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1 Useful mathematics, notation and definitions

Fact 1.1. If we multiply a column vectorx by a matrix Amxn , this defines afunction: Amn : R

n ! Rm

Notation 1. A = [a(1)...a(n)], ai 2 Rm refers to columns 1 through n of matrix

A.

Notation 2. A =

264

a1...

am1

375, ai 2 Rn refers to the rows of A.

Definition 1.1. C(A) = ha(1),...,a(n)i refers to the space spanned by thecolumns of A.

Proposition 1.1. D = AB and B is nonsingular =) rank(D) = rank(A)

Proposition 1.2. Let Amn, then, Rank(A) {m, n}Proposition 1.3. D = AB =) Rank(D) min{Rank(A), Rank(B)}

Proposition 1.4. Let Amn =) rank(A) = rank(AA0) = rank(A0A).

1.1 Some Vector Calculus

For S() = min

(Y-X)(Y-x) where Yn1, Xnk and k1

Fact 1.2. Derivative w.r.t. a column vector

@S()

@=

0BB@

@S()@1

...@S()

@k

1CCA

Fact 1.3. Derivative w.r.t. a row vector

@S()

@0=

@S()@1

, . . . ,@S()@k

Fact 1.4. Let() : Rk ! Rm, then () =

0BBB@

1()2()

...

m()

1CCCA is a vector valued

mapping, then

@()@0

=0BB@

@1()@ 0

...@m()@ 0

1CCA

where each element is a row vector as in fact 1.3 and@()@ 0

is a m k matrix1.

1This symbol 0 means transpose.

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Fact 1.5. Let () : Rk

! Rm

, () =

0BBB@

1()2()

...

m()

1CCCA , be a vector valued

mapping, then

@ 0()

@=

@1()

@, . . . ,

@m()

@

where each element is a column vector as in fact 1.2 and@0()@

is an k mmatrix.

Fact 1.6. Lety = Ax where A does not depend on x, then @y@x0

= A where @y@x0

is an m n matrix.

Fact 1.7. Lety = Ax where A does not depend on x, then

@y

@x = A0

where

@y

@xis an nm matrix.

Proposition 1.5. Suppose y = Ax (), where =

0B@

1...

r

1CA, then if A does

not depend on ,@y()

@0=

@y

@x0@x

@0= A

@x ()

@0

the resulting matrix is m r.

Note 1. A@x0()@0

is not well defined, to see this :

@x0

@0=

0BBB@

@x1

@1

@x2

@1

@x3

@1 @xn@1

@x1@2

@x2@2

@x3@2

@xn@2

......

......

...@x1@r

@x2@r

@x3@r

@xn@r

1CCCArn

, but Amn!

Fact 1.8. y = x0Bx = (x0Bx)0 = x0B0x = y0, since y is a scalar.

Proposition 1.6. If Fact 1.8 holds =) we can always find a decompositionA of B s.t A is symmetric matrix.

Proof. Let A = 12 (B + B0) be a symmetric matrix, then:

y =1

2(x0Bx + x0B0x)) =

1

2[x0 (B + B0) x] =

1

2(2x0Ax) = y0

Proposition 1.7. y = x0Ax =) @y@x

= 2x0A = 2Ax

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X

0

y-X=e

x1

x2

y

Figure 1: Geometry of Least Squares.

1.2 Geometry of Least Squares

y = x11 + ... + xkk implies that y lives in the linear space generated byhx1,...,xki R

n , however when we add ", we account for all aspects of y thatdo no live in hx1,...,xki.

Definition 1.2. We call P a projection matrix if:

1. P : Rn ! Rn.

2. P =P0

3. P2 = P

Fact 1.9. In our particular caseP = X(X0X)1X

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