Question

u and v are column vectors in R^2 and A = I +uv^T. Show that if u^Tv is not equal to -1 then A^-1 = I -uv^T/(1+u^Tv)

Answer 1

if u and v are 2 column vectors in R^2, what is uv?

Answer 2

\[\binom{x_1}{y_1}\binom{x_2}{y_2}=\text{hard to do a rc operation}\]

Answer 3

uv would be a scalar, right?

Answer 4

matrix multipication can only be perforned if col1 = row2 right?

we have 1 column on the left, and 2 rows on the right

Answer 5

Your right, sorry a typo it should be u and v are column vectors in R^2 and A = I +uv^T. Show that if u^Tv is not equal to -1 then A^-1 = I -uv^T/(1+u^Tv)

Answer 6

Okay I edited the question

Answer 7

u = x1 v=x2
y1 y2

uT = x1 y1 vT = x2 y2

uvT = x1x2 x1y2
y1x2 y1y2


I + uvT = 1+x1x2 x1y2 = A
y1x2 1+y1y2


so thats one part written out

Answer 8

having trouble recalling the uv stuff. one is a dot product and the other is matrix multiplication

Answer 9

uv dot product would be a scalar ... x1x2 + y1y2

Answer 10

youre pretty much trying to manipulate A into A^-1

and show there respective parts. for which i believe we need a determinant for

Answer 11

if you have a picture of the problem, that might clear up some stuff about the post

Answer 12

Sorry there is no picture, just a word problem.

Answer 13

having trouble deciphering the format of the post as given. Usually there is some notation and such given in the word problem to help distinguish some nuances

Answer 14

uTv and uvT seems to relate to matrix multiplication and result in a new matrix

uv would indicate a dot product that produces a scalar

Answer 15

if the phrase: uTv = -1 referes to a determinant .... than it would seem apparent that 1+uTv cannot be zero ... or something

Answer 16

A = 1+x1x2 x1y2
y1x2 1+y1y2

det(A) = (1+x1x2)(1+y1y2)-(x1y2)(x2y1)

A^-1 = (some moving about of the components of A)/det(A)

Answer 17

does the moving about of the components of A equate to 1-uvT ? and does the det(A) = (1+det(uTv)) ??

Answer 18

i think the rule of a 2x2 is what, swap a d and negate b c

Answer 19

yes

Answer 20

then compare the swap to: I - uvT


I-uvT = 1-x1x2 -x1y2
-y1x2 1-y1y2