Find Steady State Vector Calculator

Calculate Steady State Vector

Enter the transition probabilities for your 2×2 Markov chain matrix P:

What is a Steady State Vector?

A steady state vector represents the long-term probability distribution of being in each state of a Markov chain, regardless of the initial state. For a regular Markov chain (one where it's possible to get from any state to any other state, possibly through intermediate states, and it's not periodic), as time goes on, the probability of being in any particular state settles down and approaches a fixed value. These fixed values form the components of the steady state vector.

In simpler terms, if you have a system that transitions between different states based on certain probabilities (defined by the transition matrix), the steady state vector tells you the proportion of time the system will spend in each state in the long run.

This steady state vector calculator is designed for 2×2 transition matrices, making it easy to find these long-term probabilities for two-state systems.

Who Should Use It?

This steady state vector calculator is useful for:

• Students learning about Markov chains and linear algebra.
• Researchers and analysts modeling systems that change states over time (e.g., market share, population dynamics, machine states).
• Engineers and scientists working with probabilistic models.

Common Misconceptions

A common misconception is that the system reaches the steady state after a fixed number of steps. In reality, the state probabilities *approach* the steady state values as the number of steps increases, getting closer and closer but theoretically only reaching it at infinity. However, for practical purposes, the system gets very close to the steady state after a reasonable number of transitions if the chain is regular.

Steady State Vector Formula and Mathematical Explanation

For a Markov chain with a transition probability matrix P, a steady state vector v is a probability vector (its components are non-negative and sum to 1) such that:

vP = v

This means that once the system reaches the steady state distribution v, the probability distribution for the next step remains v.

For a 2×2 transition matrix:

P =

where p11 + p12 = 1 and p21 + p22 = 1.

Let the steady state vector be v = [v1, v2], where v1 + v2 = 1.

The equation vP = v translates to:

[v1, v2] * = [v1, v2]

This gives us a system of equations:

1. v1 * p11 + v2 * p21 = v1
2. v1 * p12 + v2 * p22 = v2
3. v1 + v2 = 1

From equation 1: v2 * p21 = v1 * (1 – p11) = v1 * p12.

So, v2/v1 = p12/p21 (if p21 is not 0).

Using v2 = 1 – v1, we get (1 – v1)/v1 = p12/p21, which simplifies to v1 = p21 / (p12 + p21), provided p12 + p21 is not zero. If p12+p21=0, then p12=0 (so p11=1) and p21=0 (so p22=1), and the matrix is the identity, leading to multiple steady states.

For a unique steady state (when p12+p21 ≠ 0):

v1 = p21 / (p12 + p21) = p21 / ((1 – p11) + p21)

v2 = p12 / (p12 + p21) = (1 – p11) / ((1 – p11) + p21)

The steady state vector calculator uses these formulas.

Variables Table

Variable	Meaning	Unit	Typical Range
P	Transition Probability Matrix	Matrix	2×2 with non-negative entries, rows sum to 1
p11	Probability of transitioning from State 1 to State 1	Probability	0 to 1
p12	Probability of transitioning from State 1 to State 2	Probability	0 to 1 (1-p11)
p21	Probability of transitioning from State 2 to State 1	Probability	0 to 1
p22	Probability of transitioning from State 2 to State 2	Probability	0 to 1 (1-p21)
v	Steady State Vector	Vector	Components are probabilities (0 to 1), sum to 1
v1	Steady state probability of being in State 1	Probability	0 to 1
v2	Steady state probability of being in State 2	Probability	0 to 1

Practical Examples (Real-World Use Cases)

Example 1: Market Share

Two brands, A and B, compete in a market. Each month, some customers switch brands. Suppose:

• Brand A retains 90% of its customers (p11=0.9), and 10% switch to B (p12=0.1).
• Brand B retains 80% of its customers (p22=0.8), and 20% switch to A (p21=0.2).

So, p11 = 0.9, p21 = 0.2.

Using the steady state vector calculator with p11=0.9 and p21=0.2, we find p12=0.1, p22=0.8. The denominator (1-0.9)+0.2 = 0.1+0.2=0.3.

v1 = 0.2 / 0.3 = 2/3 ≈ 0.667

v2 = 0.1 / 0.3 = 1/3 ≈ 0.333

In the long run, Brand A will have approximately 66.7% market share, and Brand B will have 33.3%.

Example 2: Machine State

A machine can be in one of two states: "Working" (State 1) or "Broken" (State 2). Each hour:

• If it's Working, there's a 5% chance it breaks down (p12=0.05), so it stays Working with 95% probability (p11=0.95).
• If it's Broken, it gets repaired within the hour with 60% probability (p21=0.6), so it remains Broken with 40% probability (p22=0.4).

p11 = 0.95, p21 = 0.6.

