Advanced Transition Matrix Calculator

Transition Matrix Inputs

Matrix size

Stochastic convention

Transition steps, n

Decimal precision

Numerical tolerance

Maximum convergence iterations

Simulation steps

Simulation runs

Optional random seed

State Names

Transition Probabilities

Enter probabilities from zero through one.

Initial State Distribution

Simulation starting state

Number format

Diagram probability threshold

Chart display

Paste matrix data

Import CSV

Automatically normalise before calculating

Show calculation steps

Formula Used

Row convention: v_n = v₀Pⁿ

Column convention: v_n = Pⁿv₀

Stationary distribution: πP = π or Pπ = π

Absorbing chain: N = (I − Q)⁻¹, B = NR

A transition matrix models probabilities between defined states. Each row usually totals one. Valid matrices make long-term Markov analysis clear and reliable.

How to Use

Choose a size and enter every transition probability. Add an initial distribution and step count. Then calculate, inspect warnings, simulate paths, and export results.

Select the matrix size and convention.
Name states and enter all probabilities.
Provide an initial state distribution.
Choose calculation and simulation settings.
Calculate and review every result section.

Example Data

From state	State A	State B	State C
State A	0.70	0.20	0.10
State B	0.30	0.40	0.30
State C	0.20	0.30	0.50

This example represents three connected Markov states. Every row totals exactly one. It supports powers, simulation, classification, and steady-state analysis immediately.

Frequently Asked Questions

What is a transition matrix?

It stores probabilities for movements between several states. Entries describe one-step transitions. Rows or columns must total one for valid models.

What does P raised to n mean?

It represents transition probabilities after multiple consecutive steps. Matrix multiplication combines possible paths. Larger powers reveal long-term behaviour and convergence patterns clearly.

What is a stationary distribution?

It remains unchanged after another transition occurs. It describes long-run state proportions. Regular chains usually converge toward one unique stationary distribution.

What is an absorbing state?

An absorbing state cannot be left after entry. Its self-transition probability equals one. Absorbing analysis estimates destinations and expected waiting times accurately.

Can columns sum to one instead?

Yes, column-stochastic conventions are fully supported here. The vector multiplication direction changes. Select the correct convention before entering matrix values carefully.

Why does normalisation matter?

Probabilities must form valid distributions before multiplication. Normalisation rescales positive totals. It prevents invalid totals from distorting every later result.

What does irreducible mean?

Every state can eventually reach every other state. Such states communicate together. Irreducibility supports stronger long-run conclusions for Markov chains.

How does simulation differ from theory?

Theory calculates exact model probabilities using matrix operations. Simulation samples random paths. More runs usually produce closer agreement with theory.

Can I export the calculation?

Yes, results can be copied or downloaded. CSV preserves tabular values. PDF and printing create convenient reports for later use.