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I got confused when I saw this proposition from a paper. Here is the problem. If $\mathscr{N}_l$ denotes the set of real symmetric matrices of size $n$ with nullity $l$. Then the set $\mathscr{N}_l$ has dimension $$ \dim{\mathscr{N}_l}= \frac{n(n+1)}{2}-\frac{l(l+1)}{2}. $$ I don't even know what the dimension means since the set $\mathscr{N}_l$ may not be a linear space. Thank you if you can help me figure it out.

little mushroom
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  • what is the paper's definition of nullity? – Exodd Jun 12 '20 at 13:20
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    Your set is a "manifold", which is a kind of bent or deformed version of a linear space. Such sets have dimensions, too, and are studied in "differential geometry". Naively, your question asks how many entries in your matrix may be freely filled in, such that the others are determined by what you've filled in already and by the nullity conditions. – kimchi lover Jun 12 '20 at 14:02
  • Nullity is dimension of the kernel. – little mushroom Jun 12 '20 at 14:27
  • Thank you, Kimchi Lover. – little mushroom Jun 12 '20 at 14:33
  • To the OP. On this website, usage is, if one is satisfied with an answer, then one upvotes it and (or) one gives the green ticket or, if one is not satisfied, then one writes why –  Jun 20 '20 at 10:17

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$\mathscr{N}_l$ is an algebraic set. Its dimension is the maximum of its local dimensions (yes, it's not obvious).

The orthogonal group $O(n)$ acts on $S_n$, the vector space of $n\times n$ real symmetric matrices by $(P,A)\mapsto P^{-1}AP$. Consider $A\in \mathscr{N}_l$; then there is $P\in O(n)$ s.t. $P^{-1}AP=diag(\lambda_1,\cdots,\lambda_{n-l},0_l)$, where $\lambda_i\not= 0$.

i) The number of parameters is $u=dim(O(n))+n-l=n(n-1)/2+n-l.$

ii) The stabilizer of the diagonal is

$T=\{R\in O(n); diag(\lambda_1,\cdots,\lambda_{n-l},0_l)R=Rdiag(\lambda_1,\cdots,\lambda_{n-l},0_l)\}$.

The required dimension is $u-dim(T)$; then we search a matrix $A$ that realizes $\min(dim(T))$. It suffices to choose $A$ with distinct eigenvalues; in this case,

$T_{min}=\{diag(\pm 1,\cdots,\pm 1,V);\;V\in O_{l}\}$.

Thus $dim(T_{min})=(l^2-l)/2$.

Finally $dim(\mathscr{N}_l)=u-(l^2-l)/2$, as required. $\square$