Similar to the difference between diagonalization and diagonalization

Similar diagonalization means that m is taken from the nth order square matrix in the exchange body k, and the diagonalization of m is to determine a diagonal matrix d and a reversible square matrix p, so that m=pdp-1. Let f be the automorphism of the KN corresponding to M and diagonalize M to determine a basis of KN so that the matrix corresponding to F in this basis is a diagonal matrix.

Not all quadrilaterals.

The diagonal lines are complementary, but the angle that complements each other must be a quadrilateral quadrilateral inner circle.

Proof: Know: Quadrilateral ABCD, BAD+BCD=180° Verification: Quadrilateral ABCD in the wheel pick.

Proof: Suppose the quadrilateral ABCD is not connected in a circle, at the three points b, a, and d, o, and c is not o. Peso.

1) If the contact exchange of o, p point is across o, connector dp, bp, c point wiki.

apd>∠acd，∠apb>∠acb

apd +∠apb>∠acd+∠acb

of the DPB> within the BCD

Western ABPD connection o, second-hand bad+ bpd= 180° bad + bcd<180° this is known.

Bad+bcd= 180° is contradictory, it is not possible outside of point C.

2) If the point c is at o, the connection ac and the extension cross o at the point q, the connection dq, cq, c in a similar way is not provable o] br> by equations (1) and (2) only know at point c o, i.e. the assumption is not true.

The quadrilateral is connected to the circle ABCD. Peso (see Temple Geometry Textbook) - Middle East.

Two matrices that are similar may not necessarily be diagonal, but one of them can be diagonalized and the other can be diagonalized.

The two matrices have similar sufficient and necessary conditions.

It's that they have the same invariant factor, or they have the same determinant factor, or they have the same elementary factor, or they have the same standard form.

In mathematics, a matrix is a set of complex or real numbers arranged in a rectangular array, which originally came from a square matrix composed of coefficients and constants of a system of equations. This concept was first proposed by the 19th-century British mathematician John Kelly.

Matrices are a common tool in advanced algebra and are also commonly found in applied mathematics disciplines such as statistical analysis. In physics, matrices have applications in circuits, mechanics, optics, and quantum physics; Computer science.

, 3D animation.

Crafting also requires the use of matrices. The operation of the matrix is numerical analysis.

important questions in the field.

Decomposing matrices into combinations of simple matrices can simplify the operation of matrices in theory and practical applications. For some widely used and special matrices, such as sparse matrices and quasi-diagonal matrices, there are specific fast operation algorithms of Huiyu oak. For the development and application of matrix-related theory, please refer to the above "Matrix Theory".

in astrophysics.

In the field of quantum mechanics, infinite-dimensional matrices will also appear, which is a kind of generalization of matrices.

The n-order phalanx can be diagonalizedSufficient and necessary conditionsYes: There are n linearly independent eigenvectors in n-order squares.

Corollary: If this nth-order square matrix has n different eigenvalues, then there must be a similar matrix in the matrix.

If there are duplicate eigenvalues in the order n square, the number of linearly independent eigenvectors for each eigenvalue is exactly equal to the number of repetition of heavy rocks that the eigenvalue is coarse.

Diagonal matrix.

There is an important value because of the diagonal matrix.

It is particularly easy to deal with: their eigenvalues and eigenvectors are known, and a matrix is elevated to its power by simply lifting the diagonal elements to the same power.

Any two 3rd order matrices a,b are similar:

1. Find the feature polynomial first.

f(λ)e-a|，g(λ)e-b|。

2. If f( )g( ) then the matrices a and b are not similar.

3. If f( )g( ) and there are 3 different roots, then the matrices a and b are similar.

4. If f( )g( ) has 2 different roots, i.e., f( )g( )a) 2( -b),(ae-a)(be-a)=(ae-b)(be-b)=0, then the matrices a,b are similar.

Diagonalization is generalized, just to turn the matrix into a diagonal matrix, and there is no requirement for the value of the diagonal element (it is not required that the sock collapse is not zero). In this sense, symmetry matrices must be similarly diagonalized, and this is true.

How do you achieve similar diagonalization? In fact, similarity diagonalization is to find an orthogonal array t

Make t'at=t (-1)at=diag (each i has its geometric multiplicity).

Here's how to do it: find all the values of a and find the eigenvector i1,.. corresponding to the eigenvalues of the whole clothisi (si is the geometric weight of i).

,.. for each set of i1The ISI performs Schmidt orthogonalization respectively, and then the R group of vectors after Schmidt orthogonalization is arranged into a matrix according to the source circle in order, which is denoted as T, and T is the request.

The concept of diagonalization is specific to matrices, and the diagonalization of matrices comes from the simplification of linear transformations, so it is best to know the correspondence between linear transformations and linear transformations and matrices.

Let the linear transformation a, the matrix under the basis m be a, the matrix under the basis n be b, and the transition matrix from m to n be x, then it can be proved that the crack b=x-1ax

Then the definition: a, b are 2 matrices. If there is an invertible matrix x, satisfying b=x-1ax, then a is said to be similar to b (an equivalence relation).

If there is an invertible matrix x that makes a similar to a diagonal matrix b, then a is said to be diagonalizable.

Correspondingly, if the matrix of the linear transformation A under the basis m is A, and A is similar to the diagonal matrix B, then the basis n can be obtained by making x the transition matrix, and the matrix of A is linearly transformed into a diagonal matrix under n, thus achieving simplification.

Assuming that the matrix is a, then the condition is sufficient and necessary.

is: a has n linearly independent eigenvectors.

The minimal polynomial of a has no double root.

Sufficient but not necessary:

aThere are no heavy eigenvalues.

a*a^h=a^h*a

Necessary non-sufficient condition: f(a) is diagonal, where f is any analytic function of the spectral radius of convergence greater than a.

Real symmetry can be similar to diagonalization because the eigenvalues of the real symmetry matrix are all real numbers, so the n-order matrix has n eigenvalues (including multiples) in the real number domain, and the repetition of each eigenvalue of the real symmetry matrix is the same as the number of irrelevant eigenvectors, so that the n-order matrix has n irrelevant eigenvectors, so it can be diagonalized.

The main properties of the matrix are real:

1. The eigenvectors corresponding to the different eigenvalues of the real symmetry matrix a are orthogonal.

2. The eigenvalues of the real symmetry matrix a are all real numbers, and the eigenvectors are all real limb vectors.

3. The nth-order real symmetry matrix a must be diagonalized, and the elements on the similar diagonal matrix, i.e., the forest bureau, are the eigenvalues of the matrix itself.

4. If 0 has k-weight eigenvalues, there must be k linearly independent eigenvectors, or there must be rank r(0e-a)=n-k, where e is the identity matrix.