I dunno, there are some semi-useful things you can do.
For example, the transform from (Alive, Bob, Carol, Dan) to (Male, Female) is linear -- it's another matrix that you can compose with the individual-ownership one you have here.
Or, call your individual-ownership matrix A, and say that P is the covariance of daily changes to prices of the four stocks listed. Then A P A' is the covariance of daily changes to the peoples' wealths. The framing as linear algebra hasn't been useless.
I kinda get what you're saying though. Like, why would powers of this matrix be useful? It only makes sense if there's some implicit transform between prices and people, or vice versa, that happens to be an identity matrix.
You can make up a story. Say the people can borrow on margin some fraction of their wealth. Then say that they use that borrowing to buy stock, and that that borrowing affects prices. Composing all these transforms, you could get from price to price, and then ask what the dynamics are as the function is iterated.
But, ok, "I'm just going to do an SVD of the matrix and put it in a slide" isn't going to tell anybody much.
Maybe there's a use for a rank-one approximation to this system? Like, "this is pretty close to a situation where there's a single ETF with those stocks in these proportions, and where the people own the following numbers of shares in the ETF"? Maybe if you have millions of people and millions of stocks and wanted to simulate this "stock market" at 100Hz on a TI-83?
For example, the transform from (Alive, Bob, Carol, Dan) to (Male, Female) is linear -- it's another matrix that you can compose with the individual-ownership one you have here.
Or, call your individual-ownership matrix A, and say that P is the covariance of daily changes to prices of the four stocks listed. Then A P A' is the covariance of daily changes to the peoples' wealths. The framing as linear algebra hasn't been useless.
I kinda get what you're saying though. Like, why would powers of this matrix be useful? It only makes sense if there's some implicit transform between prices and people, or vice versa, that happens to be an identity matrix.
You can make up a story. Say the people can borrow on margin some fraction of their wealth. Then say that they use that borrowing to buy stock, and that that borrowing affects prices. Composing all these transforms, you could get from price to price, and then ask what the dynamics are as the function is iterated.
But, ok, "I'm just going to do an SVD of the matrix and put it in a slide" isn't going to tell anybody much.
Maybe there's a use for a rank-one approximation to this system? Like, "this is pretty close to a situation where there's a single ETF with those stocks in these proportions, and where the people own the following numbers of shares in the ETF"? Maybe if you have millions of people and millions of stocks and wanted to simulate this "stock market" at 100Hz on a TI-83?
I dunno. You can make up stories.