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On page 21 of Matrix Differential Calculus by Magnus and Neudecker (3rd ed, ISBN:0-471-98632-1), the book states, without any apparent justification, that to prove the statement:

If $A$ has $r$ non-zero eigenvalues, then rank($A$) $\geq r$.

We start by

...using [Schur Decomposition], $S^*AS=M$. We partition $M = \left( \begin{array}{cc}M_1 && M_2 \\ 0 && M_3 \end{array} \right)$ where $M_1$ is a non-singular upper triangular $r \times r$ matrix and $M_3$ is strictly upper triangular.

where $M$ is the upper diagonal matrix and $S$ is the unitary matrix resulting from Schur Decomposition. This statement is later repeated for Theorem 21.

Just looking at an arbitrary upper triangular matrix such as

$\left(\begin{array}{ccccc} 1 & 2 & 3 & 4 & 5\\ 0 & 0 & 1 & 2 & 3\\ 0 & 0 & 4 & 5 & 6\\ 0 & 0 & 0 & 7 & 8\\ 0 & 0 & 0 & 0 & 9 \end{array}\right)$

I can't see how this is generally possible. Is there a theorem regarding Schur Decomposition that I'm missing?

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  • $\begingroup$ One thought is that we can manipulate $S$ in a way that "rearranges $M$". But I'm not sure how this would work. $\endgroup$
    – lycus
    Mar 27, 2016 at 21:54
  • $\begingroup$ I don't understand the Schur decomposition stuff. So, I give you another proof of this fact. The rank of a matrix $A$ is the dimension of its column space and so, the dimension of the image of $A$. If $A$ has $r$ non-zero eigenvalues, let $x_1,\ldots,x_r$ be corresponding eigenvectors. We know that $\{x_1,\ldots,x_r\}$ is linearly independent. If we now restrict $A$ (as a linear map) to the $r$-dimensional subspace spanned by the $x_k$ we easily see that this restriction is injective. So, $\operatorname{rank}(A)\ge\operatorname{rank}(A') = r$, where $A'$ is the restriction. $\endgroup$
    – Friedrich Philipp
    Mar 27, 2016 at 21:58
  • $\begingroup$ @FriedrichPhilipp You may have an eigenvalue with multiplicity $k>1$ yet an eigenspace of dimension $<k$. That's the point of using the Schur decomposition I think: it always exists. $\endgroup$
    – Jean-Claude Arbaut
    Mar 27, 2016 at 22:08
  • $\begingroup$ Sure, you are right. Sometimes I am too fast. $\endgroup$
    – Friedrich Philipp
    Mar 27, 2016 at 22:18
  • $\begingroup$ @lycus In your example, replace the second basis vector by $f = e_2+4e_3$. Then $Af = 14e_1 + 4f$ and $Ae_3 = 3e_1+f$. So, you swapped the diagonal entries $0$ and $4$. You can do this again and again until you have $0$ in the lower right corner. $\endgroup$
    – Friedrich Philipp
    Mar 27, 2016 at 23:00

1 Answer 1

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You can do as follows. Let $E=\bigoplus_{i|\lambda_i\not=0}\ker((A-\lambda I)^n),F=\ker(A^n)$; note that $A_{|E}\in L(E),A_{|F}\in L(F)$ and $E\bigoplus F=\mathbb{C}^n$. We choose an orthonormal basis of $E$ that triangularizes $A_{|E}$ and an orthonormal basis of $E^{\perp}$. The matrix $A$ becomes $B=\begin{pmatrix}M_1&N_2\\0&N_3\end{pmatrix}$ where $M_1$ is invertible triangular and $N_3$ is nilpotent. It remains to choose another orthonormal basis of $E^{\perp}$ that triangularizes $N_3$.

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