**§ ****5 ****Quadratic and Hermitian types**

One and two

[ Bilinear ] If 2 *n* real (or complex) variables , a quadratic homogeneous polynomial_{}_{}

_{} ( 1 )

is called bilinear type, where

_{},_{}

_{}

[ quadratic ] a quadratic homogeneous polynomial with respect to *n* real (or complex) variables_{}

_{} ( 2 )

is called the quadratic form, where is the symmetric part of matrix *A* , that is , the elements of ._{}_{}_{}_{}

The necessary and sufficient conditions for expression ( 2 ) to be always equal to zero are: *A* is antisymmetric ._{}

When the matrix *A* is symmetric, the quadratic form is said to be symmetric . When the matrix *A* is real ( a real number), the quadratic form is said to be real . From ( 2 ), we can see that each quadratic form can be is symmetrical ._{}

A real symmetric quadratic form is said to be positive definite, negative definite, semi-positive definite or semi-negative definite for each set of real numbers that are not all zeros such that , or , respectively . All other real symmetric quadratic forms is called indeterminate (that is , the sign of and is related to) or constant equal to zero .
_{}_{}_{}_{}_{}_{}_{}

[ Change the quadratic type to the standard type ]

1 ^{o} a linear transformation

_{} _{} ( 3 )

or
_{}

_{}

turn each quadratic ( 2 ) into a quadratic with respect to the new variable_{}

_{} ( 4 )

in
_{}

or
_{}

If *A* is symmetric, then it is also symmetric; if both *A* and *T* are real, they are also real ._{}_{}

2 ^{o} For each real symmetric quadratic form, there is a linear transformation ( 3 ) with real coefficients such that the matrix in ( 4 ) is a diagonal matrix, so_{}_{}

_{} ( 5 )

In the formula ( 5 ), the number *r* of the coefficients not equal to zero is independent of the diagonal transformation adopted, and is equal to the rank of the known matrix A, and *r **is* called the rank of the quadratic form . The positive coefficient of the coefficient in the formula ( 5 ) The difference between a number and a negative number is also independent of the transformation applied to the diagonalization (i.e. the Jacobi - Sylvester law of inertia), which is called the sign difference of the quadratic form ._{}_{}

3 ^{o} In particular, for each real symmetric quadratic form, there is a linear transformation corresponding to the real orthogonal matrix *T* , which can transform the quadratic form into a standard form, that is,

_{} ( 6 )

where the real numbers are the eigenvalues of the known matrix *A.*_{}

4 ^{o} Perform the transformation again , and the expression ( 6 ) becomes_{}

_{}

where is equal to 1 , or 0 , corresponding to the eigenvalue being positive, negative or zero , respectively ._{}_{}_{}

[ Simultaneous simplification of two quadratic forms ] Given two real symmetric quadratic forms , , which are positive definite, we can find a real transformation ( 3 ), which can simultaneously transform , into the standard form . In particular There exists a real transformation ( 3 ) such that_{}_{}_{}_{}_{}

_{}_{}

_{}

The real numbers are the eigenvalues of matrices , which are algebraic equations of degree *n*_{}_{}

_{}

the root

[ Method for determining positive definiteness, etc. ]

1 ^{o} The necessary and sufficient conditions for a real symmetric quadratic form to be positive definite, negative definite, semi-positive definite, semi-negative definite, indefinite or always equal to zero are: the eigenvalues of the matrix (which must be real) are both positive and negative, respectively , all non-negative, all non-positive, different signs, or all equal to zero ._{}

2 ^{o} A sufficient and necessary condition for a real symmetric quadratic form to be positive definite or semi-positive definite is: every principal subform of_{}

_{}

are either positive or non-negative .

3 ^{o} The necessary and sufficient conditions for a real symmetric quadratic form to be negative definite or semi-negative definite are: positive definite or semi-positive definite, respectively ._{}

4 ^{oThe} necessary and sufficient conditions for a real matrix *A* to be positive semi-definite are: . If *B* is nonsingular, then *A* is positive definite ._{}

5 ^{o} If both *A* and *B* are positive definite or negative definite, then *AB* is also positive definite or negative definite . Each positive definite matrix *A* has a unique pair of square *roots **Q* , ._{}_{}

2. Hermit ( *H* ) type

[ *H* -form ] A quadratic form with respect to *n* real (or complex) variables_{}

_{}

is called a Hermitian type ( *H* type), where *A* is an *n* -order Hermitian matrix (Chapter IV, § 2 , IV), that is ._{}

An *H* -form is called positive definite, negative definite, semi-positive definite, or semi-negative definite if , for any set of complex numbers that are not all zero , such that , , or , respectively . All other *H* forms are called indefinite *(* i.e. the sign is related to ) or is always equal to zero ._{}_{}_{}_{}_{}_{}_{}

[ Change *H* type to standard type ]

^{1o} A linear transformation ( 3 ) turns each *H* -shape into a new *H* -shape with respect to the new variable_{}

_{}

in the formula
_{}

or
_{}

2o ^{For} each type *H* , there is a linear transformation ( 3 ) such that

_{} ( 7 )

In formula ( 7 ), the number *r* of coefficients not equal to zero has nothing to do with the diagonal transformation adopted, and is equal to the rank of the known matrix A , *r* is called the rank of the *H* type ._{}

3 ^{o} In particular, there is a linear transformation corresponding to the diagonal unitary matrix *T for each **H type, which can be **transformed* into a standard type

_{} ( 8 )

where the real arrays are the eigenvalues of the known matrix *A.*_{}

4 ^{o} Perform the transformation again , and the expression ( 8 ) becomes_{}

_{}

where is equal to 1 , or 0 , corresponding to the eigenvalue being positive, negative or zero , respectively ._{}_{}_{}

[ Simultaneous simplification of two *H* -forms ] Given two *H* -forms and , which are positive definite, there exists a transformation ( 3 ) such that _{}_{}_{}

_{}

_{}

The real numbers are the eigenvalues of matrices , which are algebraic equations of degree *n*_{}_{}

_{}

's root .

[ Method for determining positive definiteness, etc. ]

1 ^{o} A sufficient and necessary condition for an *H* type to be positive definite, negative definite, semi-positive definite, semi-negative definite, indefinite or always equal to zero is: the eigenvalues of matrix *A* (which must be real) are respectively positive, all negative, all are non-negative, both are non-positive, have different signs, or are both equal to zero .

2 ^{o} A necessary and sufficient condition for a Hermitian matrix *A* (and the corresponding *H* -form) to be positive definite or positive semi-definite is that every principal subform of is positive or nonnegative .
_{}

3 ^{o} The necessary and sufficient conditions for a Hermitian matrix *A* (and the corresponding *H* -form) to be negative definite or semi-negative definite are: - *A* is positive definite or semi-positive definite, respectively .

4 ^{o} The necessary and sufficient conditions for a matrix *A* to be a positive semi-definite Hermitian matrix are:

_{}

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