0
votes
1answer
29 views

Rank Nullity and Dimension relation

How would one prove the relations: $rank S◦T = rankT-dim(kerS ∩ ImT)$ and $nullity S◦T = nullityT+dim(kerS ∩ ImT)$ I understand that the use of rank nullity theorem is required but am confused by ...
1
vote
0answers
116 views

positive definite binary matrix

What are the conditions for a binary matrix $A$ (matrix with elements 0 or 1) to be positive definite (not even symmetric), i.e. $\forall x\neq 0, x^TAx>0, A_{ij}\in \{0,1 \}$ Put it another way, ...
0
votes
1answer
63 views

The productrelation matrix is equivalent to the product of the matrix of the relations

I didn't get this thing. Let $R, S$ be two relations from $A \to A$ with $A$ being an arbitrary set. And $M_R$ and $M_S$ their relation matrixes defined as: $(M_R)_{ij}=\left\{ \begin{array}{l l} ...
1
vote
1answer
123 views

Pairwise comparisons of maxima of differences between ordered n-tuples?

I have some ordered tuples $a,b,c$, and I am interested in the following relation: $$ a\succ b \Leftrightarrow \max_i \{a_i-b_i\} >\max_i\{b_i-a_i\} $$ That is, I'm interested in the maximum ...