【高代】the Transpose and Determinnants of Matrices

05-10

Today we are going to summarize the left fundemental knowledege about matrices.

Theres nothing more deserving mentioning about the matrix transpose,but it will play an important role in algebra:

At this stage,we just need to know the following things:

$(AB)^T=B^TA^T;(A+B)^T=A^T+B^T;\ (cA)^T=cA^T;(A^T)^T=A$

We have known that $detegin{pmatrix} a&b\ c&d end{pmatrix}=ad-bc$ ,and it can have a geometric interpretation. Left multiplication by A maps the space $mathbb R^2$ of real two-dimensional column vectors to itself, and the area of the parallelogram（平行四邊形） that forms the image of the unit square via this map is the absolute value of the determinant of A. Just like this:

Then we talk the $n imes n$ -dimensional matrix. Its straight forward to see determinants as a function from $n imes n$ space to the real numbers:

$det:mathbb R^{n imes n} ightarrowmathbb R$ .

A particular fomula should be added to define what the determinant is, or it may be too confusing. Thus expansion by minors is introduced, in which the determinants of $(n-1) imes(n-1)$ submatrices are called minors(餘子式）.

Thats to say, $det A=sum_{v=1}^{n}{(-1)^{v+j}a_{vj} det A_{vj}}$ .

Its fine to say that determinants are unique, although it needs some proof. Accounting for the complexity and length of the proof(though important), I just skip it.

Other properties of the formula of dorminants are trivial, and I bet its unnecessary to repeat it.

Permutations(全排列）

Def. A permutation of a set S is a bijective map $p$ from a set $S$ to itself.

It supports the compositon like $qcirc p$ .

The set of all permutations of the indices ${1,2,...,n}$ is called the symmetric gruop, and is denoted by $S_n$ .We will discuss it in the next chapter.(I feel some kind of excited to learn it)

We denote $p(3)=4,p(4)=1,p(1)=3$ as $mathbf (3,4,1)$ or

A 2-cycle is also called a transposition.When a number is fixed, we usually omit it in the cycle.

e.g. $p=(5 2)(1 3 4)$ ; $p=(1 3 2 5)$

Then here comes the permutation matrix （置換矩陣）P. For instance:

$PX=egin{pmatrix}0 0 1\1 0 0\0 1 0end{pmatrix}egin{pmatrix}x_1\x_2\x_3end{pmatrix}=egin{pmatrix}x_3\x_1\x_2end{pmatrix}$

Moreover, if $P=sum_{i}{e_{pi,i}},X=sum{e_jx_j}$ ,then $PX=sum_{i}{e_{pi}x_i}$ .

A sign of a permutation is its $det P=pm 1$ ,and a permutation is even if its sign is +1,and odd if sign is -1.Thats too familiar.

Using this ,we have another formula of determinants:

$det A=sum_{perm p}{(sign p)a_{1,p1}...a_{n,pn}}$ ,this is the formula that is first introduced when I first learnt matrix.

The Cofactor Matrix(伴隨矩陣）

The cofactor matrix of an $n imes n$ matrix A is the $n imes n$ matrix cof(A) whose i,j entry is

$cof(A)_{ij}=(-1)^{i+j}detA_{ji}$

Be careful thats j column,i row; not i column,j row.

Its mainly used to compute a inverse of a matrix, using the following theorem:

Thm. Let A be an $n imes n$ matrix, let C=cof(A) be its cofactor matrix, and let $alpha=det A$ .If $alpha e0$ ,then A is inversible, and $A^{-1}=alpha^{-1}C$ .

In my eyes, the calculation makes me feel sick......

If any chance, I will add some proof to some theorems, but may be not.

T.B.C.

(Next well learn group!)