## Saturday 10 September 2011

### Determinants I: cyclic VS symmetrical

Recently I just heard someone saying that the factorization of determinants has homogeneous, cyclic and symmetrical properties. Now I'm going to clarify the above claim. In part 1 (this one), we will discuss the properties of function and in part 2 we will look at those determinants.

1) The definition of homogeneous is simple unique in this case. We have a unique factorization of a certain determinants, but why?

- The properties of determinants is based on the definition of the determinants.
- Each properties can be induced from definition.
- The properties are invertible.

Now if we are going to factorize $s=Det(X)$, through properties $X$ and $Y$ respectively,
The expression through process $X$ is $X(s)$ while that of $Y$ is $Y(s)$, we have $X(s) = Det(X) = Y(s)$.
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The factorization result is equivalent each other, so the answer is homogeneous. If the expression is real number or real variable, by the fundamental theorem of algebra, it has an unique factorization over real number.

P.S. this is not a rigor proof on the homogeneous property but more like a guide.

2) An expression is cyclic if the function value is invariant upon cyclic shift of variables. For a function of three variables that means $f(a,b,c) = f(b,c,a) = f(c,a,b)$. For example, $f(a,b,c) = ab+bc+ca$ is cyclic.

3) An expression is symmetrical if the function value is invariant upon swaps of variables. For example, $f(a,b,c) = a^2+b^2+c^2$ is symmetric.

4) A symmetrical function must be a cyclic function. This is obvious since symmetrical function permutes all variable in the function in a random way but cyclic function changes their order in a specified way in which it can also be reached by symmetrical function.

5) A 2-variable cyclic function must be symmetrical, because $f(a,b) = f(b,a)$ is the only criteria to show that it's cyclic and symmetrical.

6)(5) is NOT the case that for more variable, cyclic is equivalent to symmetrical. Consider $f(a,b,c) = a^2b+b^2c+c^2a$ which is cyclic but not symmetric.

7) Elemental function are the general form of composing a symmetrical function. Moreover, ALL symmetrical expression can be expressed in terms of sum and product of elemental function (if it's constant free)
Definition: elemental function $\sigma _i$ of $n$ variable is equal to the sum of all possible product of i-out-of-n distinct variable.
For example, 2 variable:
$\sigma _1 = a+b, \sigma _2 = ab$
3 variable:
$\sigma _1 = a+b+c, \sigma _2 = ab+bc+ca, \sigma _3 = abc$

Any symmetric expressions can be expressed in terms of elemental functions of the same amount of variables. For example, 2 variables:
$a^2 + b^2 = (a+b)^2 - 2ab = \sigma _1^2 - 2\sigma _2$
$a^4 + b^4 = (a+b)^4 - 2ab(2(a+b)^2-ab) = \sigma _1^4 - 4\sigma _1^2\sigma _2 + 2\sigma _2^2$
3 variables:
$a^2+b^2+c^2 = (a+b+c)^2 - 2(ab+bc+ca) = \sigma _1^2 - 2\sigma _2$
$a^2b^2 + b^2c^2+c^2a^2 = (ab+bc+ca)^2 - 2abc(a+b+c) = \sigma _2^2 - 2\sigma _1 \sigma _3$

8) Lastly a group of isolated function with the loss of beauty, asymmetrical function are functions with value sign flipped upon a swap of neighboring variable. For two variables that is $f(a,b) = -f(b,a)$.