Before revision:  First, we show that a polynomial matrix P(x) with chordal sparsity is positive semidefinite for all x\in R^n if and only if there exists a sum-of-squares (SOS) polynomial \sigma(x) such that \sigma (x)P (x) can be decomposed into a sum of sparse SOS matrices , each of which is zero outside a small principal submatrix .
After revision:  First, we establish that a polynomial matrix P(x) with chordal sparsity is positive semidefinite for all x\in R^n if and only if there exists a sum-of-squares (SOS) polynomial \sigma(x) such that \sigma (x)P (x) can be decomposed into a sum of sparse SOS matrices , each of which is zero outside a small principal submatrix .