Compute log(I+A) for 2x2 matrices using Taylor series
Enter 2x2 Matrix A
Result
Step-by-Step Derivation
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. For matrices close to identity, we use 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.
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]]
Applications
Control TheoryLie GroupsDifferential EquationsSignal ProcessingMachine 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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