Principal axis transformation

Japanese: 主軸変換 - しゅじくへんかん

Given ^the quadratic formula f ( x , y ) = ax2 ⁺ 2hxy + by2 + c
If we rotate the coordinate system by an appropriate angle to create a new coordinate system,
f ( X , Y ) = lX ² + mY ² + n
The xy terms can be eliminated as shown below. A rotation that eliminates the product of the variables in a quadratic formula is called a principal axis transformation. When a curve expressed by a quadratic formula, i.e., a quadratic curve, is symmetric with respect to a line, this line is called the principal axis. An ellipse and a hyperbola have two principal axes that are perpendicular to each other, and a parabola has one principal axis. Therefore, performing a principal axis transformation on a quadratic formula is nothing more than taking a line parallel to the principal axis of the quadratic curve as the coordinate system. A principal axis transformation is also used to diagonalize a symmetric matrix, that is, to select an appropriate orthogonal matrix U for a symmetric matrix S and make U ^-1 SU the diagonal matrix. A linear map f that maps a plane onto itself can be expressed as a matrix whenever an orthonormal basis is determined for the plane. When the matrix is symmetric, it is known that if a new appropriate orthonormal basis is selected, the matrix representation of f with respect to this becomes diagonal. In this case, the orthogonal transformation that maps the old orthogonal basis to the new orthogonal basis is called the principal axis transformation of f .

Although the above two types of principal axis transformations appear to be different, the quadratic equation ax ² +2 hxy + by ² can be transformed into a symmetric matrix

If we consider it in correspondence with f, we can see that it is essentially the same thing. From the standpoint of linear mapping, the new orthonormal basis mentioned above is the eigenvector of f .

[Ryoichi Takagi]

[Reference item] | Eigenvalue

Principal axis transformation example
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Principal axis transformation example

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

与えられた二次式
　　f(x, y)＝ax²＋2hxy＋by²＋c
に対して、座標系を適当な角度だけ回転して新しい座標系をつくれば、
　　f(X, Y)＝lX²＋mY²＋n
のようにxyの項がなくなるようにできる。このように二次式の変数の積の項をなくすような回転移動を主軸変換（主軸への変換という意味）という。二次式で表される曲線、すなわち二次曲線が、ある直線に関して対称になっているとき、この直線を主軸という。楕円(だえん)と双曲線は直交する2本の主軸をもち、放物線は1本の主軸をもつ。したがって二次式の主軸変換をすることは、この二次曲線の主軸に平行な直線を座標系にとることにほかならない。また対称行列を対角化すること、すなわち対称行列Sに対し、適当に直交行列Uを選び、U^-1SUを対角行列とすることを主軸変換ということもある。平面を自分自身に写す線形写像fは、平面に一つの正規直交基底を決めるたびに行列で表現できる。それが対称行列のときは、新たに適当な正規直交基底を選ぶと、これに関してfを行列により表現したものが対角形になることが知られている。このとき、旧正規直交基底を新正規直交基底に写すような直交変換をfの主軸変換という。