Preface

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Introduction to Linear Algebra with Mathematica

Glossary

Theorem: If C is a nonsingular n × n matrix, then there is an n × n matrix B (which may be complex) such that eB = C. If C is a nonsingular real n × n matrix, then there is a real n × n matrix B such that eB = C².
If S is a nonsingular n × n matrix such that S-1CS = J is in Jordan canonical form, and if eK = J, then SeKS-1 = C. As a result, \( e^{{\bf S}\,{\bf K}\,{\bf S}^{-1}} = {\bf C} \) and B = SKS-1 is the desired matrix. Thus, it suffices to consider the nonsingular matrix C or C² to be a Jordan block.

For the first statement of the theorem, assume that C = λI + N, where N is nilpotent; that is, Nm = 0 for some integer m with 0 ≤ m < n. Because C is nonsingular, λ ≠ 0 and we can write \( {\bf C} = \lambda \left( {\bf I} + (1/\lambda ){\bf N} \right) . \) A computation using the series representation of the function t &maps; ln(1 + t) at t = 0 shows that, formally (that is, without regard to the convergence of the series), if B = (lnλ)I + M, where

\[ {\bf M} = \sum_{j=1}^{m-1} \frac{(-1)^{j+1}}{j \lambda^j}\, {\bf N}^j . \]
Then eB = C. The series is finite because N is a nilpotent matrix. Thus, the formal series identity is an identity. This proves the first statement of the theorem.

The Jordan blocks of C² correspond to the Jordan blocks of C. The blocks of C² corresponding to real eigenvalues of C are all of the type rI + N, where r > 0 and N is real nilpotent. For a real matrix C all the complex eigenvalues with nonzero imaginary parts occur in complex conjugate pairs; therefore, the corresponding real Jordan blocks of C² are block diagonal or “block diagonal plus block nilpotent” with 2 × 2 diagonal subblocks of the form

\[ \begin{pmatrix} \alpha & -\beta \\ \beta & \phantom{-}\alpha \end{pmatrix} . \]
Some of the corresponding real Jordan blocks for the matrix C² might have real eigenvalues, but these blocks are again all block diagonal or “block diagonal plus block nilpotent” with 2 × 2 subblocks. For the case where a block of C² is rI + N where r > 0 and N is real nilpotent a real “logarithm” is obtained by the matrix formula given above. For block diagonal real Jordan block, write
\[ {\bf R} = r \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \phantom{-}\cos\theta \end{pmatrix} , \]
where r > 0, and note that a real logarithm is given by
\[ \ln r{\bf I} + \begin{pmatrix} 0 & -\theta \\ \theta & \phantom{-}0 \end{pmatrix} . \]
Finally, for a “block diagonal plus block nilpotent” Jordan block, factor the Jordan block as follows:
\[ {\cal R} \left( {\bf I} + {\cal N} \right) \]
where \( {\cal R} \) is block diagonal with R along the diagonal and \( {\cal N} \) has 2 × 2 blocks on its super diagonal all given by R−1. Note that we have already obtained logarithms for each of these factors. Moreover, it is not difficult to check that the two logarithms commute. Thus, a real logarithm of the Jordan block is obtained as the sum of real logarithms of the factors.

Theorem can be proved without reference to the Jordan canonical form.

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