Evaluate matrix validity, row sums, column sums, powers. Track future states with reliable transition checks. Use precise tools for stronger probability matrix analysis today.
This calculator checks non-negativity, row sums, column sums, matrix powers, and state transitions. It also estimates a stationary distribution when the matrix is square and row stochastic.
Use spaces, commas, or semicolons between values. Put each matrix row on a new line.
Example input 0.50 0.30 0.20 0.25 0.50 0.25 0.20 0.40 0.40
This example shows a row stochastic transition matrix with three states.
| From \ To | State 1 | State 2 | State 3 | Row Sum |
|---|---|---|---|---|
| State 1 | 0.50 | 0.30 | 0.20 | 1.00 |
| State 2 | 0.25 | 0.50 | 0.25 | 1.00 |
| State 3 | 0.20 | 0.40 | 0.40 | 1.00 |
All entries are non-negative, and every row sums to 1.
A matrix is row stochastic when every entry is non-negative and each row sum equals 1.
Condition: pij ≥ 0 and Σj pij = 1
A matrix is column stochastic when every entry is non-negative and each column sum equals 1.
Condition: pij ≥ 0 and Σi pij = 1
A matrix is doubly stochastic when it satisfies both row stochastic and column stochastic rules.
If P is a square transition matrix, then Pn gives the n-step transition behavior.
For a row-based state vector s0, the future state after n steps is:
sn = s0Pn
A stationary distribution π satisfies πP = π. This page estimates it iteratively when the matrix is square and row stochastic.
A stochastic matrix contains non-negative values and represents probability transitions. Depending on the definition used, each row or each column must sum to 1. Square stochastic matrices often model Markov chains and state movement over time.
A row stochastic matrix has rows that sum to 1. A column stochastic matrix has columns that sum to 1. Both require non-negative entries. Some matrices satisfy both rules and become doubly stochastic.
Floating-point calculations can produce tiny rounding differences. Tolerance lets the calculator treat values like 0.999999999 and 1.000000001 as effectively equal when checking whether sums match the required probability total.
Repeated multiplication of the same matrix requires the number of rows and columns to match. That is why powers such as P² or P³ only work with square matrices. Transition matrices for Markov chains are usually square.
It estimates how an initial distribution changes after several transitions. If your starting vector describes current state probabilities, multiplying by the transition matrix power shows the distribution after the chosen number of steps.
A stationary distribution remains unchanged after multiplication by the transition matrix. It can describe long-run behavior in a Markov process. Some matrices converge to one stable distribution, while others do not converge cleanly.
They help you compare the original matrix against probability-ready versions. Row normalization forces each row to sum to 1. Column normalization forces each column to sum to 1. This is useful when your source matrix is close to stochastic.
You can enter decimals, but negative values prevent a matrix from being stochastic because probabilities cannot be negative. The calculator still evaluates the matrix mathematically and clearly reports whether it passes stochastic conditions.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.