A real matrix factors into orthogonal transformations and independent axis scalings. A unit circle becomes an ellipse whose semiaxis lengths are the singular values; its principal directions are the left singular vectors.实矩阵可以分解为正交变换与各坐标轴上的独立缩放。单位圆由此变为椭圆，半轴长度是奇异值，主轴方向是左奇异向量。
The marked circle first rotates by , stretches by factors 2 and 0.6, then rotates by . The moving seam makes the otherwise invisible first rotation visible.带标记的圆先经 旋转，再按 2 和 0.6 缩放，最后经 旋转。移动的标记线使圆轮廓上不易察觉的第一步旋转可见。