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).

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.