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Spielman says in Lecture 3: Laplacians and Adjacency Matrices

Fiedler’s Theorem will follow from an analysis of the eigenvalues of tri-diagonal matrices with zero row-sums. These may be viewed as Laplacians of weighted path graphs in which some edges are allowed to have negative weights. Proposition 3.1.3. Let M be a symmetric matrix such that $M \mathbf{1}=\mathbf{0}$

Then, $M =\sum_{i\neq j}−M (i, j)L(i,j).$

Proof. The expression on the right-hand side of (3.1) clearly agrees with M in all off-diagonal entries. Given all the off-diagonal entries, the diagonal entries are determined by the constraint M 1 = 0, which the right-hand side of (3.1) satisfies as well because L(i,j)1 = 0 for all i = j.

My question is What does M equals a summatory sign means? I think L is the Laplacian Matrix of M, but i'm not sure either.

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$M$ does not equal a summation sign. $M$ equals the sum, over all pairs $(i,j)$ with $i\ne j$, of the quantity $-M(i,j)L(i,j)$. – Gerry Myerson Feb 24 '12 at 4:03
I Understand that M is equal the sum. But I don't understand how a Matrix M is equal a summatory, since a $ (M(i,j) \in R$ and $L(i,j) \in R$, then $\sum_{i \neq j} (M(i,j) L(i,j))\in R$ and M is a Matrix – lucianolorenti Feb 24 '12 at 18:42
First, notation: what appears as $L(i,j)$ in the question is $L_{(i,j)}$ in the Spielman notes. Now as to the meaning, it is clear from the context that $L_{(i,j)}$ is a matrix, not a number. Exactly which matrix it is, I don't know - maybe the definition is given in one of the earlier lectures. – Gerry Myerson Feb 24 '12 at 23:00

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