Questions and Answers II (45%)
Find the singular value decomposition of matrix A=\begin{bmatrix} 2\\3 \end{bmatrix}. (A = UDV^T) Suppose Q = A^TA
Write out the calculation process
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1. (10%) Find the eigenvalues ??, ?? of Q, where ?? < ??
?? =?
?? =?
2. (10%) Find the eigenvectors E?, and E?, of Q, where E? = span\begin{bmatrix} a\\c \end{bmatrix}=?
and
a+b=?
c+d=?
E? = span\begin{bmatrix} b\\d \end{bmatrix}=?
3. (5%) Find a set of normal orthogonal bases \(\vec{v}_1\) and \(\vec{v}_2\) of R² so that the vectors
L(\vec{v}_1\) and L(\vec{v}_2\) are orthogonal, where V = \begin{bmatrix} \vec{v}_1\\\vec{v}_2 \end{bmatrix}=?
det(V)=?
4. (5%) Find the diagonal matrix D=?
tr(D)=?
5. (5%) Find U = \begin{bmatrix} \vec{u}_1\\\vec{u}_2 \end{bmatrix}=?
det(U)=?
6. (10%) Find the singular value ??, ?? of A, where ?? < ??
?? =?
?? =?
