Matrix Norm and Condition Number: the Largest Stretch
An interactive live film: A maps the unit sphere to an ellipsoid; the induced 2-norm is its longest semi-axis and the condition number the longest over the shortest. Turn a unit vector x to find the direction A stretches most.
Narration
A three by three matrix A sends the unit sphere, in grey, to an ellipsoid, in blue. Every unit vector x lands at A x on its surface.
The induced two norm of A is the largest stretch: the supremum of the length of A x over all unit vectors x. It is the longest semi axis of the ellipsoid, the largest singular value, the square root of the largest eigenvalue of A transposed A.
The rest of the narration is in the book's Companion.
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Preview2:29Four Views of Matrix MultiplicationChapter 2
Preview1:44Column Space: Every Output Ax Lies on One PlaneChapter 2
Preview1:37The Projection Matrix: Outer times Inverse InnerChapter 2