has a singular value decomposition and to find it you usually have to use a machine. But if you like, this problem will show you how to find an SVD for 
by hand. An SVD for 
will have the form 
.a. Start with the vectors
and 
. For any value of 
these vectors are orthogonal. Find a value of 
so that 
and 
are orthogonal. I.e., compute 
and 
, find their dot product, and then find a value of 
that makes the dot product zero.b. Once you've found
, you can get 
and 
by normalizing (making unit vectors out of) 
and 
.c. You can get
and 
by normalizing (making unit vectors out of) 
and 
.d. You can get
by computing how much 
stretched 
. I.e. compute 
, similarly for 
.e. Use the cross product to get a vector
to complete the orthonormal basis of 
.