Mathematics: Ohm's Law in Matrix Form

Introduction
So I was reading on Physics Forums about Ohm's Law in vector form and also SPICE analysis using matrices.  So I decided to do some more research and write on the topic.  From my understanding, many SPICE programs use a mixture of this technique to make node voltage analysis simpler in software.  It's actually a very effective technique.

Notation
The following notation is used for complex numbers.  It makes use of matrix algebra and can be used to multiple any complex numbers together.  I'll do my best to describe as much as possible how this applies to circuit analysis.

[R] Real Value (Resistance, Real part of Voltage, etc)

[X] Reactance Value (Inductance, Phase component, etc)

This is the basic form to representing a component.  So if we have a DC voltage source at 10V, it would be represented as so:

So to form Ohm's Law simply construct the equation:

which is equivalent to

So you see, use the dot product to find the product of the complex numbers.

Conclusion

This is a simple example but this can be expanded to be used in very complex circuits including frequency dependent components and some linear models.  Anyways hope this was helpful.  Let me know if you wish for more examples.

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Justin Coulston

justin.coulston@gmail.com