Let M be a 3-connected matroid and let $\mathbb{F}$ be a field. Let A be a matrix over $\mathbb{F}$ representing M and let $(G,\mathcal{B})$ be a biased graph representing M. We characterize the relationship between A and $(G,\mathcal{B})$, settling four conjectures of Zaslavsky. We show that for each matrix representation A and each biased graph representation $(G,\mathcal{B})$ of M, A is projectively equivalent to a canonical matrix representation arising from G as a gain graph over ${\mathbb{F}}^{+}$ or ${\mathbb{F}}^{\times }$ realizing $\mathcal{B}$. Further, we show that the projective equivalence classes of matrix representations of M are in one-to-one correspondence with the switching equivalence classes of gain graphs arising from $(G,\mathcal{B})$, except in one degenerate case.

令M为 3 连通拟阵，令$\mathbb{F}$成为一个领域。设A是一个矩阵$\mathbb{F}$代表M并让$(G,\mathcal{乙})$是表示M的有偏图。我们描述了A和$(G,\mathcal{乙})$，解决了扎斯拉夫斯基的四个猜想。我们展示了对于每个矩阵表示A和每个有偏图表示$(G,\mathcal{乙})$的M , A投影等价于由G产生的规范矩阵表示，作为增益图${\mathbb{F}}^{+}$要么${\mathbb{F}}^{\times }$意识到$\mathcal{乙}$. 此外，我们证明了M的矩阵表示的投影等价类与增益图的切换等价类一一对应$(G,\mathcal{乙})$，除了一种退化的情况。