MatrixExp

The matrix exponential $e^A = \sum_{k \ge 0} A^k / k!$, for a square matrix A.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageMatrixExp!↗

MatrixExp(A)the matrix exponential of the square matrix A.

SymPyMatrix(A).exp().tolist()

MatrixExp(A, v)$e^A v$: A's matrix exponential applied to the vector v.

SymPylist(Matrix(A).exp() * Matrix(v))

Typed signatures

Type	Package	Applies
(matrix, list<number>?) -> list<number> | matrix	analytic	always

Domain: Linear algebra

Details

• NOT Exp of a matrix, which broadcasts element-wise instead -- see that entry's divergence note.
• Exact where the structure gives one: a diagonal A reduces to elementwise Exp on the diagonal; a 2×2 A reduces to a closed form in Cosh, Sinh and Sqrt of its trace and determinant (which also covers every 2×2 nilpotent and repeated-eigenvalue case, and -- written to recognize a provably negative discriminant -- every 2×2 rotation generator, in Cos and Sin instead); a nilpotent A of any size, with exact entries, reduces to its truncated Taylor series.
• Otherwise numeric only, produced under N(): scaling-and-squaring, in double precision.
• Rejects a non-square argument, same as MatrixRank's siblings Inverse, MatrixPower, and Determinant.
• MatrixExp(A, v) computes $e^A$ first (via whichever path above applies) and multiplies by v; it is not a matrix-free method, just a shorter call.

Examples

3 more kept as data, checked against the oracles

Basic #

The 2×2 closed form -- exact, not decimalized

live ↗

In$\mathrm{MatrixExp}(\bigl\lbrack\bigl\lbrack0, 1\bigr\rbrack, \bigl\lbrack1, 0\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\cosh(1), \sinh(1)\bigr\rbrack, \bigl\lbrack\sinh(1), \cosh(1)\bigr\rbrack\bigr\rbrack$

Known value · mpmath 1.3.0 expm

The exponential of the zero matrix is the identity

live ↗

In$\mathrm{MatrixExp}(\bigl\lbrack\bigl\lbrack0, 0\bigr\rbrack, \bigl\lbrack0, 0\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack1, 0\bigr\rbrack, \bigl\lbrack0, 1\bigr\rbrack\bigr\rbrack$

A diagonal matrix exponentiates entry by entry

live ↗

In$\mathrm{MatrixExp}(\bigl\lbrack\bigl\lbrack1, 0\bigr\rbrack, \bigl\lbrack0, 2\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack e, 0\bigr\rbrack, \bigl\lbrack0, e^2\bigr\rbrack\bigr\rbrack$

Scope #

A nilpotent matrix: $I + A$, since $A^2 = 0$

live ↗

In$\mathrm{MatrixExp}(\bigl\lbrack\bigl\lbrack0, 1\bigr\rbrack, \bigl\lbrack0, 0\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack1, 1\bigr\rbrack, \bigl\lbrack0, 1\bigr\rbrack\bigr\rbrack$

Known value · mpmath 1.3.0 expm

A diagonal matrix: exp of each diagonal entry

live ↗

In$\mathrm{MatrixExp}(\bigl\lbrack\bigl\lbrack1, 0, 0\bigr\rbrack, \bigl\lbrack0, 2, 0\bigr\rbrack, \bigl\lbrack0, 0, 3\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack e, 0, 0\bigr\rbrack, \bigl\lbrack0, e^2, 0\bigr\rbrack, \bigl\lbrack0, 0, e^3\bigr\rbrack\bigr\rbrack$

Known value · mpmath 1.3.0 expm

A 3×3 matrix with no diagonal, nilpotent, or 2×2 structure to exploit: numeric only, via scaling-and-squaring

live ↗

In$N(\mathrm{MatrixExp}(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack4, 5, 6\bigr\rbrack, \bigl\lbrack7, 8, 10\bigr\rbrack\bigr\rbrack))$Out$\bigl\lbrack\bigl\lbrack1\,892\,793.801\,966\,466\,5, 2\,319\,804.348\,433\,83, 2\,921\,688.794\,284\,903\,4\bigr\rbrack, \bigl\lbrack4\,267\,596.877\,957\,099, 5\,230\,361.132\,647\,167, 6\,587\,401.226\,390\,927\bigr\rbrack, \bigl\lbrack7\,065\,281.732\,605\,146, 8\,659\,197.695\,551\,049, 10\,905\,868.064\,094\,879\bigr\rbrack\bigr\rbrack$

Known value · mpmath 1.3.0 expm

A Jordan block: $e^{I + N} = e\,(I + N)$

live ↗

In$\mathrm{MatrixExp}(\bigl\lbrack\bigl\lbrack1, 1\bigr\rbrack, \bigl\lbrack0, 1\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack e, e\bigr\rbrack, \bigl\lbrack0, e\bigr\rbrack\bigr\rbrack$

