Linear Algebra-Matrix Representation of Positive Definiteness and Quadratic Form of Matrix

1. Judgment matrix positive definite method (4 kinds)

1. All eigenvalues ​​of the matrix are positive, i.e. λi>0

2. All principal elements of the matrix are positive

3. The order of the matrix is ​​positive

4. The quadratic form of the matrix is ​​positive

2. Conversion of quadratic matrix form and algebraic form

The relationship between quadratic form and real symmetric matrix is ​​as follows:

Let’s talk about the simplest case first. Only two variables are considered here, which is also the most common case.

There is a 2*2 matrix (note that the positive definiteness of the matrix in this article is discussed when the matrix is ​​a real symmetric matrix)

$$\left[\begin{array}{cc}a& b\\ b& c\end{array}\right]\text{\hspace{1em}( 1 )}$$
VectorX=
$$\left[\begin{array}{c}x\\ and\end{array}\right]\text{\hspace{1em}(2)}$$

Consider the calculation result of X'AX, namely

a ² + 2bxy + cy ²

This formula can realize the transformation of algebraic matrix quadratic form to matrix form

3. The above is a two-dimensional case, consider the n-dimensional case

1. Definition: a second-order homogeneous polynomial with n variables is called quadratic:

Be sure to note that this is a homogeneous polynomial

2. The sum sign of quadratic form

3. Example

From the obtained matrix A, we can see that the elements on the diagonal of the matrix determine the coefficients of the square term ,

The coefficients are averaged cross terms assigned to the corresponding matrix position in

According to this rule, we can directly write the corresponding matrix form according to the algebraic form of the given quadratic form.