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Inorder to perform eigen decomposition, I converted a rectangular matrix to square by multiplying with the transpose of the matrix. After decomposition, I got the component matrices. If I multiply the component matrices I would get the square matrix. I would like to know, if there is any method for reconstructing the original rectangular matrix from the square matrix.

Thanks in advance

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You are basically delving into singular value decomposition (SVD). Let $A$ be your rectangular matrix which of size $m\times n$. Let us assume $m<n$ (other way around is also same). Take $B_1=AA^T$ and $B_2=A^TA$. Take eigen decomposition of both. So that, $B_1=U\Lambda_1U^T$ and $B_2=V\Lambda_2V^T$. Now do the following

  • Note the non-zero values inside $\Lambda_1$ and $\Lambda_2$, are they related?
  • Make a $m\times m$ diagonal matrix $\Lambda_A$ with its diagonal entries as square roots of diagonal entries of $\Lambda_1$. Make the $m\times n$ block matrix $\Lambda=[\Lambda_A,0]$ where the zero part is a $m \times (n-m)$ zero matrix.
  • Now construct the matrix $C=U\Lambda V^T$. What is the relation between $A$ and $C$?
  • Now think about the other direction, $m\geq n$?.
  • Finally, read about SVD in any standard textbook on Matrices.
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