If a is singular it is customary to write a if 2

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Unformatted text preview: s in (1) if x is a multiple of a right singular vector of A corresponding to the minimal singular value σm . Likewise, the equality hold in (2) if b is a multiple of a left singular vector of A corresponding to the maximal singular value σ1 . 17 / 21 Condition number of a matrix The product A A−1 is the condition number of A (relative to the norm · ), denoted by κ(A) κ(A) = A A−1 The condition number is attached to a matrix, not a problem If κ(A) is small, A is said to be well-conditioned; if κ(A) is large, A is ill-conditioned. If A is singular, it is customary to write κ(A) = ∞ If · = · 2 , then A = σ1 , and A−1 = 1/σm , and thus κ(A) = σ1 σm in the 2-norm, which is the formula for computing 2-norm condition numbers of matrices. The ratio σ1 /σm can be interpreted as the eccentricity of the hyperellipse that is the image of the unit sphere of Cm under A 18 / 21    ‚ — g 6 ( 1V    I€    —  –  I€ € “    @ ( @8 € p   ( ‚ I Iw@ w h  6 h D  w y( w w    ‚‚ y i   a XC w 6 – €€   X CG I E UG E 8  D k 6 & k k ˆ k  ¨  (  – – 8  @–  € –u u u   6 B "30 % S!¨ "! Q 6! A %$B58 "$ n % ‚  3 ( ' )        € V €  8   6   ( b  – – 8     8 &am...
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This document was uploaded on 02/10/2014.

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