If A is a complex matrix of order n and A is nilpotent, that is, there exists a positive integer s such that A s = 0. Let's say that e A = ∑ k = 0 ∞ A k k ! . Prove that e A is similar to I n + A, where I n is the identity matrix of order n.

Question

If A is a complex matrix of order n and A is nilpotent, that is, there exists a positive integer s such that       A    s    =  0. Let&#039;s say that       e    A    =      ∑          k      =      0              ∞                  A      k              k      !      . Prove that       e    A   is similar to       I    n    +  A, where       I    n   is the identity matrix of order n.

wespee0 · Accepted Answer

It suffices to show that this holds in the case that A is a nilpotent Jordan block (of size n) with eigenvalue 0.Note that it is equivalent to show that       e    A    −  I is similar to A, and that that a matrix   B  ∈            C              n      ×      n       is similar to a nilpotent Jordan block (of size n) if and only if it is nilpotent with       B          n      −      1        ≠  0. So, let   B  =      e    A    −  I. By considering the series expansion   B  =  A      ∑          k      =      0        ∞              A      k              (      k      +      1      )      !      , note that for all exponents m, we have      B    m    =      A    m    (  I  +      a    1        A    1    +  ⋯  +      a          n      −      1            A          n      −      1        )for some coefficients       a    1    ,  …  ,      a          n      −      1        ∈      C  . Conclude therefore that       B          n      −      1        =      A          n      −      1        ≠  0 and that       B          n        =  0. Thus, B must be similar to a Jordan block of size n, which is to say that it is similar to A.

Dylan Nixon · Accepted Answer

This is equivalent to show that A is similar to       e    A    −  I  =  A  +      A          2      !        +  ⋯. We show a more general statement that A is similar to   B  =  A  +      a    2        A    2    +      a    3        A    3    +  ⋯ for arbitrary coefficients       a    2    ,      a    3    ,  ⋯It&#039;s enough to show this when A has a single Jordan block, i.e. there is a vector       v    0   such that   {      v    0    ,      v    1    =  A      v    0    ,      v    2    =  A      v    1    ,  ⋯  } form a basis of             C        n  . Now it&#039;s enough to show that       v    0    ,  B      v    0    ,      B    2        v    0    ,      B    3        v    0   are linearly independent. Note that the coefficients of       v    i   in       B    j        v    0   is       δ          i      j       for   i  ≤  j. The matrix of       v    0    ,  B      v    0    ,  ⋯ under the basis       v    0    ,      v    1    ,  ⋯ is triangular with only 1 on the diagonal, hence they are linearly independent.

Prove that e^A is similar to I_n+A, where I_n is the identity matrix of order n.

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