Matrix Logarithm Calculator

The matrix logarithm is the inverse of the matrix exponential: it is a matrix L such that exp(L) = A. This calculator targets a common practical case: a real 2×2 matrix A whose eigenvalues are real, positive, and distinct. Under those conditions, the (real) logarithm computed via eigen-decomposition is well-defined and typically matches the principal matrix logarithm (using the natural logarithm).

Introduction: When this calculator works (and when it won’t)

• Input size: 2×2 real matrices only.
• Required for this method: two distinct eigenvalues that are real and > 0.
• If an eigenvalue is negative or complex: a real-valued log(A) may not exist, and the (complex) logarithm is not unique without branch choices.
• If the eigenvalue is repeated: the logarithm may still exist, but the simple “diagonalize and apply log” approach can fail (matrix may be non-diagonalizable, or eigenvectors become ill-conditioned).

Definition and key identity

If A is invertible, a matrix logarithm is any matrix L satisfying:

The matrix logarithm is not unique in general. However, when A has no eigenvalues on the non-positive real axis and we choose the principal branch of the scalar logarithm, there is a unique principal logarithm. For this calculator, we narrow further to distinct positive real eigenvalues so the computation is straightforward and stays real.

Formulas used for 2×2 matrices

Let

A = [[a, b], [c, d]].

The characteristic polynomial is:

λ² − t λ + Δ = 0, where

• t = tr(A) = a + d (the trace)
• Δ = det(A) = ad − bc (the determinant)

The eigenvalues are

λ₁,₂ = (t ± √(t² − 4Δ)) / 2.

Diagonalization approach (what the calculator is doing)

If A is diagonalizable with eigen-decomposition:

A = V Λ V⁻¹, where Λ = diag(λ₁, λ₂),

then

log(A) = V log(Λ) V⁻¹, with log(Λ) = diag(log(λ₁), log(λ₂)).

This is valid (in real arithmetic) when λ₁ and λ₂ are positive real numbers and V is invertible (i.e., eigenvectors are linearly independent, which is guaranteed when eigenvalues are distinct).

How to interpret the result

• Units/base: This calculator uses the natural logarithm (base e). If you need base-10, convert by dividing by ln(10): log10(A) = log(A)/ln(10) (when defined).
• Meaning: If L = log(A), then A = exp(L). In continuous-time linear systems, L can be interpreted as a generator whose exponential produces A.
• Sanity check: For supported inputs, exp(log(A)) should reproduce A up to rounding error. Small discrepancies are normal due to floating-point arithmetic and conditioning of eigenvectors.

Worked example (successful case)

Take

A = [[4, 1], [1, 3]].

Compute trace and determinant:

• t = 4 + 3 = 7
• Δ = 4·3 − 1·1 = 11

Discriminant: t² − 4Δ = 49 − 44 = 5, so √5 ≈ 2.2361.

Eigenvalues:

• λ₁ = (7 + √5)/2 ≈ 4.6180
• λ₂ = (7 − √5)/2 ≈ 2.3820

Both are positive and distinct, so the method applies. The calculator forms eigenvectors (columns of V), computes log(λ₁), log(λ₂), and reconstructs log(A) = V log(Λ) V⁻¹. The resulting log(A) will be a real 2×2 matrix; diagonals are around ~1.55 for this example (exact entries depend on reconstruction and rounding).

Worked example (failure case and why)

Consider A = [[-1, 0], [0, 2]]. The eigenvalues are −1 and 2. Because one eigenvalue is negative, the real natural logarithm is not defined for that eigenvalue, and a real-valued log(A) does not exist in the usual sense. A complex logarithm exists but is multi-valued due to branch choices (log(−1) = iπ + 2kπi). This calculator will warn and stop rather than returning a misleading result.

Comparison table: common 2×2 input types

Limitations and assumptions

• 2×2 only: The UI and logic are built for four entries (a11, a12, a21, a22).
• Distinct positive real eigenvalues: Required to keep the computation simple and real-valued.
• Non-uniqueness in general: Even when log(A) exists, there can be infinitely many logarithms; this tool targets the principal-like real result under the stated constraints.
• Numerical stability: If eigenvalues are very close or eigenvectors are ill-conditioned, results can be sensitive to rounding.
• Rounding/formatting: Display rounding (e.g., to 6 decimals) can hide tiny differences; the underlying computed values may be slightly different.

If you routinely need matrix logarithms outside these constraints (repeated eigenvalues, complex spectrum, larger matrices), use a full-featured numerical linear algebra package that implements a robust matrix logarithm algorithm (e.g., Schur decomposition-based methods) and supports complex arithmetic and branch control.

How to use this calculator

1. Enter a11 using the unit or time period shown by the field.
2. Enter a12 using the unit or time period shown by the field.
3. Enter a21 using the unit or time period shown by the field.
4. Run the calculation and compare the output with a second scenario before acting on it.