Once you come into contact with linear algebra, you need to understand basic concepts and strong problem-solving abilities.
First of all, what we want to talk about is the determinant. In layman's terms, it is a value, which is an algebraic sum of the products of elements in different rows and columns of nth order.
This determinant is incredible and has its shadow everywhere. Therefore, if you want to do a good job in linear algebra, it must be the foundation among the foundations and the top priority.
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First, What is “odd number arrangement and even number arrangement”?
It refers to the column sorting of a group of elements. For example, for a11a23a32a44,
we can see that the column sorting is 1324. We can judge that this is an odd number arrangement. How can we tell? It's very simple. If there is no one in front of 1 that is bigger than it, it is recorded as 0. If there is no one in front of 3 that is bigger than it, it is also recorded as 0. By analogy, we add these numbers recorded as 1, so we say this is an odd permutation. -
Second, we enter the subject. What are the properties of the determinant?
Because as we said before, it is just a value. Among the numerical determinants we usually calculate, the fourth order is the largest we can calculate, and the third order is more common. So how do we calculate the common third order? ** There is a formula (**As for how this formula came about, you have to find a book to read if you are interested). -
The first property is
that its transposition is exactly the same as its own value. (Transpose means rotating it 90 degrees clockwise) -
The second property is
that a certain row (or column) contains the same factors and can be directly taken out and placed outside it. -
The third property:
If the elements of two rows (or two columns) in it are equal or proportional, then its value is directly 0 -
The fourth property (this old one)
says that if the elements of a certain row can be written as the addition of two numbers, then we can ruthlessly split it into two determinants and add them. -
The fifth property
- Oh, I think about it, it is the exchange of elements in two rows (or two columns), then you need to add a negative sign in front of it.
OK, Everybody, it looks very simple, but in fact you have to practice it!
We just mentioned above that there are formulas for third-order ones, which can be directly derived. However, for higher-order ones, it will not work according to that formula. So
Third, let’s come up with the concept again. What are cofactors and algebraic cofactors?
OK, let’s talk about the cofactor first. As the name suggests, we don’t know what it means. Haha, it’s actually based on one element. For example, there is a fourth-order determinant. As for this element, it is in three rows and four columns, so we It is stipulated that the third row and fourth column where it is located should be completely removed, and the remaining elements will be combined into a third-order determinant in the previous order, then the obtained value is the cofactor!
Knowing this cofactor, the algebraic cofactor is the element we mentioned above multiplied by the cofactor, and then multiplied by (-1)^(the row where the element is located + the column where the element is located). Perfect~!
After knowing the above two concepts, we can solve the calculation of the higher-order determinant just mentioned.
How to calculate it? If there is a fourth-order determinant, then we can expand it by a certain row (or by a certain column), let The corresponding element is multiplied by its algebraic cofactor, and the sum of the products is the value of the determinant (this was also discovered by a great mathematician, and you can deduce it later if you are interested). This formula is very commonly used
! ! ! ! ! ! !
Another one that is also very commonly used is that when a certain row (or column) is multiplied by an algebraic cofactor that does not belong to it, the sum of the products is 0. (This can be deduced)
Okay~ There are so many concepts, so how do we use them?
Remember some common conclusions -
For the main diagonal,
we form an upper (lower) triangle. At this time, the value of the determinant is directly multiplied by the main diagonal and taken away directly. -
For the sub-diagonal
, we form an upper (lower) triangle. At this time, the value of the determinant is direct. Can it be direct? If you want to eat p, you have to multiply the sub-diagonal and add one (-1)^( n*(n-1)/2), it is considered complete. -
Laplace
means that if you find that there are many zeros in a determinant, and through movement, the determinant of a certain order can be 0, then you have figured out this Laplace (I personally feel that Laplace Lars is a monster, a monster with a sense of justice). How to calculate it after figuring it out?
It's very simple, similar to the upper triangle or lower triangle of the main diagonal, but this time it becomes the multiplication of two determinants.
Then there is a main diagonal, and there are also sub-diagonals!
The formula of the subdiagonal is (-1)^(m+n)|A| |B|, m and n respectively correspond to the orders of the two determinants. -
Vandermonde Determinant
This is quite difficult, I think, but its proof still needs to be mastered (induction method).
The conclusion is too complicated. You can just read it in a book, but it is difficult to type it out.
Well, we have given the above formulas, but when actually giving the question, we will not give you this form. We need our sharp eyes to convert it into these formulas~ How to convert
?
There are three types. The first one is that we can add k times the first row (or first column) to other rows
. The second one is that we can add multiple rows (multiple columns) to the first row
. The third one is that we can add it row by row. Addition, (the first row is added to the second row, the second row is added to i and the third row, and so on)
Anyway, these are the types. It is easy to say but difficult to do.
To be continued...