【高代】Matrices:The Basic Operations

05-12

Id like to review what I learned last year, but Im reluctant to read the same fundamental book once again. So I decided to read another book written by Artin in order that I could pick up what I have learned as well as new knowledge related to abstract algebra.

Its just a try...There is nothing that needs to be paid attention to in todays note. Its even valueless for an algebra learner.

Thanks for your reading.

$egin{pmatrix} a_{11}& ... & a_{1n}\ vdots&&vdots \ a_{m1}&...&a_{mn}\ end{pmatrix}$

An m $imes$ n matrix is a collection of mn numbers arranged in a rectangular array.It has n columns and m rows.The number in a matrix is called the matrix entries,which may be denoted by $a_{ij}$ .When m=n, its called a square matrix.

A 1 $imes$ n matrix, also called n-dimensional row vector and similarly, a n $imes$ 1 matrix, namely n-dimensional column vector is common and essential:

$(a_1,...,a_n)$ or $egin{pmatrix} b_1\ vdots\ b_m end{pmatrix}$

Addition of matrices, scalar multiplication, matrix multiplication is so familiar to us that I decide to skip it.

Next, here comes the one of the most important forms that we should obtain, the system of equations

$egin{equation} egin{cases} a_{11}x_1&+...+&a_{1n}x_n&=&b_1\ a_{21}x_1&+...+&a_{2n}x_n&=&b_2\ vdots & &vdots&& vdots \ a_{m1}x_1&+...+&a_{mn}x_n&=&b_m end{cases} end{equation}$ can be written in matrix notation as $AX=B$ .

Go futher, $M=[A|B]$ are called augmented matrix(增廣矩陣）.

Its been well known how to use ditributive laws and associative laws and except for commutative matrices, that commutative laws does not hold. Meanwhile,the definition of upper triangular, invertible matrix is simple as well. I just want to mention the following equality that may be useful:

$egin{pmatrix} a&b\ c&d end{pmatrix} ^{-1}=frac{1}{ad-bc} egin{pmatrix} d&-b\ -c&a end{pmatrix}$

The denominator ad-bc is the determinant（行列式） of the matrix.

It should be added that the set of all invertible n $imes$ n matrices is called the n-dimensional general group.

Then lets talk about block multiplication.

To begin with, let M and M be m $imes$ n and n $imes$ p matrices. Then decompose them into blocks as follows:

M=[A|B]; M=[A|B],where A has r columns and A has r rows.

Then MM=AA+BB.

Moreover, we denote that

$egin{pmatrix} A&|&B\ hline C&|&D end{pmatrix} egin{pmatrix} A&|&B\ hline C&|&D end{pmatrix} = egin{pmatrix} AA+BC&|&AB+BD\ hline CA+DC&|&CB+DD end{pmatrix}$

Now its fine to give the definition of matrix units.

Actually,for the space of all m $imes$ n matrices, the set of matrix untis is called a basis; Analogously,the set ${e_1,...e_n}$ forms what is called the standard basis of the n-dimensional space $mathbb R^n$ . ( $e_i=(0,...,stackrel{i}{1},...,0)$ )

T.B.C.