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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Rectangular number table can be formed
from the coefficients
The rectangular number table
with m rows and n columns
constructed by mn numbers in
a certain order is called a mn
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.
C and so on. For instance
The matrix is called a square
matrix if its row number
is equal to its column number,
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)
They are all called trapezoid matrixes.
Are they Trapezoid matrixes?
Please remember the characteristics of trapezoid matrix and follow its definition.
Trapezoid matrix is the most frequently-used matrix.
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.,
elements are the same
Type is the same
Obviously, A+B=B+A (A+B)+C=A+(B+C)
Negative matrix: the negative matrix of
We write-A, i.e.,
Is called the product of number and matrix，
Briefly number multiple。write：kA
In general，we have
It is the difference
between Matrix and
1.The multiplication of matrixes
does not satisfy commutative law;
2. does not satisfy cancellation law；
3. Has nonzero zero factor.
This is another
notice property 5.
The conclusion does
not hold if the orders of
A and B are not equal.
Transposition of 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.