[Linear algebra] 1. Determinant

 

Chapter 1 determining factors

§ 1 second-and third-order Determinant ----------------> concept of the determinant
2 full sorting and Reverse Order
§ 3 Definition of the rank-N Criterion
§ 4 interchange --------------------------------> the nature and calculation of the determinant
§ 5 nature of the determinant
§ 6 determine the number of rows (columns ).
§ 7 karat Mo's law ----------------------> solving linear equations.

1.1 binary linear equations and second-order determining factors

PS: multiply the diagonal lines by a second-order criterion.

1.2. Third-Order Determinant

PS: Parallel diagonal calculation rule

1.3. Full arrangement and number of reverse orders

Full arrangement: n! Situation

Reverse Order: when the order of two elements is different from the standard order, these two elements form a reverse order.

Number of reverse orders: the total number of all reverse orders in the arrangement is called the number of reverse orders in this arrangement. (odd arrangement \ even arrangement)
Algorithm for calculating the number of reverse orders: regular comparison

1.4. Order N Determinant

PS: det (AIJ)

• A total of N-level determining factors are n! Item.
• Each item is the product of n elements in different columns of different rows.
• Each item can be written as a1p1a2p2... anpn (except positive and negative numbers), where P1P2... PN is 1, 2 ,..., N.
• When P1P2... PN is an even arrangement, the corresponding item is taken as the positive number;
• When P1P2... PN is an odd arrangement, the negative number is used for the corresponding item.

1.5. Four Special determinant Calculation

1.6 change Definitions

Definition: During the arrangement, any two elements are called and the remaining elements are not moved.

Adjacent swap: swap two adjacent elements, called adjacent swap

• Adjacent swap is a special case of swap.
• The general SWAP can be achieved through a series of adjacent pairs.
• If the same swap is performed two times in a row, the arrangement is restored.

1.7 relationship between exchange and parity

Theorem 1 is the parity of changing the arrangement.

It is inferred that the odd order is the odd number of SWAPs in the standard arrangement, and the even order is the even number of SWAPs in the standard arrangement.

This is because after the positions of any two elements in ai1j1, ai2j2, ai3j3,... and ainjn are switched, the odd parity of the sum of the row-label arrangement and the reverse Number of the column-label arrangement remains unchanged.

Therefore:

Therefore, the Order N determinant can be written in the following form:

1.8. Nature of the determinant

Property 1 is equivalent to its transpose determinant, that is, D = DT.

Property 2: two rows (columns) of the interchangeable determinant, and the number of the determinant.
It is inferred that if there are two rows (columns) that are identical, then this determinant is zero.
All elements in a row (column) of attribute 3 are multiplied by the same factor K, which is equal to multiplying the number K by the factor.
The common factor of all elements in a row (column) of the inferred determinant can be referred to outside of the determinant symbol.

Property 4 if there are two rows (columns) in a factor proportional to the element, then this factor is zero.

Property 5 if the elements in a column (ROW) of a determining factor are the sum of two numbers, they can be split into two forms of the addition of the two determining factors.

Property 6: multiply the elements of a column (ROW) of the determinant by the same multiple and add them to the elements corresponding to the other column (ROW). The determinant remains unchanged.

PS: generalized lower Triangle Matrix

PS: use the property to convert the determinant into the top triangle, so as to calculate the value of the determinant.

1.9. Remainder and algebra Remainder

Remainder formula: After row I and column J where the element AIJ is located are crossed out, n-1-1 rank criterion left behind is called the remainder of the element AIJ, which is recorded as mij.

Algebra: Call AIJ = (-1) I + jmij an algebraic Remainder of the element AIJ.

PS: each element in the determinant corresponds to a remainder and an algebraic remainder respectively.

The theorem is a rank-n deciding factor. If all the elements in the first row are zero except AIJ, This determining factor is equal to the product of AIJ and Its Algebraic remainder formula, that is, D = aijaij.

1.10. Spread by row (column) in a single clause [a lower-level determinant represents a Higher-Order Determinant]

Theorem 3 determine the sum of the product of each element of any row (column) and its corresponding algebraic remainder formula

PS: fandmond (vandremonde) Determinant

The sum of the products of any one row (column) element and the corresponding element of the other row (column) is equal to zero.

PS: the proof of the above inference is very subtle !!!

1.11. kramo's law [solutions to linear equations]

Linear equations:

When D! = 0:

There is a solution and the solution is unique:

PS: Where, DJ replaces the element in column J of the coefficient determining factor D with the constant term on the right of the equations, and then obtains the rank n determining factor.

Theorem 4 if the coefficient determinant of the linear equations (1) is not equal to zero, the linear equations must have solutions and the solutions are unique.

Theorem 4' if the linear equations have no solution or two different solutions, its coefficient determinant must be zero.

1.12 homogeneous and non-homogeneous equations

Definition: Linear Equations with zero constants are called homogeneous linear equations; otherwise, they are called non-homogeneous linear equations.

Conclusion: homogeneous linear equations always have solutions, because (0, 0 ,..., 0) is a solution called zero solution. Therefore, homogeneous linear equations must have zero solutions, but not necessarily non-zero solutions.

Theorem 5 if the coefficient of the homogeneous linear equations is D! = 0, then the homogeneous linear equations only have zero solutions, and there is no non-zero solutions.

Theorem 5' if the homogeneous linear equations have a non-zero solution, its coefficient determinant must be zero.

 

 

LZ description:As we need to study Kalman filtering next, let's review the linear algebra PPT. In order to make it easier to find it for future use, I sorted out the main content and wrote a blog. The knowledge in this chapter is mainly used to solve the simplification and calculation of the determining factor, and to calculate the linear equations using the determining factor. Well, it's not too early. Let's write the first chapter today. Tomorrow is the matrix and its operations ~ Http://www.cnblogs.com/zjutlitao/