearly indipendent and eigenvectors of eigenvalue $0$. – InsideOut Sep 16 '17 at 07:07
  • What is the diagonal matrix that A is represented by? –  Sep 16 '17 at 07:50
  • Is a diagonal matrix where the first $r$ diagonal entries are $1$ and the remaining $n-r$ diagonal entries are zeros. All non-diagonal entries are zeros as well. – InsideOut Sep 16 '17 at 07:59