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In the mathematical fields of geometry and linear algebra, a principal axis is a certain line in a Euclidean space associated to an ellipsoid or hyperboloid, generalizing the major and minor axes of an ellipse. The principal axis theorem states that the principal axes are perpendicular, and gives a constructive procedure for finding them.## Motivation

The equations in the Cartesian plane R^{2}:
^{2}. In other words, they are obtained from the original coordinates by the application of a rotation (and possibly a reflection). As a consequence, one may use the c_{1}, c_{2} coordinates to make statements about length and angles (particularly length), which would otherwise be more difficult in a different choice of coordinates (by rescaling them, for instance). For example, the maximum distance from the origin on the ellipse c_{1}^{2} + 9c_{2}^{2} = 1 occurs when c_{2}=0, so at the points c_{1}=±1. Similarly, the minimum distance is where c_{2}=±1/3.## Formal statement

The principal axis theorem applies to quadratic forms in R^{n}; that is polynomials Q(x) which are homogeneous of degree 2. Any quadratic form can be put in the form
## References

Mathematically, the principal axis theorem is a generalization of the method of completing the square from elementary algebra. In linear algebra and functional analysis, the principal axis theorem is a geometrical counterpart of the spectral theorem. It has applications to the statistics of principal components analysis and the singular value decomposition. In physics, the theorem is fundamental to the study of angular momentum.

- $frac\{x^2\}\{9\}+frac\{y^2\}\{25\}=1$

- $\{\}frac\{x^2\}\{9\}-frac\{y^2\}\{25\}=1$

- $5x^2+8xy+5y^2=1.$

- $u(x,y)^2+v(x,y)^2=1qquadtext\{(ellipse)\}$

- $u(x,y)^2-v(x,y)^2=1qquadtext\{(hyperbola)\}$

The trick is to write the equation in the following form:

- $5x^2+8xy+5y^2=$

To orthogonally diagonalize A, one must first find its eigenvalues, and then find an orthonormal eigenbasis. Calculation reveals that the eigenvalues of A are

- $lambda\_1\; =\; 1,quad\; lambda\_2\; =\; 9$

- $mathbf\{v\}\_1\; =\; begin\{bmatrix\}1-1end\{bmatrix\},quad\; mathbf\{v\}\_2=begin\{bmatrix\}11end\{bmatrix\}.$

- $mathbf\{u\}\_1\; =\; begin\{bmatrix\}1/sqrt\{2\}-1/sqrt\{2\}end\{bmatrix\},quad\; mathbf\{u\}\_2=begin\{bmatrix\}1/sqrt\{2\}1/sqrt\{2\}end\{bmatrix\}.$

Now the matrix S = [u_{1} u_{2}] is an orthogonal matrix, since it has orthonormal columns, and A is diagonalized by:

- $A\; =\; SDS^\{-1\}\; =\; SDS^T\; =$

This applies to the present problem of "diagonalizing" the equation through the observation that

- $5x^2+8xy+5y^2=mathbf\{x\}^TAmathbf\{x\}=\; (S^Tmathbf\{x\})^TD(S^Tmathbf\{x\})=1left(frac\{x-y\}\{sqrt\{2\}\}right)^2+9left(frac\{x+y\}\{sqrt\{2\}\}right)^2.$

It is tempting to simplify this expression by pulling out factors of 2, however it is important not to do this. The quantities

- $c\_1=frac\{x-y\}\{sqrt\{2\}\},quad\; c\_2=frac\{x+y\}\{sqrt\{2\}\}$

It is possible now to read off the major and minor axes of this ellipse. These are precisely the individual eigenspaces of the matrix A, since these are where c_{2} = 0 or c_{1}=0. Symbolically, the principal axes are

- $$

- The equation is for an ellipse, since both eigenvalues are positive. (Otherwise, if one were positive and the other negative, it would be a hyperbola.)
- The principal axes are the lines spanned by the eigenvectors.
- The minimum and maximum distances to the origin can be read off the equation in diagonal form.

Using this information, it is possible to attain a clear geometrical picture of the ellipse: to graph it, for instance.

- $Q(mathbf\{x\})=mathbf\{x\}^TAmathbf\{x\}$

The first part of the theorem is contained in the following statements guaranteed by the spectral theorem:

- The eigenvalues of A are real.
- A is diagonalizable, and the eigenspaces of A are mutually orthogonal.

In particular, A is orthogonally diagonalizable, since one may take a basis of each eigenspace and apply the Gram-Schmidt process separately within the eigenspace to obtain an orthonormal eigenbasis.

For the second part, suppose that the eigenvalues of A are λ_{1}, ..., λ_{n} (possibly repeated according to their algebraic multiplicities) and the corresponding orthonormal eigenbasis is u_{1},...,u_{n}. Then

- $Q(mathbf\{x\})\; =\; lambda\_1c\_1^2+lambda\_2c\_2^2+dots+lambda\_nc\_n^2,$

where the c_{i} are the coordinates with respect to the given eigenbasis. Furthermore,

- The i-th principal axis is the line determined by the n-1 equations c
_{j}= 0, j ≠ i. This axis is the span of the vector u_{i}.

- Strang, Gilbert (1994).
*Introduction to Linear Algebra*. Wellesley-Cambridge Press.

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Last updated on Saturday February 16, 2008 at 10:10:52 PST (GMT -0800)

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