Hello,
I have to show invariance with linear transformation \(\displaystyle X \to AX, A ∈ \mathbb{R} ^{d×d} \)
Show that y is invariant under this transformation and w is not.
\(\displaystyle w = (XX^T)^{-1}y^T\\
y = w^TX \)
\(\displaystyle X ∈ \mathbb{R} ^{d×n}, y ∈ \mathbb{R} ^{1×n}\\\)
My beginning:
\(\displaystyle for\ y\\
y = w^T(AX)\\
y = (((AX)(AX)^T)^{-1}(AX)y^T)^TAX\\
y = (((AX)(X^TA))^{-1}(AX)y^T)^TAX\\
y = (((X^TA)^{-1}(AX)^{-1}(AX)y^T)^TAX\\
y = (((X^TA)^{-1}y^T)^TAX
\)
what should I do now?
It should be \(\displaystyle y = (((XX^T)^{-1})Xy^T)^TX\) at the end I think
I have to show invariance with linear transformation \(\displaystyle X \to AX, A ∈ \mathbb{R} ^{d×d} \)
Show that y is invariant under this transformation and w is not.
\(\displaystyle w = (XX^T)^{-1}y^T\\
y = w^TX \)
\(\displaystyle X ∈ \mathbb{R} ^{d×n}, y ∈ \mathbb{R} ^{1×n}\\\)
My beginning:
\(\displaystyle for\ y\\
y = w^T(AX)\\
y = (((AX)(AX)^T)^{-1}(AX)y^T)^TAX\\
y = (((AX)(X^TA))^{-1}(AX)y^T)^TAX\\
y = (((X^TA)^{-1}(AX)^{-1}(AX)y^T)^TAX\\
y = (((X^TA)^{-1}y^T)^TAX
\)
what should I do now?
It should be \(\displaystyle y = (((XX^T)^{-1})Xy^T)^TX\) at the end I think