Math Review Hooks

Development

The product is rows against columns,

If $\mu$ is a distribution and $P$ is row-stochastic, then $(\mu P)_j=\sum_i\mu_i P_{ij}\ge 0$ and

so $\mu P$ is again a distribution. A stationary distribution satisfies $\pi P=\pi$ together with $\sum_i\pi_i=1$: it is a left eigenvector for eigenvalue $1$, normalized and nonnegative. For a generator the analogous equation is $\pi G=0$. When the series converges, the transition matrices are $P(t)=\exp(tG)=\sum_{k\ge 0}(tG)^k/k!$. The truncation $I+tG$ can have negative entries, so it need not be a transition matrix.

Worked examples

Take $P=\begin{bmatrix}0.7&0.3\\0.4&0.6\end{bmatrix}$ and $\mu=(1,0)$. Then $\mu P=(0.7,0.3)$. One more step gives

Both rows of $P$ sum to $1$, and both images are distributions. The stationary equation $0.4\pi_2=0.3\pi_1$ with $\pi_1+\pi_2=1$ yields $\pi=(4/7,3/7)$. Direct multiplication returns $\pi P=\pi$.

For the generator $G=\begin{bmatrix}-2&2\\1&-1\end{bmatrix}$, rows sum to $0$. Solving $\pi G=0$ with $\pi_1+\pi_2=1$ gives $\pi=(1/3,2/3)$. The truncation $I+G=\begin{bmatrix}-1&2\\1&0\end{bmatrix}$ has a negative entry, so it is not a transition matrix even though $G$ is a legal generator.

Where this is used

A one-step forecast of a rating or a regime is the product $\mu P$. Continuous-time chains store the generator, not a single $P$, and the matrix exponential builds $P(t)$. Long-run capital or inventory calculations solve $\pi P=\pi$ or $\pi G=0$ before any path is simulated. A step $I+tG$ with $t$ too large can assign negative probability.

Pitfalls

• Coding $\mu P$ when the stored matrix is column-stochastic, or the reverse. The row-sum convention fixes which side the vector sits on.
• Accepting a generator whose rows do not sum to $0$. The diagonal must cancel the off-diagonal rate.
• Treating an arbitrary eigenvector as $\pi$. Nonnegativity and total mass $1$ are part of the definition, not optional scaling.

References

Optional further reading (open references — MatheLinux does not host third-party course files):

• Matrix multiplication (Wikipedia)
• Stochastic matrix (Wikipedia)
• Transition rate matrix (Wikipedia)