1. I. Quadratic Forms and Canonical Forms Def 1 ： Definition 2 ： If linear operations

2. Its matrix is If it is invertible, then call it an invertible linear operation. If it is orthogonal, then call it an orthogonal operation. Def 3 ： The degree of all unknowns in the quadratic form is 2, that is then we call it canonical form of quadratic form(or standard form). II. Matrix of quadratic form ：

3. Matrix of quadratic form

4. Matrix of quadratic form （ obviously, a real symmetric matrix ） Def 4 ： Given quadratic form we say the rank of symmetric matrix A is the rank of quadratic form f. III. Matrix of quadratic form after invertible operations So:

5. Theory 1 IV. Change quadratic to canonical form by orthogonal transform. Question 1: Matrix of canonical form = ？ What’s on earth the problem of changing quadratic into canonical form? Question 3 ： Can the quadratic be changed into canonical forms? Yes! Any a real symmetric matrix is orthogonal congruent with a diagonal matrix. Question 2 ：

6. Theory 2 Process of changing quadratic into canonical form. (i) Write down the matrix of quadratic form: A ；

9. Learn by yourself ： Changing quadratic into canonical forms by square and inertial theory.

11. It’s an elliptic cylinder.