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ⁿ

Matrix Powers

Aⁿ = A·A·…·A (n times). A⁰ = I. A⁻ⁿ = (A⁻¹)ⁿ when A is invertible. For diagonalizable A = PDP⁻¹: Aⁿ = PDⁿP⁻¹.

Concept Fundamentals
I (identity)
A⁰
(A⁻¹)ⁿ
A⁻ⁿ
Dⁿ = diag(λᵢⁿ)
Diagonal
Aᵏ=0 for some k
Nilpotent

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Diagonal: Dⁿ = diag(λ₁ⁿ,…,λₙⁿ). Nilpotent: Aᵏ = 0 for some k. (Aⁿ)ᵐ = Aⁿᵐ.

Key quantities
I (identity)
A⁰
Key relation
(A⁻¹)ⁿ
A⁻ⁿ
Key relation
Dⁿ = diag(λᵢⁿ)
Diagonal
Key relation
Aᵏ=0 for some k
Nilpotent
Key relation

Ready to run the numbers?

Why: Matrix powers model discrete dynamics: Markov chains, recurrence relations, graph walks.

How: Repeated multiplication for positive n. A⁰ = I. A⁻ⁿ = (A⁻¹)ⁿ. Diagonalization: Aⁿ = PDⁿP⁻¹.

Diagonal: Dⁿ = diag(λ₁ⁿ,…,λₙⁿ).Nilpotent: Aᵏ = 0 for some k.
Sources:Gilbert Strang, Linear AlgebraWolfram MathWorld

Run the calculator when you are ready.

Compute AⁿPositive, zero, or negative integer power

Examples

A2 =
[0, 0]
[0, 0]
Dimension: 2×2
det(A): 0.0000
Screenshot card — share your result

Result Matrix Entries

Entry Sign Distribution

Steps

[SETUP]Matrix A: [0, 0] [0, 0]
[SETUP]Power n: 2
[STEP]A^2: [0, 0] [0, 0]
[RESULT]Result A^n: [0, 0] [0, 0]

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

ⁿ

AⁿAᵐ = Aⁿ⁺ᵐ

📐

(Aⁿ)ᵐ = Aⁿᵐ

Key Takeaways

  • • A^n = A × A × ... × A (n times). A^0 = I, A^1 = A.
  • • A^(-n) = (A^(-1))^n requires A invertible (det(A) ≠ 0).
  • • Diagonal matrices: diag(a,b)^n = diag(a^n, b^n).
  • • Identity: I^n = I. Nilpotent: some A^k = 0.
  • • Rotation matrix (90°): powers cycle every 4.

Did You Know?

📐Matrix powers model repeated linear transformations: A^n x applies the transformation n times.Source: Geometry
🔢Fibonacci: F_n computed via [[1,1],[1,0]]^n in O(log n) matrix multiplications.Source: Recurrence
⚛️For diagonalizable A = PDP⁻¹, A^n = PD^nP⁻¹ — eigenvalues raised to n.Source: Eigenvalues
📊Markov chains: P^n gives n-step transition probabilities.Source: Probability
🔬Nilpotent matrices satisfy A^k = 0 for some k; index = smallest such k.Source: Algebra
📜Binary exponentiation: A^n in O(log n) multiplications via squaring.Source: Algorithms

How It Works

1. A^0 = I

By definition, any square matrix to power 0 is the identity matrix.

2. A^1 = A

First power returns the matrix unchanged.

3. A^n for n > 1

Multiply A by itself n times: A × A × ... × A.

4. A^(-n)

Compute A⁻¹ (adjugate/det), then raise to power |n|.

Expert Tips

Diagonal First

For diagonal matrices, A^n = diag(a₁ⁿ, a₂ⁿ, ...). No multiplication needed.

Check det Before A⁻¹

Negative powers need det(A) ≠ 0. Check determinant first.

Nilpotent Shortcut

If A^k = 0, then A^n = 0 for all n ≥ k. Example: [[0,1],[0,0]]² = 0.

Diagonalization

A^n = PD^nP⁻¹ for diagonalizable A — eigenvalues to power n on diagonal.

Comparison Table

FeatureThis CalculatorNumPyManual
Positive & negative powers✅✅⚠️
Step-by-step for small n✅❌✅
Inverse check (det)✅✅✅
Up to 5×5✅✅⚠️
Charts✅❌❌

FAQ

What is A^0?

A^0 = I (identity matrix) for any square matrix A, by convention.

Can I use negative powers?

Yes, if A is invertible (det(A) ≠ 0). A^(-n) = (A^(-1))^n.

What if det(A) = 0?

The matrix is singular. Negative powers are undefined. A^0 and positive powers still work.

How is A^n computed?

Repeated multiplication: A × A × ... × A. For large n, binary exponentiation is faster.

What is a nilpotent matrix?

A matrix with A^k = 0 for some k. Example: [[0,1],[0,0]] has A² = 0.

Why does rotation^4 = I?

90° rotation matrix R satisfies R⁴ = I (four 90° rotations = 360°).

Diagonal matrix powers?

diag(a,b,c)^n = diag(a^n, b^n, c^n). Very efficient.

Identity matrix powers?

I^n = I for any n (positive, negative, or zero).

Stats

A^0
I
Naive complexity
O(n³·k)
Binary exp
O(log k)
A^m · A^n
A^(m+n)

Sources

  • • Strang, G. Linear Algebra and Its Applications
  • • Horn & Johnson, Matrix Analysis
  • • Khan Academy — Linear Algebra
  • • MIT 18.06 — Linear Algebra
  • • Wolfram MathWorld — Matrix Power
  • • 3Blue1Brown — Essence of Linear Algebra
Disclaimer: This calculator is for educational purposes. Matrix powers use JavaScript floating-point arithmetic. For production or large matrices, use numerical libraries (NumPy, MATLAB).
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