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Published equation contexts

Ij=∑iVi GijI_j = \sum_i V_i\, G_{ij}

Why this formula appears here

The multiplication step follows directly from two textbook circuit laws. Apply a voltage ViV_i simultaneously to each row i, representing the i-th component of an input vector. By Ohm’s law, the cell at row i and column j draws a current ViV_i times its conductance GiG_ij. By Kirchhoff’s current law, every current flowing into a given column sums automatically on that column’s wire, so the total current collected at the bottom of column j is Ij=∑iVi GijI_j = \sum_i V_i\, G_{ij}. which is precisely the j-th component of the matrix-vector product between the input vector and the conductance matrix stored across the array — computed in one step, in parallel across every cell, using nothing but the physics of the…

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ii

Starting index or lower bound: i

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Ij=∑iVi GijI_j = \sum_i V_i\, G_{ij}

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

The multiplication step follows directly from two textbook circuit laws. Apply a voltage ViV_i simultaneously to each row i, representing the i-th component of an input vector. By Ohm’s law, the cell at row i and column j draws a current ViV_i times its conductance GiG_ij. By Kirchhoff’s current law, every current flowing into a given column sums automatically on that column’s wire, so the total current collected at the bottom of column j is Ij=∑iVi GijI_j = \sum_i V_i\, G_{ij}. which is precisely the j-th component of the matrix-vector product between the input vector and the conductance matrix stored across the array — computed in one step, in parallel across every cell, using nothing but the physics of the…

Meanings in this article

  • IjI_j: the total current collected at the bottom of column j.
  • ii: the cell at row.
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