### Matrix calculus: Conclusion of Matrix calculus

### End of Matrix Calculus

In this chapter we have become acquainted with properties of predominantly square matrices #A# that help to describe the corresponding linear mapping #L#. Because the rank, determinant, characteristic polynomial, and minimal polynomial of #A# are equal to those of any conjugate of #A#, these quantities and polynomials do not depend on the matrix used to describe #L# (and thus not on the chosen basis for the underlying vector space).

These quantities and polynomials will help us to find a special matrix, which exhibits the main features of #L# without need for further computation. For most linear maps, this is the so-called diagonal form, but for the exceptions there is the Jordan normal form; both are treated in the chapter *Invariant subspaces of linear maps*.

Determinants were popular toys among the mathematicians of the nineteenth century. The name determinant stems from the French mathematician Augustin-Louis Cauchy (1789-1846). All kinds of identities between determinants were detected.

Cramer's rule goes back to Gabriel Cramer (1704-1752), though Cramer himself gave no proof of the formula named after him. Nowadays the importance of determinants in different mathematical branches is obvious. In Calculus, for example, determinants appear in the substitution rule for multiple integrals. For instance, in the transition to new variables in a double integral, a #(2\times 2)#-determinant appears in the new integral; in the transition to polar coordinates #x=r\cos( \phi)#, #y=r\sin(\phi)# it looks like this:

\[

\begin{array}{rcl}

\int\int f(x,y)\, \dd x\, \dd y & =& \int\int f(r\cos (\phi), r\sin (\phi))

\left|\begin{array}{cc}

\cos (\phi) & \sin(\phi) \\ -r \sin(\phi) & r\cos (\phi) \\

\end{array}\right| \, \dd r\, \dd\phi\\

&=&\int\int f(r\cos (\phi), r\sin (\phi)) r \, \dd r\, \dd\phi

\end{array}

\] The permutations used for the definition of a determinant are useful in the description of all kinds of symmetries, such as the 48 symmetries of the cube. The proof that there exist no formulas for determining zeros of polynomials of degree at least #5#, also makes essential use of the theory of permutations.

Determinantal functions are special cases of certain multilinear expressions, also called tensors. Alternating multilinear forms play a role in the integration theorems by Gauss, Green, and Stokes.

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