About Similarity Matrix Definition of matrix similarity

p^(-1)ap=b

then the matrix A is said to be similar to B and is denoted as A B.

aa represents the result of a linear combination of vectors in a according to the components of a as coefficients (it is clear to multiply a by column and block), but when x and y are regarded as the representation matrix of a linear transformation f, f(a) = ax, f(b) = by, that is, the image of f is expressed according to the original basis, naturally there will be no f(a) = f(b), and the meaning of each quantity should be paid attention to when analogy.

If you want to derive the relationship between similar transformations and transition matrices, just take advantage of it.

axa=f(a)a=f(aa)=f(bb)=f(b)b=byb, combined with the definition of the transition matrix a=bp, pa=b, and substitution can get px=yp.

Interpretation of matrices.

matrix]

One of a rectangular arrangement of mathematical elements, such as the coefficients of a simultaneous linear equation, obeys special algebraic laws.

The word decomposition Explanation of moment moment ǔ Draw right angle or square tool: Moment ruler (curved ruler). Rectangle (rectangle).

Moment (in physics, the force that rotates an object multiplied by the distance to the axis of rotation). Propriety. Laws, rules :

Be in the trace. radical : arrow; Explanation of the array array (formation) è the situation laid out by the army when it fights:

Front. Formation. Embattled.

Battlefield: Positional. Fall.

Charge into battle. A measure word that refers to a passage in which an event or action has passed: paroxysmal.

Travail. It rained for a while. Radical: 阝.

First, find the eigenvalues of the similarity matrix, substitute the eigenequations respectively, solve the eigenvectors respectively, and form the matrix p, then we can get p (-1)ap=d, where d is the diagonal matrix composed of all eigenvalues.

In sexual algebra, a similarity matrix is a matrix in which there is a similarity relationship. Let a and b be nth-order matrices, and if there is an nth-order invertible matrix p such that p (-1)ap=b, then the matrix a is said to be similar to b, and it is denoted as a b. The operation of the pair is called the similarity transformation, and the invertible matrix is called the similarity transformation cavity Sakura matrix.

Matrices A are similar to B, i.e., there are invertible matrices.

p, satisfies p -1ap = b

Basic conclusion: Eigenpolynomial for similarity matrices.

Same. Corollary: Eigenvalues of similarity matrices.

The same beats Lee, determinant.

The same is the same, and the traces are the same (this inference is commonly used, need to be remembered) two common conclusions: the determinant of a is equal to the product of all eigenvalues of a.

The trace of a is equal to the sum of all the eigenvalues of a.

Calculate the eigenvalue of b: |b-λe|=1- )2(1+) So the eigenvalues of b are: 1,1,-1

From the adherence of a and b, it is known that the eigenvalues of a are 1,1,-1, so the eigenvalues of a-2e are 1-2=-1,1-2=-1, -1-2=-3

Therefore a-2e is reversible. [A reversible sufficient necessary strip of reeds to sell parts.]

One is that none of the eigenvalues of a are 0].

The eigenvalues of a-e are: 1-1=0, 1-1=0, -1-1 = 2

Therefore r(a-e) =1 [ don't ask why, just use it, its rank is equal to the number of its non-zero eigenvalues ]

So r(a-2e)+r(a-e) =3+1 = 4

Find the eigenvalue of b, let |re-b|=r 3-r 2-1=0, knowing that 1 and 2 are not eigenvalues of b.

A is similar to B, and the eigenvalues of the stool hail of the collapsed rock are the same as those of A and B. 1,2 is also not a eigenvalue of a.

Therefore, a-2e, a-e are reversible.

So r(a-2e)=3,r(a-e)=3,r(a-2e)+r(a-e)=6.

p^(-1)ap=b

then the matrix A is said to be similar to B and is denoted as A B.

A sufficiently necessary condition for a matrix A to be similar to a diagonal matrix is that matrix A has n linearly independent eigenvectors.

Note: The process of proving the theorem has actually given a way to diagonally the square matrix.

If the matrix can be diagonalized, it can be achieved by following these steps:

1) Find all the eigenvalues;

2) For each eigenvalue, let its weight be k, then the basic solution system of the corresponding homogeneous equation system is composed of k vectors, that is, the corresponding linearly independent eigenvectors;

3) The eigenvectors obtained above happen to be the linearly independent eigenvectors of the matrix.

Let a and b be nth-order matrices, and if there is a nth-order nonsingular matrix p that makes p (-1)*a*p=b true, then the matrix a is said to be similar to b, and it is denoted as a b

This is advanced mathematics, you don't need to memorize the definition, just remember the nature.

Let a and b be nth-order matrices, if there is an invertible matrix.

p, such that p-1 (the inverse matrix of p) ap=b, says a is similar to b, and is denoted as a b.

If a b, then.

Eigenpolynomials of matrices a,b.

Same as eigenvalue.

a|=|b|

r(a)=r(b)

tr(a)=tr(b)

Similarity is the same, so 2+x = 1+y

The similar determinant is equal, so -15x = -20y

Solution: x=-4, y=-3