qSingular values are good
measures of the “size” of the matrix singular vectors are good
indications of strong/weak input or output directions.
• Geometrically,
the singular values of a matrix A are precisely the lengths of the semi-axes
of the hyperellipsoid E defined by
•E={y: y=Ax, x Î Cn, ||x||=1}.
• Thus v1 is the
direction in which ||y||
is largest for all ||x||=1, while vn is the direction in which ||y|| is smallest for all ||x||=1
•v1(vn) is the highest (lowest) gain input direction
• u1(um) is the highest (lowest) gain observing direction
• e.g.,
•
•
• A maps a unit disk to an
ellipsoid with semi-axes of s1 and
s2.