Linear algebra Notes (feature problems are similar to matrices)

1) unary polynomial, polynomial, i-th coefficient, constant term, first term, first coefficient, n polynomial, 0 polynomial, 0 polynomial, polynomial equal, polynomial plus and minus.---
2) The vector representation of the polynomial: the elements of the vector represent the coefficients of the polynomial, and the number of times is implied in the order of the elements, such as f (x) =x^2+2x+1 can be expressed as (1,2,1). N-th polynomial is a vector with n+1 components.
3) Polynomial division, complex root, root, 0 points, real roots, heavy-cause type, heavy root, single-cause type, mono, n-th polynomial has a maximum of n different roots;
4) The basic definition of algebra: In the plural domain, n-th algebraic equations have at least one root;
5) On the complex field, f (x) can always be decomposed into K (X-C1) (X-C2) ... (X-CN), and the only (non-sequential) k equals the reciprocal of the first coefficient;
6) In the complex field, n-th polynomial, just have n root, if the two n-th polynomial is only a non-0 constant factor (corresponding proportional), then two polynomial root exactly the same;
7) Complex C is the root of f (x) =0, then its conjugate complex is also the root of f (x); the singular real coefficient polynomial has at least one real roots;
8) Characteristic value: V is the linear space on the number domain K, T is the linear transformation of V, if the existence number λ0 belongs to K, nonzero vector α0 belongs to V, so that tα0=λ0α0 is established, it is called λ0 is the characteristic value of T; α0 is the characteristic vector of T which belongs to λ0;
9) The eigenvalues of the linear transformation are exactly the same as the eigenvalues of the Matrix, and the eigenvalues and eigenvectors of the similarity matrix are identical;
10) characteristic matrix, characteristic polynomial. Λe-a is the characteristic matrix of A, and F (λ) =|λe-a| is the characteristic polynomial of A;
11)?? Λ0 is the necessary and sufficient condition for the eigenvalue of N-order matrix A. Λ0 is a characteristic polynomial f (λ) =|?? The root of the λe-a|; this theorem reveals a path to the eigenvalues and eigenvectors of a matrix A; the algorithm can be found in mymathlib.
12) Characteristic space: A is n-order matrix, λ0 is a characteristic value, then the solution space of the λe-a x=0 is a characteristic space that belongs to λ0;
13) Properties of matrix eigenvalues:
A) The sum of the eigenvalues equals the trace of matrix A: λ1+λ2+...+λn=a11+a22+. +ann=tra (A):
B) | A|=λ1λ2 ... Λn
C) | A| is not equal to 0, the inverse eigenvalues of a and a are reciprocal;
D) if λ is a eigenvalue of a, then λ^k is the eigenvalues of a^k;
E) A and its transpose matrix have the same characteristic value;
The sufficient and necessary condition for a n-order matrix similar to a diagonal matrix is that it has n linearly independent eigenvectors;
The N-order matrix has n different eigenvalues, then it must be similar to the diagonal matrix; for complex fields, n-order matrices with no eigenvalues must be similar to diagonal matrices;
16) The real symmetric matrix must be similar to the diagonal matrix;
17) The Lambda matrix, the elementary transformation of the λ matrix, the characteristic matrix of n-order matrix A (??). ΛE-A) must be equivalent to a diagonal shape?? λ matrix;
18) Elementary factor:?? Use?? The elementary transformation of the lambda matrix will be a characteristic matrix?? The λe-a is transformed into a diagonal matrix, and then each diagonal element is decomposed into the product of a non-identical one-power-exponent, then all of these power-of-one-powers are referred to as the elementary factors of A;
The sufficient and necessary condition for the similarity of N-order matrices A and B is that their elementary factors are the same;
20) about when the matrix, about when the block, about when the standard-shaped;
The necessary and sufficient condition of n-order matrices similar to diagonal matrices is that the primary factor of a is all at once;
?

????

Linear algebra Notes (feature problems are similar to matrices)