Configure A, t, numerical method, and state response
Paste, upload, transform, and restore matrices
Formula used
The matrix exponential is defined by an infinite power series. It extends the ordinary scalar exponential to square matrices. Every matrix power uses matrix multiplication, not entrywise multiplication.
The default numerical method uses scaling and squaring. It first scales the matrix into a safer norm range. A Padé rational approximation is then evaluated. Repeated squaring restores the original scale.
The calculator also includes an adaptive Taylor method. A specialized closed form handles many two-dimensional systems. Verification compares independent numerical paths whenever possible.
Key identities
$e^{A\cdot0}=I$
$(e^{At})^{-1}=e^{-At}$
$\det(e^{At})=e^{t\operatorname{tr}(A)}$
$\frac{d}{dt}e^{At}=Ae^{At}$
How to use this calculator
Select a dimension from one through twelve. The entry grid updates immediately.
Use decimals, fractions, scientific notation, or complex values.
Use one for e raised to A. Negative and complex times also work.
Automatic mode is recommended. Experts may select Padé, Taylor, or closed form.
This enables the solution x(t) equals e raised to At times x(0).
Inspect residuals, plots, analysis, and code exports after calculation.
Why matrix exponentials matter
Matrix exponentials describe continuous linear evolution. They appear in control systems, electrical circuits, chemical kinetics, population models, mechanics, and stochastic processes. The exponential maps a system generator into a finite-time transition operator.
For the homogeneous system $x'(t)=Ax(t)$, the state is $x(t)=e^{At}x(0)$. The same matrix also forms the continuous-time state-transition matrix. Its eigenvalues often reveal decay, growth, oscillation, or instability.
Direct elementwise exponentiation gives the wrong result. Matrix multiplication couples rows and columns across every power. The difference becomes especially important when off-diagonal terms represent interaction between states.
Supported numerical strategies
Padé scaling and squaring
This approach evaluates a rational approximation after scaling the matrix. It is the preferred general-purpose method in this application. It usually handles moderate matrix norms more reliably than a raw Taylor sum.
Adaptive Taylor expansion
The Taylor method builds successive matrix powers and factorial scaling. It stops when the latest term becomes small relative to the accumulated result. Users can control tolerance and maximum terms.
Analytic two-by-two form
Two-dimensional systems admit a compact trace-determinant formula. The calculator uses complex arithmetic when the discriminant becomes negative. This naturally handles rotations and damped oscillations.
Practical limits and numerical notes
This single-file build targets education and moderate workloads. It accepts matrices through twelve by twelve. Very large, sparse, or highly nonnormal systems require specialized scientific libraries.
The app estimates eigenvalues only for smaller real matrices. It gives exact analytic eigenvalues for two-by-two matrices. General symbolic Jordan forms are not computed because they are fragile numerically.
Forced response uses $A^{-1}(e^{At}-I)Bu$ for constant inputs. That compact formula requires an invertible matrix. Singular systems need augmented matrices or dedicated integral methods.
Frequently asked questions
Is e raised to At calculated entry by entry?
No. Matrix powers use row-by-column multiplication. Entrywise exponentiation represents a different operation.
What does t equal one mean?
Setting t to one calculates the ordinary matrix exponential e raised to A.
Can the matrix contain fractions?
Yes. Enter values such as 1/2, -3/4, or 5/8.
Can the calculator process complex matrices?
Yes. Use i or j notation, such as 2+3i or -i.
Which method should I choose?
Automatic mode is best for most users. Padé is the general numerical default.
Why can two methods differ slightly?
Floating-point rounding and truncation affect every numerical algorithm. Residuals quantify those differences.
What is the state-transition matrix?
For a constant matrix A, the state-transition matrix is e raised to At.
How is stability classified?
Continuous systems are stable when every eigenvalue has a negative real part.
Does e raised to A always have an inverse?
Yes. Its inverse is e raised to negative A, even when A is singular.
Can I export results?
Yes. CSV, JSON, print-to-PDF, LaTeX, Python, and MATLAB formats are included.
Can I evaluate several times?
Yes. Enter a comma-separated time list to build state and norm charts.