Orthogonality PDF slang: we say 'u1;:::;uk

6.3 Orthogonal and orthonormal vectors

6.3 Orthogonal and orthonormal vectors. Definition. We say that 2 vectors are orthogonal if they are perpendicular to each other.

5. Orthogonal matrices

R × has orthonormal columns if its Gram matrix is the identity matrix: a square real matrix with orthonormal columns is called orthogonal.

21. Orthonormal Bases

In addition to being orthogonal each vector has unit length. Suppose T = {u1

Orthogonal but not Orthonormal

https://empslocal.ex.ac.uk/people/staff/reverson/uploads/Site/procrustes.pdf

Lecture 4 Orthonormal sets of vectors and QR factorization

slang: we say 'u1

Orthogonal and orthonormal sets

24-Feb-2015 Note that if S is orthonormal then o ? S

Math 115A - Week 9 Textbook sections: 6.1-6.2 Topics covered

Orthonormal bases. • Gram-Schmidt orthogonalization. • Orthogonal complements. *****. Orthogonality. • From your lower-division vector calculus you know

Orthogonality

slang: we say 'u1;:::;uk are orthonormal vectors' but orthonormality (like I (you'd think such matrices would be called orthonormal not orthogonal).

Orthonormal Bases in Hilbert Space APPM 5440 Fall 2017 Applied

02-Dec-2017 Let (ek) be an orthonormal sequence in an inner product space X. Let x ? X. The quantities. ?ekx? are called the Fourier coefficients of x ...

Orthonormal Sets • Bessel Inequality • Total Orthonormal Sequences

EL3370 Orthogonal Expansions - 5. BESSEL INEQUALITY. Theorem (Bessel Inequality). If !! {e n. } is an orthonormal sequence in an inner product space V

Orthogonality

Stephen Boyd and Sanjay Lall

EE263

Stanford University

Orthonormal set of vectors

?normalizedif????? ?,?? ??????? (??are calledunit vectorsordirection vectors) ?orthogonalif?????for???? ?orthonormalif both setof vectors, not vectors individually T???? 2

Orthonormality

an orthonormal set of vectors is independent ?to see this, multiply??? ?by?T ?warning: if? ? ?then??T???(since its rank is at most?) (more on this matrix later ...) 3

Orthonormal basis for??

A matrix?is calledorthogonalif?is square and?T???

?(you"d think such matrices would be calledorthonormal, notorthogonal) ?it follows that?????T, and hence also??T??,i.e., ??T??? 4

Expansion in orthonormal basis

suppose?is orthogonal, so????T?,i.e., ?????T????? ??T??is called thecomponentof?in the direction?? ????T?resolves?into the vector of its??components ?????reconstitutes?from its??components ???is called the (??-)expansionof? 5

Complementary subspaces

if???? ????and?is orthogonal thenrange????andrange????are calledcomplementary subspaces, becauserange???? ?range?????

?they are orthogonali.e., every vector in the first subspace is orthogonal to every vector in the second

subspace ?every vector in??can be expressed as a sum of two vectors, one from each subspace ?each subspace is theorthogonal complementof the other 6

Complementary subspaces

range???? ?range????? ?range????range????because?T???? ? ?to showrange????range????, suppose???range????, then?T??? ?, and since?????T??? ??T??we have?????T??and so??range?? 7

Geometric interpretation

if?has orthonormal columns then transformation???? ?we say?isisometric, it preserves distances 8

Example: Rotation

rotation by?in??is given by 9

Example: Reflection

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[PDF] Orthogonality slang: we say 'u1;:::;uk

Orthogonality

Stephen Boyd and Sanjay Lall

Stanford University

Orthonormal set of vectors

Orthonormality

Orthonormal basis for??

A matrix?is calledorthogonalif?is square and?T???

Expansion in orthonormal basis

Complementary subspaces

Complementary subspaces

Geometric interpretation

Example: Rotation

Example: Reflection