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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

*e*

^{B}=

**C**. If

**C**is a nonsingular real

*n*×

*n*matrix, then there is a real

*n*×

*n*matrix

**B**such that

*e*

^{B}=

**C**².

**S**is a nonsingular

*n*×

*n*matrix such that

**S**

^{-1}

**C**

**S**=

**J**is in Jordan canonical form, and if

*e*

^{K}=

**J**, then

**S**

*e*

^{K}

**S**

^{-1}=

**C**. As a result, \( e^{{\bf S}\,{\bf K}\,{\bf S}^{-1}} = {\bf C} \) and

**B**=

**S**

**K**

**S**

^{-1}is the desired matrix. Thus, it suffices to consider the nonsingular matrix

**C**or

**C**² to be a Jordan block.

For the first statement of the theorem, assume that **C** = λ**I** + **N**, where **N** is nilpotent; that is, **N**^{m} = **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

*e*

^{B}=

**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 **C**² correspond to the Jordan blocks of **C**. The blocks of **C**² corresponding to real eigenvalues of **C** are all of the type *r***I** + **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 **C**² are block diagonal or “block diagonal plus block nilpotent” with 2 × 2 diagonal subblocks of the form

**C**² 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

**C**² is

*r*

**I**+

**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

*r*> 0, and note that a real logarithm is given by

**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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