The derivation of the characteristic polynomial of a linear map from the characteristic equation

Question

In the book of linear algebra by Werner Greub, at page $121$, it is given that

ButI couldn't understand what exactly the author is doing and trying to doing.For example, what is $S_p$, or what does he mean by "Collect in each term of $S_p$ the indexes ...", or what is $$z_{v_1} = \phi x_{v_1} ... z_{v_p} = \phi x_{v_p}.$$I mean even explaining the notation would help.

Generally, the most confusing part is between "Expanding the left hand-side..." and "$\Delta(z_1,...,z_n)= ...$"

I would appreciated if somebody could explain it to me.

Edit:

Actually the main problem is the author explains things as if I'm in his head, which makes almost impossible for me to understand the text with those missing commas, so please be explicit as much as possible in your answers.

Edit 2:

Thanks to the @LeeMosher's answer, I got it the first part, but I have still some questions about the rest.For example, why the permutations $\sigma$ in $4.46$ have the restriction $$\sigma (1) < ... < \sigma (p) \quad \& \quad \sigma (p+1) < ... < \sigma (n),$$

and how $4.47$ is derived from $4.46$.

Answer 1

Let me write this out in the case $n=2$.

Consider the left hand side of (4.45): $$\Delta(\phi x_1 - \lambda x_2, \phi x_2 - \lambda x_2) $$ In the $n \times n$ case, $\Delta$ is an $n$-linear function, as said in the answer of @ChristianBlatter, which means that it is linear separately in each of its $n$ arguments. In this case where $n=2$, that means $\Delta$ is linear separately in each of its $2$ arguments.

As instructed, "Expand the left hand side..." \begin{align*} \Delta(\phi x_1 - \lambda x_2,\phi x_2 - \lambda x_2) &= \Delta(\phi x_1,\phi x_2 - \lambda x_2) + \Delta (-\lambda x_2,\phi x_2 - \lambda x_2) \\ (*) \qquad &= \underbrace{\Delta(\phi x_1, \phi x_2)}_{\text{in $S_2$}} + \underbrace{\Delta(\phi x_1,-\lambda x_2) + \Delta(-\lambda x_1,\phi x_2)}_{\text{in $S_1$}} \, + \underbrace{\Delta(-\lambda x_1,-\lambda x_2)}_{\text{in $S_0$}} \end{align*} Notice: on the first line, I used linearity of $\Delta$ in its 1st argument; and then, in each of the two $\Delta$ expressions on the right hand side of the first line, I applied linearity of $\Delta$ in its 2nd argument.

Notice, as said,

• "... we obtain a sum of $2^2=4$ terms each of the form $$\Delta(z_1,z_2) $$ where every argument [that is, each of the arguments $z_i$ for $i=1,2$] is either $\phi x_1$ or $-\lambda x_1$ ..."

To put this in a wordier manner: in each of the four terms,

• the first argument $z_1$ is either $\phi x_1$ or $-\lambda x_1$,

• the second argument $z_2$ is either $\phi x_2$ or $-\lambda x_2$.

Now let's see what this says about $S_p$. It says:

• "$S_p$ $(0 \le p \le 2)$ is the sum of terms in which $p$ of the arguments are equal to $\phi x_i$ and $2-p$ of the arguments are equal to $-\lambda x_i$".

In the display above, you will see that I have labelled each of the four terms depending on whether it is in $S_0$, $S_1$, or $S_2$, where:

• $S_0$ is the sum of terms (of $(*)$) in which $0$ of the arguments are are equal to $\phi x_i$ and $2$ of the arguments are equal to $-\lambda x_i$.
• $S_1$ is the sum of terms (of $(*)$) in which $1$ of the arguments is equal to $\phi x_i$ and $1$ of the arguments is equal to $-\lambda x_i$.
• $S_2$ is the sum of terms (of $(*)$) in which $2$ of the arguments are equal to $\phi x_i$ and $0$ of the arguments are equal to $-\lambda x_i$.

And now we can explicitly write them out: \begin{align*} S_0 &= \Delta(-\lambda x_1,-\lambda x_2) \\ S_1 &= \Delta(\phi x_1,-\lambda x_2) + \Delta(-\lambda x_1,\phi x_2)\\ S_2 &= \Delta(\phi x_1,\phi x_2) \end{align*}

Answer 2

Note 1: There are dozens of commas missing in the formulas of this page!

Note 2: When Greub's book first appeared it was the "definitive" book on linear algebra at its time. But it was not meant to be a textbook for an introductory linear algebra course.

The quantity at stake is $$\Psi:=\Delta(\phi x_1-\lambda x_1,\ldots,\phi x_n-\lambda x_n)\ .$$ Since $\Delta(\ldots)$ is an $n$-linear function of $n$ vector variables the expanded expression $\Psi$ is a sum of $2^n$ terms of "monomial" character $\Delta(\ldots)$. For each $p\in[0, n]$ there are ${n\choose p}$ terms containing $p$ entries of the form $\phi x_j$ and $n-p$ entries of the form $-\lambda x_j$. The sum of these ${n\choose p}$ terms Greub calls $S_p$.

ad "Collect in each term $\ldots$":

Now he looks at a typical term in $S_p$. Such a term is $\Delta(\ldots)$ of an $n$-tuple ("list") $\ell$ of $n$ vectors in a particular order. He then says: Let $\nu_1<\nu_2<\ldots<\nu_p$ be the positions of the entries in the list $\ell$ of the form $\phi x_j$, and let $\nu_{p+1}<\ldots<\nu_n$ be the positions of the entries in the list $\ell$ of the form $-\lambda x_j$.

ad "Generally the most confusing$\ldots$":

He now goes on to rearrange the entries in the list $\ell$ such that he first writes the $p$ vectors of the form $\phi x_{\nu_k}$ and then the $n-p$ vectors of the form $-\lambda x_{\nu_k}$. Since $\Delta(\ldots)$ has the well known properties with respect to permutations of its entries he formally introduces the permutation $\sigma:\>i\mapsto\nu_i$ and denotes its sign by $\epsilon_\sigma$. In the final formula $(4.46)$ the common factor $(-\lambda)^{n-p}$ is taken out of the sum.