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Equation 3 · Part 5 · How Post-CMOS, Neuromorphic, Photonic, and Quantum AI Compute Actually Works

=

Ij=∑iVi GijI_j = \sum_i V_i\, G_{ij}
=

What this part means

The expressions on both sides represent the same quantity under the stated assumptions.

Its job in the formula

The equals sign connects the complete expression on the left with the complete expression on the right. Both sides must have compatible units.

The passage around this formula

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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Learn the underlying idea

An equals sign says that the expression on its left and the expression on its right have the same value under the stated definitions and assumptions.

Open the illustrated equality: what the equals sign claims guide →

Sources cited in the surrounding passage

These citations provide research context; check each source for the exact claim it supports.