Quadratic Forms. Matrix of a Quadratic Form. Positively Definite Quadratic Forms, Sylvester's Criterion.
Literature: Collection of Problems in Mathematics. Part 1. Edited by A.V. Efimov, B.P. Demidovich.
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 $$A(x, x)=\sum\limits_{i, j=1}^n a_{i,j}x_ix_j,$$ where $$A=(a_{ij})=\begin{pmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\a_{21}&a_{22}&\cdots&a_{2n}\\\vdots&\vdots&\ddots&\vdots\\a_{n1}&a_{n2}&\cdots&a_{nn}\end{pmatrix}$$ - the matrix of the quadratic form, and $x=x_1e_1+...+x_ne_n$.
The quadratic form $A(x, x)$ defined in the real linear space $L_n$ is called positive (negative) definite if for any $x\in L_n$ $(x\neq 0)$ $$A(x, x)>0 \qquad (<0).$$
Let $A=(a_{ij})$ be the matrix of the quadratic form $A(x, x)$, and $$D_1=a_{11}, \quad D_2=\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix},\,\,\cdots,\,\, D_n=\begin{vmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\a_{21}&a_{22}&\cdots&a_{2n}\\\vdots&\vdots&\ddots&\vdots\\a_{n1}&a_{n2}&\cdots&a_{nn}\end{vmatrix} -$$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 $(-1)^kD_k > 0$ 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.
4.218. $x_1^2+26x_2^2+10x_1x_2.$
Solution.
The matrix of the quadratic form is given by
$$A=\begin{pmatrix}1&5\\5&26\end{pmatrix}.$$
Let's compute the leading minors of matrix $A$:
$$D_1=1>0,$$
$$D_2=\begin{vmatrix}1&5\\5&26\end{vmatrix}=1\cdot 26-5\cdot 5=1>0.$$
Thus, all principal minors of its matrix $A$ were positive, which means that the given quadratic form is positively definite.
Answer: positively definite.
4.219. $-x_1^2+2x_1x_2-4x_2^2.$
Solution.
The matrix of the quadratic form has the form $$A=\begin{pmatrix}-1&1\\1&-4\end{pmatrix}.$$
We will compute the leading minors of the matrix $A$:
$$D_1=-1<0,$$
$$D_2=\begin{vmatrix}-1&1\\1&-4\end{vmatrix}=-1\cdot (-4)-1\cdot 1=3>0.$$
Thus, the inequalities $(-1)^kD_k>0,,,k=1, 2, \cdots, n,$ are satisfied, which means that the given quadratic form is negatively definite.
Answer: negatively definite.
4.223. $2x_4^2+x_1x_2+x_1x_3-2x_2x_3+2x_2x_4.$
Solution.
The matrix of the quadratic form is given by:
$$A=\begin{pmatrix}0&0,5&0,5&0\\0,5&0&-1&1\\0,5&-1&0&0\\0&0&0&2\end{pmatrix}.$$
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.
4.220. $x_1^2-15x_2^2+4x_1x_2-2x_1x_3+6x_2x_3.$
Answer: of general form.
4.221. $12x_1x_2-12x_1x_3+6x_2x_3-11x_1^2-6x_2^2-6x_3^2.$
Answer: negative definite.
4.222. $9x_1^2+6x_2^2+6x_3^2+12x_1x_2-10x_1x_3-2x_2x_3.$
Answer: positive definite.
4.224. $x_1^2+4x_2^2+4x_3^2+8x_4^2+8x_2x_4.$
Answer: positive definite.
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