Why are resistance matrices always symmetric?

My book says when using nodal analysis to find the voltages and currents in resistive circuits, the resulting matrix (of the system of equations) is always symmetrical. Why?

The exact quote is:

The node equations for networks containing only resistors and independent current sources can always be written in this symmetrical form

Also the diagonal elements are always positive and the off-diagnonal ones are always negative. Why?

A resistor of R ohms between A and B will show up in the matrix as

+1/R -1/R
-1/R +1/R

The rows represent the voltage on A and B; the columns represent the net current into nodes A and B.

The columns must be equal and opposite because the current flowing into one node of the resistor must be equal and opposite the current flowing out the other node. The two numbers in each column must be equal and opposite because current should only flow in the resistor in response to a voltage difference between the two nodes. The value of the the upper-left cell is 1/R because an increase on A's voltage of 1 volt should increase the current flowing into A by 1/R.

• Could you please add in a circuit diagram? It's confusing without a diagram to refer to. – dfg Mar 18 '14 at 17:52
• Assume there are two nodes in a schematic labeled A and B; each has an associated row which says how much current net flows into or out of the node (typically for equation to be solved, such rows must total zero), and an associated column (which represents the unknown voltage on that node). Adding a resistor will add the above values to the intersections of the rows associated with A and B, and the columns associated with A and B. If you need a diagram A --\/\/\-- B. – supercat Mar 18 '14 at 18:18

Symmetry follows from a property called Reciprocity of linear resistor networks.

The opposite signs of the diagonal and off-diagonal non-zero entries in the matrix follows from the discrete laplacian associated with combining a linear consitutive relation (Ohm's law) with a conservation law (kirchoff's law). Laplacians are always symmetric positive definite.