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

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 studies of angular momentum and birefringence.

Conteúdo 1 Motivação 2 Declaração formal 3 Veja também 4 References Motivation The equations in the Cartesian plane R2: {estilo de exibição {começar{alinhado}{fratura {x^{2}}{9}}+{fratura {^{2}}{25}}&=1\[3pt]{fratura {x^{2}}{9}}-{fratura {^{2}}{25}}&=1end{alinhado}}} definir, respectivamente, an ellipse and a hyperbola. In each case, the x and y axes are the principal axes. This is easily seen, given that there are no cross-terms involving products xy in either expression. No entanto, the situation is more complicated for equations like {estilo de exibição 5x^{2}+8xy+5y^{2}=1.} Here some method is required to determine whether this is an ellipse or a hyperbola. The basic observation is that if, by completing the square, the quadratic expression can be reduced to a sum of two squares then the equation defines an ellipse, whereas if it reduces to a difference of two squares then the equation represents a hyperbola: {estilo de exibição {começar{alinhado}você(x,y)^{2}+v(x,y)^{2}&=1qquad {texto{(ellipse)}}\você(x,y)^{2}-v(x,y)^{2}&=1qquad {texto{(hyperbola)}}.fim{alinhado}}} Desta forma, in our example expression, the problem is how to absorb the coefficient of the cross-term 8xy into the functions u and v. Formalmente, this problem is similar to the problem of matrix diagonalization, where one tries to find a suitable coordinate system in which the matrix of a linear transformation is diagonal. The first step is to find a matrix in which the technique of diagonalization can be applied.

The trick is to write the quadratic form as {estilo de exibição 5x^{2}+8xy+5y^{2}={começar{bmatriz}x¥d{bmatriz}}{começar{bmatriz}5&4\4&5end{bmatriz}}{começar{bmatriz}x\yend{bmatriz}}= mathbf {x} ^{textsf {T}}Amathbf {x} } where the cross-term has been split into two equal parts. The matrix A in the above decomposition is a symmetric matrix. Em particular, by the spectral theorem, it has real eigenvalues and is diagonalizable by an orthogonal matrix (orthogonally diagonalizable).

It is tempting to simplify this expression by pulling out factors of 2. No entanto, it is important not to do this. The quantities {estilo de exibição c_{1}={fratura {x-y}{quadrado {2}}},quad c_{2}={fratura {x+y}{quadrado {2}}}} have a geometrical meaning. They determine an orthonormal coordinate system on R2. Em outras palavras, they are obtained from the original coordinates by the application of a rotation (and possibly a reflection). Consequentemente, one may use the c1 and c2 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, por exemplo). Por exemplo, the maximum distance from the origin on the ellipse c12 + 9c22 = 1 occurs when c2 = 0, so at the points c1 = ±1. De forma similar, the minimum distance is where c2 = ±1/3.

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

Formal statement The principal axis theorem concerns quadratic forms in Rn, which are homogeneous polynomials of degree 2. Any quadratic form may be represented as {estilo de exibição Q(mathbf {x} )= mathbf {x} ^{textsf {T}}Amathbf {x} } where A is a symmetric matrix.

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.

Em 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 λ1, ..., λn (possibly repeated according to their algebraic multiplicities) and the corresponding orthonormal eigenbasis is u1, ..., un. Então {estilo de exibição Q(mathbf {x} )= lambda _{1}c_{1}^{2}+lambda _{2}c_{2}^{2}+dots +lambda _{n}c_{n}^{2},} where the ci are the coordinates with respect to the given eigenbasis. Além disso, The i-th principal axis is the line determined by the n − 1 equations cj = 0, j ≠ i. This axis is the span of the vector ui. See also Sylvester's law of inertia References Strang, Gilbert (1994). Introduction to Linear Algebra. Wellesley-Cambridge Press. ISBN 0-9614088-5-5. Categorias: Theorems in geometryTheorems in linear algebra

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