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Matrix Logarithm Calculator

Calculate matrix log and log(A) for 2x2 matrices using Taylor series

Enter 2x2 Matrix A

Matrix Logarithm Formula

log(I + X) = X - X^2/2 + X^3/3 - X^4/4 + ...

Converges when ||X|| < 1

exp(log(A)) = A (when defined)

log(A) = sum_{k=1}^{∞} (-1)^(k+1) (A-I)^k / k

The matrix logarithm is the inverse operation of matrix exponential. People search for it as matrix log, log matrix, log of matrix, log of a matrix, or logarithm of a matrix. For matrices close to identity, this calculator uses Taylor series expansion for computation.

⚠ Logarithm exists only when all eigenvalues have positive real parts. For best accuracy, input matrix should be close to identity.

How to Find the Log of a Matrix

For a matrix near the identity matrix, write A = I + X and use log(I + X) = X - X^2/2 + X^3/3 - X^4/4 + ... . The more terms used, the better the approximation when the series converges.

What is Matrix Logarithm?

The matrix logarithm is a matrix function that is the inverse of the matrix exponential. If A = exp(X), then X = log(A). It's used in various areas of mathematics and engineering.

Taylor Series

log(I+X) = X - X²/2 + X³/3 - X⁴/4 + ... converges for ||X|| < 1.

Eigenvalues

If A has eigenvalues λ, log(A) has eigenvalues log(λ).

Convergence

Series converges when matrix norm is less than 1.

Properties

log(AB) ≠ log(A)+log(B) in general, unlike scalar case.

Teaching Example: A = I + X where X = [[0.5,0.2],[0.3,0.4]]
1. log(I+X) ≈ X - X²/2 + X³/3 - X⁴/4
2. X² = [[0.31, 0.18], [0.27, 0.22]]
3. X³ = [[0.204, 0.126], [0.189, 0.147]]
4. log(I+X) ≈ [[0.408, 0.162], [0.243, 0.312]]

Matrix Log vs Matrix Exponential

Operation	Purpose	Important Note
Matrix logarithm log(A)	Finds a matrix X such that exp(X)=A	Existence depends on eigenvalues and branch choices.
Matrix exponential exp(A)	Builds continuous-time transformations and system solutions	Always defined for square matrices.
Scalar log rules	log(ab)=log(a)+log(b)	This does not generally hold for matrices because multiplication may not commute.

Related Matrix Tools

Matrix Exponential Matrix Power Eigenvalue Calculator Power Series Calculator

Applications

Control Theory Lie Groups Differential Equations Signal Processing Machine Learning

Frequently Asked Questions

What is matrix logarithm?▼

Matrix logarithm is the inverse of matrix exponential: exp(log(A)) = A. Defined for matrices with no eigenvalues <= 0.

How to compute log(A)?▼

Use Taylor series: log(I+X) = X - X^2/2 + X^3/3 - X^4/4 + ... for ||X|| < 1.

When does log(A) exist?▼

When A is invertible and all eigenvalues have positive real parts (no eigenvalues on negative real axis).

Matrix log vs scalar log?▼

Matrix log shares properties: log(AB) != log(A)+log(B) generally. But exp(log(A)) = A when defined.

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