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MATRIX. CONCEPT OF MATRIX. 1.Definition of matrix Equations. Rectangular number table can be formed from the coefficients. This is a matrix. The rectangular number table with m rows and n columns constructed by m n numbers in a certain order is called a m n

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concept of matrix
CONCEPT OF MATRIX
  • 1.Definition of matrix Equations

Rectangular number table can be formed

from the coefficients

slide3

This is a matrix

The rectangular number table

with m rows and n columns

constructed by mn numbers in

a certain order is called a mn

Matrix (briefly matrix).

The horizontals are called the rows of matrix and

the verticals are called the columns of matrix

Is called the element located row i and column j.

The matrix is called real matrix if

its elements are real numbers.From now on,

we only discuss the real matrix.

slide4

Matrix is usually denoted with capital A、B、

C and so on. For instance

Simply write:

Column

matrix

Footmark

Row matrix

slide5

The main

diagonal line

The matrix is called a square

matrix if its row number

is equal to its column number,

i.e., m=n.

slide7

6.Trapezoid matrix Let

If

holds for i>j (i<j), and the number of zero before (behind)

the first (last) nonzero element is growing larger (less) as

the rows increase, then we call A a upper (lower)

trapezoid matrix.

They are all called trapezoid matrixes.

slide9

???

No!

Are they Trapezoid matrixes?

Please remember the characteristics of trapezoid matrix and follow its definition.

Trapezoid matrix is the most frequently-used matrix.

slide10

Operation of matrix

一、linear operation

1.Equality: equality of two matrixes means that

their number of rows and number of columns

are respectively equal and their corresponding

elements are the same. i.e.,

=

corresponding

elements are the same

same type

Type is the same

slide11

2.addition、subtraction

define

Let

and

Obviously, A+B=B+A (A+B)+C=A+(B+C)

A+O=O+A=A A-A=O

Negative matrix: the negative matrix of

is

We write-A, i.e.,

slide12

3.number multiple

Is called the product of number and matrix,

Briefly number multiple。write:kA

slide14

=

In general,we have

slide15

what condition can enable A and B to multiply?

= O

Obviously

It is the difference

between Matrix and

number.

slide16

Example2

But

Please remember:

1.The multiplication of matrixes

does not satisfy commutative law;

2. does not satisfy cancellation law;

3. Has nonzero zero factor.

This is another

difference between

matrix

and number.

slide17

Please particularly

notice property 5.

The conclusion does

not hold if the orders of

A and B are not equal.

slide18

Positive integer power of the square matrix.

Transposition of matrix.

Please remember!

slide19

=

=

That is

slide20

Symmetric and anti-symmetric matrix

Any square matrix can be decomposed

to the sum of a symmetric matrix

and an anti-symmetric matrix.

The determinant of an odd

order anti-symmetric matrix

equals to zero.

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