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Orthogonal matrices, change of basis, rank Math Tools for - - PowerPoint PPT Presentation

Orthogonal matrices, change of basis, rank Math Tools for Neuroscience (NEU 314) Spring 2016 Jonathan Pillow Princeton Neuroscience Institute & Psychology. Lecture 6 (Thursday 2/18) accompanying notes/slides todays topics


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Math Tools for Neuroscience (NEU 314) Spring 2016 Jonathan Pillow

Princeton Neuroscience Institute & Psychology. accompanying notes/slides Lecture 6
 (Thursday 2/18)

Orthogonal matrices, change of basis, rank

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today’s topics

  • orthonormal basis
  • change of basis
  • orthogonal matrix
  • rank
  • column space and row space
  • null space
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basis

1

v

1

v

2

v

2

v

1

v

  • set of vectors that can “span” (form via linear

combination) all points in a vector space Two different (orthonormal) bases for the same 2D vector space 1D vector space (subspace of R2)

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  • rthonormal basis
  • basis composed of orthogonal unit vectors
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Change of basis

  • Let B denote a matrix whose columns form an
  • rthonormal basis for a vector space W

If B is full rank (n x n), then we can get back to the

  • riginal basis through

multiplication by B

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Change of basis

  • Let B denote a matrix whose columns form an
  • rthonormal basis for a vector space W

Vector of projections of v along each basis vector

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Orthogonal matrix

  • In this case (full rank, orthogonal columns), B is an
  • rthogonal matrix

Properties: length- preserving

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Orthogonal matrix

  • 2D example: rotation matrix

1

^ e ) ^ ( 2 e

Ο

) ^ ( 1 e . .g e

2

^ e

=

Ο Ο Ο =

sin θ cosθ cosθ sin θ

] [

( )

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SLIDE 9

Rank

  • the rank of a matrix is equal to
  • the rank of a matrix is the dimensionality of the vector

space spanned by its rows or its columns

  • # of linearly independent columns
  • # of linearly independent rows

(remarkably, these are always the same) equivalent definition: for an m x n matrix A:

rank(A) ≤ min(m,n)

(can’t be greater than # of rows or # of columns)

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column space of a matrix W:

n × m matrix

vector space spanned by the columns of W

c1 cm

  • these vectors live in an n-dimensional space, so the

column space is a subspace of Rn

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SLIDE 11

row space of a matrix W:

n × m matrix

vector space spanned by the rows of W

  • these vectors live in an m-dimensional space, so the

column space is a subspace of Rm

r1 rn

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SLIDE 12

null space of a matrix W:

  • the vector space consisting of

all vectors that are orthogonal to the rows of W

  • the null space is therefore entirely orthogonal to the row

space of a matrix. Together, they make up all of Rm.

r1 rn

  • equivalently: the null space of W is the vector space of all

vectors x such that Wx = 0.

n × m matrix

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null space of a matrix W:

1

v

1 D v e c t

  • r

s p a c e s p a n n e d b y v 1 W = ( )

v1

n u l l s p a c e basis for null space