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January 6th, 2018, 12:03 PM   #11
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Rank deficiency has nothing to do with diagonalizability.*

Diagonalize $\displaystyle A=\begin{bmatrix}
1 & 2\\
0& 0
\end{bmatrix}$

eigenvalues 0,1

eigenvectors $\displaystyle \begin{bmatrix}
-2\\ 1 \end{bmatrix},\begin{bmatrix}
1\\0\end{bmatrix}$

$\displaystyle V=\begin{bmatrix}
-2 &1 \\
1 & 0
\end{bmatrix}, V^{-1}=\begin{bmatrix}
0& 1\\
1& 2
\end{bmatrix},
V^{-1}AV=\begin{bmatrix}
0 &0 \\
0 &1
\end{bmatrix}
$

Background

If $\displaystyle A$ has a complete set of eigenvalues it is diagonalizable:
Let columns of $\displaystyle V$ be eigenvectors of $\displaystyle A$. Since $\displaystyle V$ is non-singular, find $\displaystyle V^{-1}$. Then:
$\displaystyle V^{-1}AV = V^{-1}DV = DV^{-1}V = D$

If $\displaystyle A$ is diagonalizable it has a complete set of eigenvectors:
$\displaystyle V^{-1}AV = D, AV = VD$
If $\displaystyle c_{i}$ is a column of $\displaystyle V$, $\displaystyle Ac_{i}=c_{i} \lambda_{i}$,
$\displaystyle D$ consists of eigenvalues of $\displaystyle A$ and the $\displaystyle c_{i}$ are eigenvectors of $\displaystyle A$.

*A defective matrix is one for which geometric multiplicity is less than algebraic multiplicity, in which case it doesn't have a complete set of eigenvalues. see:
Eigenvectors of a distinct eigenvalue

A great video on diagonalization is:
https://www.google.com/search?ei=4hl...93.g16PkcYC8RU
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January 6th, 2018, 02:26 PM   #12
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Quote:
Originally Posted by zylo View Post
Rank deficiency has nothing to do with diagonalizability.
He's not talking about whether A is rank deficient, but whether the matrices whose columns are eigenvectors of A are rank deficient. In your example, you relied on the fact that the matrix V formed by eigenvectors of A is invertible (i.e. is not rank deficient) to diagonalize A.

The condition that an n x n matrix A doesn't have n linearly independent eigenvectors can be rephrased by saying that every matrix whose columns are eigenvectors of A is rank deficient. Both of these are equivalent to A not being diagonalizable.
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January 8th, 2018, 04:22 AM   #13
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"In linear algebra, a defective matrix is a square matrix that does not have a complete basis of eigenvectors, and is therefore not diagonalizable. In particular, an n × n matrix is defective if and only if it does not have n linearly independent eigenvectors."

"A defective matrix always has fewer than n distinct eigenvalues, since distinct eigenvalues always have linearly independent eigenvectors. In particular, a defective matrix has one or more eigenvalues λ with algebraic multiplicity m > 1 (that is, they are multiple roots of the characteristic polynomial), but fewer than m linearly independent eigenvectors associated with λ."

https://en.wikipedia.org/wiki/Defective_matrix

A matrix A whose eigenvectors form a rank deficient matrix is quite obtuse. If you didn't have a full complement of eigenvectors, you couldn't , or wouldn't, form such a matrix in the first place.
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January 8th, 2018, 11:21 AM   #14
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Hmmm. For some reason I had wrong link to great video on diagonalization. Try again:

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