Quadratic Forms with Symmetric Matrices
When you encounter a general quadratic equation, the best way to understand it is by looking at its structure in matrix form. Imagine you have a symmetric matrix with elements:
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When you encounter a general quadratic equation, the best way to understand it is by looking at its structure in matrix form. Imagine you have a symmetric matrix with elements:
This matrix has eigenvalues and corresponding orthonormal eigenvectors . Something interesting happens when we use coordinate transformation through the matrix .
For coordinate transformation, we use . In the new coordinates , the equation becomes:
For both variables with , the completing the square process is performed separately:
The result of this process gives a simpler form:
By determining the center point and the constant , we obtain:
For , various curve forms can emerge depending on the signs of the eigenvalues.
If and , then the conic section formed is an ellipse:
With semi-axis lengths in the direction of and in the direction of .
When and , the conic section formed is a hyperbola:
With semi-axis lengths in the direction of and in the direction of .
A special condition occurs when and . Completing the square gives:
The conic section formed is a parabola:
Conic sections in satisfy the general quadratic equation:
Which can be written in matrix form as:
For a symmetric matrix , vector , and scalar , the quadratic surface is defined as the solution set of the general quadratic equation:
Which can be written in explicit form:
If is symmetric and is an orthonormal basis of eigenvectors with , then the orthonormal matrix enables diagonalization or .
In the new coordinate basis and , the quadratic surface has diagonal form:
In the orthonormal basis of eigenvectors, the quadratic form has a diagonal structure. This transformation is called principal axis transformation because the new coordinate axes are aligned with the directions of the matrix eigenvectors.