Question

Let A ? ?^{m×n}, m ? n. The pseudoinverse via SVD or Moore-Penrose inverse is defined in the following manner. $$A^{dagger} = VSigma^{dagger}U^{T},$$ where $$Sigma^{dagger} = egin{bmatrix} frac{1}{sigma_1} & 0 & 0 & dots & 0 & dots & 0 \ 0 & frac{1}{sigma_2} & 0 & dots & 0 & dots & 0 \ 0 & 0 & frac{1}{sigma_3} & dots & 0 & dots & 0 \ 0 & 0 & 0 & dots & frac{1}{sigma_r} & dots & 0 \ 0 & 0 & 0 & dots & 0 & dots & 0 \ vdots & vdots & vdots & dots & vdots & dots & vdots \ 0 & 0 & 0 & dots & 0 & dots & 0 end{bmatrix}$$ and ?_i, for i = 1...r are the singular values. Recall that the SVD of matrix A = U?V^T. Verify that A^{dagger} satisfies the following properties a) AA^{dagger}A = A b) A^{dagger}AA^{dagger} = A^{dagger} c) (AA^{dagger})^T = AA^{dagger} d) (A^{dagger}A)^T = A^{dagger}A

          Let A ? ?^{m×n}, m ? n. The pseudoinverse via SVD or Moore-Penrose inverse is defined
in the following manner.
$$A^{dagger} = VSigma^{dagger}U^{T},$$
where
$$Sigma^{dagger} = egin{bmatrix} frac{1}{sigma_1} & 0 & 0 & dots & 0 & dots & 0 \ 0 & frac{1}{sigma_2} & 0 & dots & 0 & dots & 0 \ 0 & 0 & frac{1}{sigma_3} & dots & 0 & dots & 0 \ 0 & 0 & 0 & dots & frac{1}{sigma_r} & dots & 0 \ 0 & 0 & 0 & dots & 0 & dots & 0 \ vdots & vdots & vdots & dots & vdots & dots & vdots \ 0 & 0 & 0 & dots & 0 & dots & 0 end{bmatrix}$$
and ?_i, for i = 1...r are the singular values. Recall that the SVD of matrix A = U?V^T. Verify
that A^{dagger} satisfies the following properties
a) AA^{dagger}A = A
b) A^{dagger}AA^{dagger} = A^{dagger}
c) (AA^{dagger})^T = AA^{dagger}
d) (A^{dagger}A)^T = A^{dagger}A

$Let A ? ?^m×n, m ? n. The pseudoinverse via SVD or Moore-Penrose inverse is defined in the following manner. A^dagger = VSigma^daggerU^T, where Sigma^dagger = eginbmatrix frac1sigma1 0 0 dots 0 dots 0 0 frac1sigma2 0 dots 0 dots 0 0 0 frac1sigma3 dots 0 dots 0 0 0 0 dots frac1sigmar dots 0 0 0 0 dots 0 dots 0 vdots vdots vdots dots vdots dots vdots 0 0 0 dots 0 dots 0 endbmatrix and ?i, for i = 1...r are the singular values. Recall that the SVD of matrix A = U?V^T. Verify that A^dagger satisfies the following properties a) AA^daggerA = A b) A^daggerAA^dagger = A^dagger c) (AA^dagger)^T = AA^dagger d) (A^daggerA)^T = A^daggerA$

Added by Gregory D.

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