Matrix exponential

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the matrix exponential is a <a href="/facts/Matrix_function/T2e9sFm9">matrix function</a> on <a href="/facts/Square_matrix/5TP9mNNn">square matrices</a> analogous to the ordinary <a href="/facts/Exponential_function/ewlYxvR2">exponential function</a>. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the <a href="/facts/Exponential_map_(Lie_theory)/30I1poVM">exponential map</a> between a matrix <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> and the corresponding <a href="/facts/Lie_group/a9jCQyjF">Lie group</a>.
Let X be an n × n <a href="/facts/Real_number/R02gw5Pb">real</a> or <a href="/facts/Complex_number/2w7mqI7e">complex</a> <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a>. The exponential of X, denoted by eX or exp(X), is the n × n matrix given by the <a href="/facts/Power_series/vYh6sTps">power series</a>

 
 
 
 
 e
 
 X
 
 
 =
 
 ∑
 
 k
 =
 0
 
 
 ∞
 
 
 
 
 1
 
 k
 !
 
 
 
 
 X
 
 k
 
 
 
 
 {\displaystyle e^{X}=\sum _{k=0}^{\infty }{\frac {1}{k!}}X^{k}}

where 
 
 
 
 
 X
 
 0
 
 
 
 
 {\displaystyle X^{0}}
 
 is defined to be the identity matrix 
 
 
 
 I
 
 
 {\displaystyle I}
 
 with the same dimensions as 
 
 
 
 X
 
 
 {\displaystyle X}
 
, and ⁠
 
 
 
 
 X
 
 k
 
 
 =
 X
 
 X
 
 k
 −
 1
 
 
 
 
 {\displaystyle X^{k}=XX^{k-1}}
 
⁠. The series always converges, so the exponential of X is well-defined.
Equivalently,

e
          
            X
          
        
        =
        
          lim
          
            k
            →
            ∞
          
        
        
          
            (
            
              I
              +
              
                
                  X
                  k
                
              
            
            )
          
          
            k
          
        
      
    
    {\displaystyle e^{X}=\lim _{k\rightarrow \infty }\left(I+{\frac {X}{k}}\right)^{k}}

for integer-valued k, where I is the n × n <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a>. 
Equivalently, given by the solution to the differential equation

 
 
 
 
 
 d
 
 d
 t
 
 
 
 
 e
 
 X
 t
 
 
 =
 X
 
 e
 
 X
 t
 
 
 ,
 
 
 e
 
 X
 0
 
 
 =
 I
 
 
 {\displaystyle {\frac {d}{dt}}e^{Xt}=Xe^{Xt},\quad e^{X0}=I}

When X is an n × n <a href="/facts/Diagonal_matrix/tjIAHrE6">diagonal matrix</a> then exp(X) will be an n × n diagonal matrix with each diagonal element equal to the ordinary <a href="/facts/Exponential_function/ewlYxvR2">exponential</a> applied to the corresponding diagonal element of X.

Matrix exponential open-in-new

Matrix exponential