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


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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