Question

Find the singular value decomposition of A = [ (3 2 2), (2 3 2) ] and determine the angle of rotation induced by U and V . Also, write the rank 1 decomposition of A in terms of the columns of U and rows of V . Can we do dimensionality reduction in this case?

how to find angle of rotation induced by U and V?

please provide the detailed process for above.

Answer 1

A=USVT

U and V are orthogonal matrix while S is a diagonal matrix.

AAT =$\begin{bmatrix} 3 & 2 & 2\\ 2& 3 & 2 \end{bmatrix}$   
2 3 2 2 3 2
=$\begin{bmatrix} 17 & 16 \\ 16 & 17 \end{bmatrix}$   

det(AAT - kI )= 0

$\begin{vmatrix} 17-k&16 \\ 16& 17-k \end{vmatrix}$=0

(17-k)2 - 162 =0 i.e   289 + k2 -34k -256 =0

k2 -34k +33 =0

simplifying this we get k=33 and k=1

so singular values are sqrt(33) and 1

eigen vector for 33 =$\begin{bmatrix} 1\\ 1 \end{bmatrix}$ after normalization=u1

1/sqrt(2) [1/(sqrt(2))

and for 1 =
-1 1
after normalization
1/sqrt(2) (-1/(sqrt(2))
=u2

S=
sqrt33 0 0 1
=
5.74 0 0 1

U=

1/sqrt(2) 1/sqrt(2) 1/ sqrt(2) -1/(sqrt(2))

A=USVT

322 233
=
1/sqrt(2) 1/sqrt(2) 1/ sqrt(2) -1/(sqrt(2))

5.74 0 0 1
VT= $\begin{bmatrix} 4.06&0.707 \\ 4.06&-0.707 \end{bmatrix}$ VT

VT is of order 2x3

VT=$\begin{bmatrix} 0.615&0.615 \\ -0.707&0.707\\ 0.348&0.348 \end{bmatrix}$

V=$\begin{bmatrix} 0.615&-0.707&0.348 \\ 0.615&0.707&0.348 \end{bmatrix}$

No the dimensionality reduction is not possible as A is not a diagonal matrix