$($a$)$ Compute $A^T$A and its eigenvalues and unit eigenvectors $v_1$ and $v_2.$ Find ${}_1.$

Rank one matrix

$A=([1,2],[3,6])$

$($b$)$ Compute $A{A}^T$ and its eigenvalues and unit eigenvectors $u_1$ and $u_2.$

$($c$)$ Verify that $A{v}_1={}_1{u}_1.$ Put numbers into the SVD:

$])$

$([3,6])$

$([,0$

The vectors $(2,2,-1)$ and $(-1,2,2)$ are orthogonal. Divide them by their lengths to find orthonormal vectors $q_1$ and $q_2.$ Put those into the columns of $Q$ and multiply $Q^{T}Q$ and $Q{Q}^T.$