Using the steady state vector calculator: p12=0.05, p22=0.4. Denominator (1-0.95)+0.6 = 0.05+0.6=0.65

v1 = 0.6 / 0.65 ≈ 0.923

v2 = 0.05 / 0.65 ≈ 0.077

In the long run, the machine will be Working about 92.3% of the time and Broken about 7.7% of the time.

How to Use This Steady State Vector Calculator

1. Enter P(1->1) (p11): Input the probability of transitioning from State 1 to State 1 in the first input field. This value must be between 0 and 1. The calculator will automatically determine P(1->2) as 1 – p11.
2. Enter P(2->1) (p21): Input the probability of transitioning from State 2 to State 1 in the second input field. This also must be between 0 and 1. P(2->2) will be 1 – p21.
3. Click Calculate: Or the results will update automatically if you change the inputs.
4. View Results: The calculator will display:
   • The primary result: The steady state vector [v1, v2].
   • The full transition matrix P.
   • Intermediate values used in the calculation.
   • A bar chart visualizing v1 and v2.
5. Reset: Click "Reset" to return to default values.
6. Copy: Click "Copy Results" to copy the main results and matrix to your clipboard.

Reading the Results

The steady state vector [v1, v2] tells you the long-term probabilities of being in State 1 (v1) and State 2 (v2), respectively. For example, if v1=0.75, it means in the long run, the system is expected to be in State 1 about 75% of the time. The steady state vector calculator provides these values clearly.

Key Factors That Affect Steady State Vector Results

The components of the steady state vector are entirely determined by the probabilities in the transition matrix P.

1. p11 (and p12): The tendency to stay in or leave State 1. A higher p11 means State 1 is "stickier," which can increase v1 if p21 is not too small.
2. p21 (and p22): The tendency to move to State 1 from State 2 or stay in State 2. A higher p21 means more flow from State 2 to State 1, generally increasing v1.
3. Relative Magnitudes: The ratio p12/p21 directly influences the ratio v2/v1. If transitions out of State 1 (p12) are much more likely than transitions into State 1 from State 2 (p21), then v2 will be larger relative to v1.
4. Irreducibility and Aperiodicity: For a unique steady state vector to exist and be approached from any starting state, the Markov chain generally needs to be irreducible (possible to get from any state to any other) and aperiodic (not trapped in cycles of states). Our 2×2 calculator implicitly assumes conditions leading to a unique steady state, except when P is the identity matrix.
5. Sum of Rows = 1: The fact that each row of P sums to 1 is fundamental. It ensures that the total probability is conserved.
6. Denominator (1-p11) + p21: If this sum is zero (i.e., p11=1 and p21=0), the matrix is reducible (the identity matrix), and there isn't a unique steady state approached from all start states. Our steady state vector calculator handles this.

Frequently Asked Questions (FAQ)

1. What is a Markov chain?

A Markov chain is a mathematical system that transitions from one state to another according to certain probabilistic rules. The key property is that the probability of transitioning to the next state depends only on the current state and not on the sequence of events that preceded it.

2. What does "steady state" mean?

Steady state refers to a situation where the probabilities of being in each state of the system no longer change over time. The system reaches an equilibrium distribution.

3. Does every Markov chain have a unique steady state vector?

Not necessarily. A finite Markov chain has a unique steady state vector if it is irreducible (you can get from any state to any other) and aperiodic (not stuck in cycles). If it's reducible or periodic, it might have multiple steady state vectors or none that are approached from all starting points. Our 2×2 steady state vector calculator is best for irreducible, aperiodic cases but will note if P=I.

4. How quickly is the steady state reached?

The system theoretically approaches the steady state as time goes to infinity. The speed of convergence depends on the eigenvalues of the transition matrix, particularly the second largest eigenvalue (in magnitude) after 1.

5. Can I use this calculator for matrices larger than 2×2?

No, this specific steady state vector calculator is designed only for 2×2 transition matrices. Finding the steady state for larger matrices involves solving larger systems of linear equations, often using methods like finding eigenvectors or matrix inversion techniques.

6. What if p11 or p21 are 0 or 1?

The calculator accepts values between 0 and 1 inclusive. If p11=1, then p12=0 (State 1 is absorbing if p21 is also 0). If p11=1 and p21=0, the matrix is [[1,0],[0,1]], and there are multiple steady states. The calculator will indicate this.

7. What if the denominator (1-p11)+p21 is zero?

This happens when p11=1 and p21=0, meaning p12=0 and p22=1. The matrix is the identity matrix, P=I. In this case, any vector [a, 1-a] is a steady state vector. The system doesn't converge to a unique distribution from all start states. The calculator flags this.

8. How is the steady state vector related to eigenvalues?

The steady state vector is the left eigenvector of the transition matrix P corresponding to the eigenvalue 1, normalized so its components sum to 1.