orth_2009_09_28_01

orth_2009_09_28_01 - 4-1 Orthogonality S Lall Stanford...

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4 - 1 Orthogonality S. Lall, Stanford 2009.09.28.01 4. Orthogonality Orthogonal sets of vectors Orthogonal matrices range-nullspace orthogonality
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4 - 2 Orthogonality S. Lall, Stanford 2009.09.28.01 Norms and Inner Products The norm measures the length of a vector bardbl x bardbl = radicalBig x 2 1 + x 2 2 + · · · + x 2 n = x T x It satisfies the Cauchy-Schwartz inequality | x T y | ≤ bardbl x bardblbardbl y bardbl The angle between two vectors is θ = ( x,y ) = cos 1 x T y bardbl x bardblbardbl y bardbl In particular, x and y are orthogonal if x T y = 0 . Write this as x y
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4 - 3 Orthogonality S. Lall, Stanford 2009.09.28.01 Orthonormal set of vectors Set of vectors { u 1 ,...,u k } ∈ R n is normalized if bardbl u i bardbl = 1 , i = 1 ,...,k ( u i are called unit vectors or direction vectors ) orthogonal if u i u j for i negationslash = j orthonormal if both slang: we say ‘ u 1 ,...,u k are orthonormal vectors’ but orthonormality (like independence) is a property of a set of vectors, not vectors individually
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4 - 4 Orthogonality S. Lall, Stanford
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