Dot diagrams and Jordan canonical forms

[psie]

Homework Statement

Why is the number of dots $r_i$ in row $i$ of a dot diagram given by $r_i=\max\{j:p_j\geq i\}$, where $p_j$ are the number of dots in column $j$?

Relevant Equations

Some familiarity with Jordan canonical forms is required I think. I'll try to explain the rest below.

We know that a Jordan canonical form is simply the matrix representation of an operator (whose characteristic polynomial splits) with respect to a special basis called a Jordan canonical basis. This basis consists of a disjoint union of cycles/chains of generalized eigenvectors. Take all the $n$ cycles that correspond to a certain eigenvalue $\lambda$ and take their union, which we denote $\gamma=\gamma_1\cup\gamma_2\cup\cdots\cup\gamma_n$. Note that $\gamma_i$ may have different lengths $p_1,p_2,\ldots,p_n$. To be concrete, suppose $n=3$ and $p_1=3, p_2=2$ and $p_3=1$. Then \begin{align*}\gamma_1&=\{(T-\lambda I)^{2}(v_1),(T-\lambda I)(v_1),v_1\};\\ \gamma_2&=\{(T-\lambda I)(v_2),v_2\};\\ \gamma_n&=\{v_3\}.\end{align*} Then the matrix representation of $T$ restricted to $\operatorname{span}(\gamma)$ is a so-called Jordan block. We can visualize a Jordan block with the help of a dot diagram as follows: $$\begin{array}{ccc}\bullet&\bullet&\bullet \\ \bullet&\bullet&\\ \bullet \end{array}$$Here the first dots in the first row are the initial vectors in the cycle; thus the bottom dots in each column are $v_1,v_2,v_3$, from left to right respectively. Dot diagrams are always ordered in decreasing lengths of cycle going from left to right.

Suppose now a dot diagram has $k$ columns and $m$ rows, with $p_j$ dots in column $j$ and $r_i$ dots in row $i$. I need to show by induction on $m=p_1$ that $p_j=\max\{i:r_i\geq j\}$ for $1\leq j\leq k$ and $r_i=\max\{j:p_j\geq i\}$ for $1\leq i\leq m$. The induction step is causing me great trouble (the base case I think I manage by myself). Any help would be very appreciated.

 

[psie]

I get it now I think. We have a dot at $(i,j)$ if and only if $p_j\geq i$ and $r_i\geq j$.

 

[WWGD]

For the sake of math trivia, a matrix with a non-trivial Jordan block is an (counter) example of a matrix that can't be diagonalized; not even over $\mathbb C$

 

[Alien101]

Homework Statement: Why is the number of dots $r_i$ in row $i$ of a dot diagram given by $r_i=\max\{j:p_j\geq i\}$, where $p_j$ are the number of dots in column $j$?
Relevant Equations: Some familiarity with Jordan canonical forms is required I think. I'll try to explain the rest below.

We know that a Jordan canonical form is simply the matrix representation of an operator (whose characteristic polynomial splits) with respect to a special basis called a Jordan canonical basis. This basis consists of a disjoint union of cycles/chains of generalized eigenvectors. Take all the $n$ cycles that correspond to a certain eigenvalue $\lambda$ and take their union, which we denote $\gamma=\gamma_1\cup\gamma_2\cup\cdots\cup\gamma_n$. Note that $\gamma_i$ may have different lengths $p_1,p_2,\ldots,p_n$. To be concrete, suppose $n=3$ and $p_1=3, p_2=2$ and $p_3=1$. Then \begin{align*}\gamma_1&=\{(T-\lambda I)^{2}(v_1),(T-\lambda I)(v_1),v_1\};\\ \gamma_2&=\{(T-\lambda I)(v_2),v_2\};\\ \gamma_n&=\{v_3\}.\end{align*} Then the matrix representation of $T$ restricted to $\operatorname{span}(\gamma)$ is a so-called Jordan block. We can visualize a Jordan block with the help of a dot diagram as follows: $$\begin{array}{ccc}\bullet&\bullet&\bullet \\ \bullet&\bullet&\\ \bullet \end{array}$$Here the first dots in the first row are the initial vectors in the cycle; thus the bottom dots in each column are $v_1,v_2,v_3$, from left to right respectively. Dot diagrams are always ordered in decreasing lengths of cycle going from left to right.

Suppose now a dot diagram has $k$ columns and $m$ rows, with $p_j$ dots in column $j$ and $r_i$ dots in row $i$. I need to show by induction on $m=p_1$ that $p_j=\max\{i:r_i\geq j\}$ for $1\leq j\leq k$ and $r_i=\max\{j:p_j\geq i\}$ for $1\leq i\leq m$. The induction step is causing me great trouble (the base case I think I manage by myself). Any help would be very appreciated.

"The highest-numbered column tall enough to reach row i is max{j : p_j ≥ i}"

If column heights are [4, 3, 3, 1]:

Row 3 gets dots from columns with height ≥ 3
That's columns 1, 2, and 3 (but not 4, since p₄ = 1 < 3)
So r₃ = 3 = max{j : p_j ≥ 3}
The botton line is - r_i counts how many columns are tall enough to reach row i, which is exactly max{j : p_j ≥ i}.