The question is :
Let A be a 2 × 2 symmetric matrix with eigenvalues 3 and −2 and corresponding unit eigenvectors (norm = 1) $u_1$ and $u_2$. Find the SVD and polar decomposition of A.
I tried to take some general values for A and expand $Au_1 = 3u_1 and Au_2 = -2u_2$, but I am not getting how to utilize the $u_1$ and $u_2$.
Even a hint will be helpful.
Since $A$ is symmetric, it is diagonalizable in an othonormal basis, i.e. $A = ODO^T$ with $O$ an orthogonal matrix. If $D$ has positive elements, this decomposition is also the SVD of $A$.
You already know the diagonal elements of $D$ and you also have a candidate for $O$, as $(u_1,u_2)$ forms an orthonormal basis. Unfortunately, one of the eigenvalue is $-2<0$.
One way to compensate for the sign is to use unitary matrices instead. Using this you will have to write $A=UD'V^H$ where $D'$ constains the absolute value of the elements in $D$ and $U,V$ are unitary matrices ($UU^H=I=VV^H$ where the exponent denotes for conjuguate transpose).
Of course, you should generate those unitary matrices from what you know : the eigenvalues and their corresponding eigenvectors.