For a (2 times 2) matrix (mathbf{A}=left(begin{array}{cc}10 & 2 -10 & 2end{array} ight)) (a) Take an eigenvalue-eigenvector decomposition, i.e., find (mathbf{X}) and (boldsymbol{Lambda}) as

For a \(2 \times 2\) matrix \(\mathbf{A}=\left(\begin{array}{cc}10 & 2 \\ -10 & 2\end{array}\right)\)

(a) Take an eigenvalue-eigenvector decomposition, i.e., find \(\mathbf{X}\) and \(\boldsymbol{\Lambda}\) as in \(\mathbf{A}=\mathbf{X} \boldsymbol{\Lambda} \mathbf{X}^{-1}\)

(b) Take a singular value decomposition, e.g., find \(\mathbf{U}\) and \(\boldsymbol{\Lambda}_{\mathrm{s}}\) and \(\mathbf{V}\) as in \(\mathbf{A}=\mathbf{U} \boldsymbol{\Lambda}_{\mathrm{s}} \mathbf{V}^{\mathrm{T}}\);

(c) Find the solution \(\mathbf{m}\) in \(\mathbf{d}=\mathbf{A m}\), where \(\mathbf{d}^{\mathrm{T}}=(1,2)\).