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In uniform scaling with a non-zero scale factor, all non-zero vectors retain their direction (as seen from the origin), or all have the direction reversed, depending on the sign of the scaling factor. In non-uniform scaling only the vectors that belong to an eigenspace will retain their direction. A vector that is the sum of two or more non ...
which is a uniform scaling and shows the meaning of special choices for : for = one gets the identity mapping, for = one gets the reflection at the center, For / one gets the inverse mapping defined by .
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin.
The two-dimensional case is the only non-trivial (i.e. not one-dimensional) case where the rotation matrices group is commutative, so that it does not matter in which order multiple rotations are performed. An alternative convention uses rotating axes, [1] and the above matrices also represent a rotation of the axes clockwise through an angle θ.
Uniform scaling. Likewise, the scale component can be removed by scaling the object so that the root mean square distance ... a non-profit organization.
These bounds are not invariant by scaling. That is, the roots of the polynomial p(sx) are the quotient by s of the root of p, and the bounds given for the roots of p(sx) are not the quotient by s of the bounds of p. Thus, one may get sharper bounds by minimizing over possible scalings. This gives
Bottom: The action of Σ, a scaling by the singular values σ 1 horizontally and σ 2 vertically. Right: The action of U , another rotation. In linear algebra , the singular value decomposition ( SVD ) is a factorization of a real or complex matrix into a rotation, followed by a rescaling followed by another rotation.
Finding an optimal solution to the above problem results in a quantizer sometimes called a MMSQE (minimum mean-square quantization error) solution, and the resulting PDF-optimized (non-uniform) quantizer is referred to as a Lloyd–Max quantizer, named after two people who independently developed iterative methods [6] [21] [22] to solve the two ...