(a) Compute A^TA and its eigenvalues and unit eigenvectors v1 and v2. Find σ1. Rank one matrix A =[ 1 2 3 6 ] (b) Compute AA^T and its eigenvalues and unit eigenvectors u1 and u2. (c) Verify that Av1= σ1u1. Put numbers into the SVD: A=UΣV^T [ 1 2 3 6 ] = [ u1 u2 ] [ σ1 0 ] [ v1 v2 ]^T.

Question

(a) Compute {A}^{T}A and its eigenvalues and unit eigenvectors {v}_{1} and {v}_{2}. Find {\sigma}_{1}. Rank one matrix \quad A =\begin{bmatrix} 1 & 2 \ 3 & 6 \end{bmatrix} (b) Compute A{A}^{T} and its eigenvalues and unit eigenvectors {u}_{1} and {u}_{2}. (c) Verify that A{v}_{1}= {\sigma}_{1}{u}_{1}. Put numbers into the SVD: A=U\Sigma{V}^{T} \quad \begin{bmatrix} 1 & 2 \ 3 & 6 \end{bmatrix}=\left[ \begin{matrix} {u}_{1}& {u}_{2}\end{matrix} ight] \begin{bmatrix} {\sigma}_{1} & \ & 0 \end{bmatrix}{ \left[ \begin{matrix} {v}_{1}& {v}_{2}\end{matrix} ight] }^{ T }.

Accepted Answer

[latex]{A}^{T}A =\begin{bmatrix} 10 & 20 \ 20 & 40 \end{bmatrix}[/latex] has [latex]\lambda [/latex]= 50 and 0,

[latex]{v}_{1} = \frac{1}{\sqrt{5}}\left[ \begin{matrix} 1 \ 2 \end{matrix} ight] [/latex],

[latex]{v}_{2} = \frac{1}{\sqrt{5}}\left[ \begin{matrix} 2 \ -1 \end{matrix} ight][/latex] ;

[latex]{\sigma}_{1}=\sqrt{50}[/latex].

Question : (a) Compute A^TA and its eigenvalues and unit eigenvectors v...