Quadratic Forms. Matrix of a Quadratic Form. Positively Definite Quadratic Forms, Sylvester's Criterion.

If a certain basis $B=(e_1,...,e_n)$ is fixed in the real linear space $L_n$, then the quadratic form $A(x,x)$ in this basis has the form $$A(x,x)=\sum _{i,j=1}^n{a}_{i,j}{x}_i{x}_j,$$ where $$A=(a_{ij})=\left(\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{n1}& a_{n2}& \cdots & a_{nn}\end{array}\right)$$ - the matrix of the quadratic form, and $x=x_1{e}_1+...+x_n{e}_n$.

The quadratic form $A(x,x)$ defined in the real linear space $L_n$ is called positive (negative) definite if for any $x\in L_n$ $(x\ne 0)$ $$A(x,x)>0(<0).$$

Let $A=(a_{ij})$ be the matrix of the quadratic form $A(x,x)$, and $$D_1=a_{11},D_2=\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|,\cdots ,D_n=\left|\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ a_{21}& a_{22}& \cdots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{n1}& a_{n2}& \cdots & a_{nn}\end{array}\right|-$$The sequence of leading minors of matrix $A$.

The criterion for the positive definiteness of a quadratic form is given by the following statement (Sylvester's criterion): for the quadratic form $A(x,x)$ to be positive definite, it is necessary and sufficient that all the leading minors of its matrix $A$ be positive, that is, $D_k>0$, for $k=1,2,\cdots ,n$.

It can be proven that for the quadratic form $A(x,x)$ to be negative definite, it is necessary and sufficient that the inequalities $(-1{)}^k{D}_k>0$ hold for $k=1,2,\cdots ,n$.

Examples.

In the following problems, determine which quadratic forms are positive or negative definite, and which are not.

1. $x_1^2+26x_2^2+10x_1{x}_2.$

Solution.

The matrix of the quadratic form is given by

$$A=\left(\begin{array}{cc}1& 5\\ 5& 26\end{array}\right).$$

Let's compute the leading minors of matrix $A$:

$$D_1=1>0,$$

$$D_2=\left|\begin{array}{cc}1& 5\\ 5& 26\end{array}\right|=1\cdot 26-5\cdot 5=1>0.$$

Thus, all principal minors of its matrix $A$ were positive, which means that the given quadratic form is positively definite.

Answer: positively definite.

2. $-x_1^2+2x_1{x}_2-4x_2^2.$

Solution.

The matrix of the quadratic form has the form $$A=\left(\begin{array}{cc}-1& 1\\ 1& -4\end{array}\right).$$

We will compute the leading minors of the matrix $A$:

$$D_1=-1<0,$$

$$D_2=\left|\begin{array}{cc}-1& 1\\ 1& -4\end{array}\right|=-1\cdot (-4)-1\cdot 1=3>0.$$

Thus, the inequalities $(-1{)}^k{D}_k>0,,,k=1,2,\cdots ,n,$ are satisfied, which means that the given quadratic form is negatively definite.

Answer: negatively definite.

3. $2x_4^2+x_1{x}_2+x_1{x}_3-2x_2{x}_3+2x_2{x}_4.$

Solution.

The matrix of the quadratic form is given by:

$$A=\left(\begin{array}{cccc}0& 0,5& 0,5& 0\\ 0,5& 0& -1& 1\\ 0,5& -1& 0& 0\\ 0& 0& 0& 2\end{array}\right).$$

We will compute the leading minors of the matrix $A$:

$$D_1=0,$$

Therefore, the quadratic form is neither positive definite nor negative definite.

Answer: of general form.

Homework:

In the following problems, determine which quadratic forms are positive definite or negative definite, and which are not.

1. $x_1^2-15x_2^2+4x_1{x}_2-2x_1{x}_3+6x_2{x}_3.$

Answer: of general form.

2. $12x_1{x}_2-12x_1{x}_3+6x_2{x}_3-11x_1^2-6x_2^2-6x_3^2.$

Answer: negative definite.

3. $9x_1^2+6x_2^2+6x_3^2+12x_1{x}_2-10x_1{x}_3-2x_2{x}_3.$

Answer: positive definite.

4. $x_1^2+4x_2^2+4x_3^2+8x_4^2+8x_2{x}_4.$

Answer: positive definite.

Tags: Matrix of a Quadratic Form, Positively Definite Quadratic Forms, Quadratic Forms, Sylvester's Criterion