Question

(Linear Algebra):
\[Let: \mathbb{S}=Span \left\{ \left[\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}\right],\left[\begin{matrix}0 & 1 \\ 1 & 0\end{matrix}\right],\left[\begin{matrix}1 & -1 \\ 0 & 1\end{matrix}\right] \right\}\] Consider \[A=\left[\begin{matrix}0 & 2 \\ -1 & 2\end{matrix}\right]\]. Find the matrix B in S such that ||A - B|| is minimized.

Answer 1

We saw something "similar" in class but it involved minimizing systems ||Ax-b|| (where x, b are vectors). I'm not sure how it works with just matrices...

Answer 2

I recall that ||A(x,y)||^2 = (0x+2y)^2 + (-1x + 2y)^2

Answer 3

so probably what you were doing in class...

Answer 4

maybe let B = (a,b)
(c,d)
so C = A-B = (0-a, 2-b)
(-1-c, 2-d)
then minimize the functional
f(x,y) = ||C(x,y)||

problem is that would include 4 unknows.... not nice...

Answer 5

actually since B is in S, then B is a linear combination of the 3 matrices, so only 3 unknowns:
\[ B = a\left[\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}\right]
+b\left[\begin{matrix}0 & 1 \\ 1 & 0\end{matrix}\right]
+c\left[\begin{matrix}1 & -1 \\ 0 & 1\end{matrix}\right]\]

\[ \quad = \left[\begin{matrix}a & 0 \\ 0 & a\end{matrix}\right]
+\left[\begin{matrix}0 & b \\ b & 0\end{matrix}\right]
+\left[\begin{matrix}c & -c \\ 0 & c\end{matrix}\right]\]

\[ \quad = \left[\begin{matrix}a+c & b-c \\ b & a+c\end{matrix}\right]\]

Answer 6

Define
\[C := A - B \]
\[\quad =\left[\begin{matrix}0 & 2 \\ -1 & 2\end{matrix}\right] - \left[\begin{matrix}a+c & b-c \\ b & a+c\end{matrix}\right]\]

\[ \quad = \left[\begin{matrix}-a-c & 2-b+c \\-1- b & 2-a-c\end{matrix}\right]\]