# Given a matrix Y in R^(mxn). Find a transformation matrix Θ in R^(nxp) such that 1/m Θ^T Y^T YΘ=Ipxp, where Ipxp is identity matrix.

Given a matrix $Y\in {\mathbb{R}}^{m×n}$. Find a transformation matrix $\mathrm{\Theta }\in {\mathbb{R}}^{n×p}$ such that
$\frac{1}{m}{\mathrm{\Theta }}^{T}{Y}^{T}Y\mathrm{\Theta }={I}_{p×p},$
where 𝐼𝑝×𝑝 is identity matrix.
My attempt: $\frac{1}{\sqrt{m}}Y\mathrm{\Theta }$ is orthogonal matrix and tried to find $\mathrm{\Theta }$ satisfies it but that doesn't work.
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Ashly Sanford
This can only work if $Y$ has full rank. Now $\mathrm{\Theta }$ could perform the basis transformation from the Gram-Schmidt procedure.