In linear algebra, especially in the theory of quadratic form, the contractual relationship between matrices is often used. The two matrices A and B are contractual, if and only if there is a reversible matrix? C, so c tac = b, then square a shrinks to matrix B.
In general, the scene of studying contract matrix in online generation problem is quadratic. The matrix used in the quadratic form is a real symmetric matrix. The necessary and sufficient condition for two real symmetric matrix contracts is that their positive and negative inertia indices are the same. It can be inferred from this condition that the contract matrix is of equal rank.
The rank of similarity matrix and contract matrix is the same.
Extended data:
Contract matrix: Let A and B be two N-order square matrices. If there is an invertible matrix c, it is called the contract between square a and b, and it is recorded as a? B.
In linear algebra, especially in the theory of quadratic form, the contractual relationship between matrices is often used. In general, the scene of studying contract matrix in online generation problem is quadratic. The matrix used in the quadratic form is a real symmetric matrix. The necessary and sufficient condition for two real symmetric matrix contracts is that their positive and negative inertia indices are the same. It can be inferred from this condition that the contract matrix is of equal rank.
The contractual relationship is an equivalent relationship, that is, it satisfies:
1, reflexivity: any matrix shrinks with itself;
2. Symmetry: If A and B contract, the contraction of B and A can be deduced;
3. Transitivity: If the contract is in B and B and the contract is in C, the contract can be introduced in C;
4. The rank of the contract matrix is the same.
Main discrimination methods of matrix contract;
1. Let A and B be N-order symmetric matrices over the complex number field, then the contraction of A and B over the complex number field is equivalent to the same rank of A and B. 。
2. If both A and B are N-order symmetric matrices in real number field, then the contract of A and B in real number field is equivalent to that A and B have the same positive and negative inertia index (that is, the number of positive and negative eigenvalues is equal).
Rotation matrix is a kind of matrix. When multiplied by a vector, it has the effect of changing the direction of the vector without changing its size. The rotation matrix does not contain inversion, it can change the right-handed coordinate system into the left-handed coordinate system or vice versa. All rotations are added and inverted to form a set of orthogonal matrices.
Mathematically, the principle of rotation matrix involves a combination design: covering design. Covering design, filling design, Steiner system and t- design are all combinatorial optimization problems in discrete mathematics. They solve the problem of how to combine elements in a collection to achieve specific requirements.
Another general application of matrix in physics is to describe linear coupled harmonic systems. The motion equation of this kind of system can be expressed in the form of matrix, that is, a mass matrix multiplied by a generalized velocity gives the motion term, and a force matrix multiplied by a displacement vector describes the interaction.
The best way to find the solution of the system is to find the eigenvector of the matrix (by diagonalization and other methods. ), the so-called system normal mode. This solution is very important in the study of molecular internal dynamics model: the vibration of atoms bonded by chemical bonds in the system can be expressed as the superposition of normal vibration modes. When describing mechanical vibration or circuit oscillation, normal mode is also needed to solve it.
References:
Baidu Encyclopedia-Contract Matrix