Sep. 21 - Lecture
= (x, Aᴴ y)
Prove (Ax,y)
=
yᴴ (Ax)
=
(yᴴ A) x
(Ax,y)
Note: yᴴ A = (Aᴴ y)ᴴ
= (Aᴴ y)ᴴ x
= (x,Aᴴ y)
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for 2-norms - when Q is orthogonal then
|| Q x || = || x ||
Q = mxn matrix that is orthogonal: Qᴴ Q = I
Notes on terminology:
A orthogonal : columns of A are orthonormal, i.e.,
Qᴴ Q = I
mxn with m >= n
Unitary = same property when m=n.
[unitary when m = n]
|| Qx ||²
= (Qx,Qx) = (Qᴴ Q x,x) = (x,x) = ||x||²
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Second triangle inequality:
|
||x|| - || y ||
| ≤ || x-y||
|| x ||
= || y + (x-y) || ≤ ||y|| + || x-y||
|| y ||
= || x + (y-x) || ≤ ||x|| + || x-y|| (b)
(a)
(b)
==> ||x|| - ||y|| ≤ || x-y ||
==> ||y|| - ||x|| ≤ || x-y ||
(a)
==> -[||y|| - ||x||] ≥- ||x-y ||
-||x-y|| ≤ ||x|| - ||y|| ≤ ||x-y||
|
||x|| - || y ||
| ≤
|| x-y||
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