Applied to a vector, $e^A v$: computed as $e^A$ (via whichever closed form applies) times $v$, not a matrix-free method

live ↗

In$\mathrm{MatrixExp}(\bigl\lbrack\bigl\lbrack0, 1\bigr\rbrack, \bigl\lbrack0, 0\bigr\rbrack\bigr\rbrack, \bigl\lbrack1, 1\bigr\rbrack)$Out$\bigl\lbrack2, 1\bigr\rbrack$

A $3\times3$ nilpotent generator: the exponential series truncates exactly, past the $2\times2$ case.

live ↗

In$\mathrm{MatrixExp}(\bigl\lbrack\bigl\lbrack0, 1, 0\bigr\rbrack, \bigl\lbrack0, 0, 1\bigr\rbrack, \bigl\lbrack0, 0, 0\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack1, 1, \frac{1}{2}\bigr\rbrack, \bigl\lbrack0, 1, 1\bigr\rbrack, \bigl\lbrack0, 0, 1\bigr\rbrack\bigr\rbrack$

Known value · mpmath 1.3.0 expm

A Jordan block (repeated eigenvalue 2): $e^{2}(I+N)$, not a diagonal exponential.

live ↗

In$\mathrm{MatrixExp}(\bigl\lbrack\bigl\lbrack2, 1\bigr\rbrack, \bigl\lbrack0, 2\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack e^2, e^2\bigr\rbrack, \bigl\lbrack0, e^2\bigr\rbrack\bigr\rbrack$

Known value · mpmath 1.3.0 expm

live ↗

In$\mathrm{MatrixExp}(\bigl\lbrack\bigl\lbrack a, b\bigr\rbrack, \bigl\lbrack0, c\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack(\frac{(a-c)\sinh(((a+c)^2-4ac)^{1/2}/2)}{\sqrt{(a+c)^2-4ac}}+\cosh(\frac{1}{2}((a+c)^2-4ac)^{1/2}))e^{(a+c)/2}, \frac{2b\sinh(((a+c)^2-4ac)^{1/2}/2)e^{(a+c)/2}}{\sqrt{(a+c)^2-4ac}}\bigr\rbrack, \bigl\lbrack0, (\frac{(c-a)\sinh(((a+c)^2-4ac)^{1/2}/2)}{\sqrt{(a+c)^2-4ac}}+\cosh(\frac{1}{2}((a+c)^2-4ac)^{1/2}))e^{(a+c)/2}\bigr\rbrack\bigr\rbrack$

live ↗

In$\mathrm{MatrixExp}(\mathrm{DiagonalMatrix}(\bigl\lbrack a, b, c\bigr\rbrack))$Out$\bigl\lbrack\bigl\lbrack e^{a}, 0, 0\bigr\rbrack, \bigl\lbrack0, e^{b}, 0\bigr\rbrack, \bigl\lbrack0, 0, e^{c}\bigr\rbrack\bigr\rbrack$

live ↗

In$\mathrm{MatrixExp}(\mathrm{IdentityMatrix}(3))$Out$\bigl\lbrack\bigl\lbrack e, 0, 0\bigr\rbrack, \bigl\lbrack0, e, 0\bigr\rbrack, \bigl\lbrack0, 0, e\bigr\rbrack\bigr\rbrack$

Applications #

A skew-symmetric generator gives a rotation matrix: the 2×2 closed form recognizes its purely imaginary eigenvalues and reduces straight to $\cos 1$ and $\sin 1$, not $\cosh i$ and $\sinh i$

live ↗

In$\mathrm{MatrixExp}(\bigl\lbrack\bigl\lbrack0, 1\bigr\rbrack, \bigl\lbrack-1, 0\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\cos(1), \sin(1)\bigr\rbrack, \bigl\lbrack-\sin(1), \cos(1)\bigr\rbrack\bigr\rbrack$

Properties #

live ↗

In$\mathrm{MatrixExp}(a)=\mathrm{MatrixFunction}(\mathrm{Exp}, a)$Out$\mathrm{True}$

Possible issues #

Rejects a non-square argument

live ↗

A generic matrix with no closed form stays symbolic without N() — use N(MatrixExp(...)) for a numeric answer.

live ↗

In$\mathrm{MatrixExp}(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack4, 5, 6\bigr\rbrack, \bigl\lbrack7, 8, 10\bigr\rbrack\bigr\rbrack)$Out$\mathrm{MatrixExp}(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack, \bigl\lbrack4, 5, 6\bigr\rbrack, \bigl\lbrack7, 8, 10\bigr\rbrack\bigr\rbrack)$

Neat examples #

Euler's identity for matrices: rotation by $\pi$ is $-I$

live ↗

In$\mathrm{MatrixExp}(\bigl\lbrack\bigl\lbrack0, -π\bigr\rbrack, \bigl\lbrack π, 0\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack-1, 0\bigr\rbrack, \bigl\lbrack0, -1\bigr\rbrack\bigr\rbrack$

About

Examples	21
Checked against	Wolfram Language 18 agree3 differSageMath 13 agree4 errorSymPy 12 agree7 error
Crosswalk	Wolfram LanguageSymPy ×2

Implementation

mappedsagepackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

Applied to a vector, e^A v, flattened as ours is.

See also: Exp, MatrixPower, Inverse, Determinant