I am quite confused about this. I know that zero eigenvalue means that null space has non zero dimension. And that the rank of matrix is not the whole space. But is the number of distinct eigenvalues ( thus independent eigenvectos ) is the rank of matrix?
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8$\begingroup$ The identity matrix has $1$ as its only eigenvalue. It has many eigenvectors. $\endgroup$– André NicolasCommented Jul 5, 2015 at 6:10
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$\begingroup$ if my matrix is of size $[n, n]$, and rank $r$, wouldn't there be $r$ eigenvalues and we can get $r$ independent eigenvectors? $\endgroup$– Charlie ParkerCommented Oct 19, 2021 at 21:25
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$\begingroup$ related: math.stackexchange.com/questions/146927/… $\endgroup$– Charlie ParkerCommented Oct 19, 2021 at 21:26
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2 Answers
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Well, if $A$ is an $n \times n$ matrix, the rank of $A$ plus the nullity of $A$ is equal to $n$; that's the rank-nullity theorem. The nullity is the dimension of the kernel of the matrix, which is all vectors $v$ of the form: $$Av = 0 = 0v.$$ The kernel of $A$ is precisely the eigenspace corresponding to eigenvalue $0$. So, to sum up, the rank is $n$ minus the dimension of the eigenspace corresponding to $0$. If $0$ is not an eigenvalue, then the kernel is trivial, and so the matrix has full rank $n$. The rank depends on no other eigenvalues.
answered Jul 5, 2015 at 6:15
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2$\begingroup$ Thanks for the answer. If I know that a matrix has 1 eigenvalue which is zero, does it mean that dimension of kernal is 1? And rank of matrix is (rank of whole space - 1)? $\endgroup$– ShifuCommented Jul 5, 2015 at 6:20
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2$\begingroup$ Well, consider the $0$ matrix. It has one eigenvalue: $0$, and the dimension of its eigenspace is $n$, since it sends everything to $0$. If you have $n$ distinct eigenvalues, one of them zero, then the eigenspace for $0$ must have dimension $1$, hence the rank is $n - 1$. $\endgroup$ Commented Jul 5, 2015 at 6:26
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3$\begingroup$ This is what I have understood. Please correct me if i am wrong. 1) If a matrix has 1 eigenvalue as zero, the dimension of its kernel may be 1 or more (depends upon the number of other eigenvalues). 2) If it has n distinct eigenvalues its rank is atleast n. 3) The number of independent eigenvectors is equal to the rank of matrix. $\endgroup$– ShifuCommented Jul 5, 2015 at 6:33
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10$\begingroup$ 2) is not right. If it has $n$ non-zero eigenvalues, then the nullity is $0$ and the rank is $n$. If one of the $n$ is $0$, then it has rank $n-1$ as I said. 1) is almost right. Yes, if $1$ of the eigenvalues is $0$, then the kernel has dimension at least $1$, maybe more. However, it doesn't just depend on the number of other eigenvalues. It is possible to have only $0$ as an eigenvalue, but still only have a nullity of $1$. 3) is again, not quite right. The rank is equal to the number of independent generalised eigenvectors. Look up Jordan Bases. $\endgroup$ Commented Jul 5, 2015 at 7:29
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4$\begingroup$ Eigenvectors/values are only applicable to square matrices. But $A^T A$ and $AA^T$ are square matrices with equal rank to $A$. $\endgroup$ Commented May 20, 2017 at 14:50
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My comment is 7 years late but I hope someone might find some useful information.
First, the number of linearly independent eigenvectors of a rank $k$ matrix can be greater than $k$. For example \begin{align} A &= \left[ \begin{matrix} 1 & 2 \\ 2 & 4 \end{matrix} \right] \\ rk(A) &= 1 \\ \end{align} $A$ has the following eigenvalues and eigenvectors $\lambda_1 = 5, \mathbf{v}_1 = [1 \ \ 2]^\top$, $\lambda_2 = 0, \mathbf{v}_2 = [-2 \ \ 1]^\top$. So $A$ has 1 linearly independent column but 2 linearly independent eigenvectors. The column space of $A$ has 1 dimension. The eigenspace of $A$ has 2 dimensions.
There are also cases where the number of linearly independent eigenvectors is smaller than the rank of $A$. For example \begin{align} A &= \left[ \begin{matrix} 0 & 1 \\ -1 & 0 \end{matrix} \right] \\ rk(A) &= 2 \end{align} $A$ has no real valued eigenvalues and no real valued eigenvectors. But $A$ has two complex valued eigenvalues $\lambda_1 = i,\ \lambda_2 = -i$ and two complex valued eigenvectors.
Another remark is that a eigenvalue can correspond to multiple linearly independent eigenvectors. An example is the Identity matrix. $I_n$ has only 1 eigenvalue $\lambda = 1$ but $n$ linearly independent eigenvectors.
So to answer your question, I think there is no trivial relationship between the rank and the dimension of the eigenspace.
