Contraction, the positive and negative of two real symmetric matrices, then these two real symmetric matrices must be contracted. Because the necessary and sufficient condition for the contraction of two real symmetric matrices is that the two real symmetric matrices have the same rank and the same positive and negative inertia indices.

Contract matrix, in linear algebra, especially in quadratic theory, often uses the contract relationship between matrices. 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.

Each symmetric matrix is reduced to a diagonal matrix, and its elements only consist of 0, 1 and-1. If the number of 1 is p, and the number of-1 is q, then given (p, q), an equivalence class about contractual relations is determined.

The number pair (p, q) is called the inertia index of the symmetric matrix (or the corresponding quadratic form), where the number p of 1 is called the positive inertia index, the number q of-1 is called the negative inertia index, and the number p-q is called the sign difference.

Extended data:

Correlation theorem of inertia index;

1, two quadratic forms can be replaced by reversible linear variables, if and only if their positive and negative inertia indices are equal. (that is, the necessary and sufficient condition for two real symmetric matrix contracts is that their positive and negative inertia indices are equal. )?

2. The positive (negative) inertia index of real symmetric matrix A is the number of its positive (negative) eigenvalues.

The necessary and sufficient condition for inferring two real symmetric matrix contracts is that the number of their positive (negative) eigenvalues is equal.

Nature of the contract:

The contractual relationship is an equivalent relationship, that is, it satisfies:

1, reflexivity, any matrix is shrinking with itself.

2. symmetry. If A signs a contract with B, the contract between B and A can be deduced.

3, transitivity, A contract in B, B contract in C, you can introduce a contract in C. ..

4. The rank of the contract matrix is the same.

The main discrimination method of matrix contract: Let A and B be N-order symmetric matrices in complex number field, then the contract of A and B in complex number field is equivalent to the same rank of A and B. ..

Baidu Encyclopedia-Contract Matrix

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