Eigenvalues of a $4\times 4$ parameters matrix Let $a,b,c,d\in\Bbb{C}$ and
$B =\begin{bmatrix}
    a & b & c & d\\
    d & a & b & c\\
    c & d & a & b\\
    b & c & d & a\\
  \end{bmatrix}$
I know that $t=a+b+c+d$ is an eigenvalue because every row's sum is $t$ (and every column) and I also know that the sum of the eigenvalues is $4a$ (because it equals the trace of $B$).
However I don't know how to continue.. and calculating the chracteristic polynomial of $B$ seems not very pleasant. Any ideas?
thanks
 A: You need to first decompose your circulant matrix:
$$B=a\left[\matrix{1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1}\right]+b\left[\matrix{0&1&0&0\\0&0&1&0\\0&0&0&1\\1&0&0&0}\right]+c\left[\matrix{0&0&1&0\\0&0&0&1\\1&0&0&0\\0&1&0&0}\right]+d\left[\matrix{0&0&0&1\\1&0&0&0\\0&1&0&0\\0&0&1&0}\right]$$
$$B=a_0\cdot B_0+a_1\cdot B_1+a_2\cdot B_2+a_3\cdot B_3$$
where $(a_0,a_1,a_2,a_3)=(a,b,c,d)$
It is easily shown that $\forall i\in[0,3], B_i=P^i$ where $P=B_1=\left[\matrix{0&1&0&0\\0&0&1&0\\0&0&0&1\\1&0&0&0}\right]$
The characteristic polynomial of $P$ is:
$$\chi_P(x)=|x\cdot I_4-P|=\left|\matrix{x&-1&0&0\\0&x&-1&0\\0&0&x&-1\\-1&0&0&x}\right|=x\cdot\left|\matrix{x&-1&0\\0&x&-1\\0&0&x}\right|-(-1)\cdot\left|\matrix{-1&0&0\\x&-1&0\\0&x&-1}\right|$$
$$\Rightarrow \chi_P(x)=x^4-1$$
Thus the eigenvalues of $P$ are the 4-th roots of unity:
$$Sp(P)=\left\{e^{i(2k\pi/4)},k\in[0,3]\right\}=\left\{e^{i(k\pi/2)},k\in[0,3]\right\}$$
Thus:
$$Sp(a_j\cdot P^j)=\left\{a_j\cdot e^{i(jk\pi/2)},k\in[0,3]\right\}$$
In addition, the set of matrices $\left\{(a_j\cdot P^j),j\in[0,3]\right\}$ commutes and $$B=\sum_{j=0}^3 a_j\cdot P^j$$
Hence
$$Sp(B)=\left\{\sum_{j=0}^3 a_j\cdot e^{i(jk\pi/2)},k\in[0,3]\right\}$$
After simplification, you get the eigenvalues of $B$:
$$\Rightarrow\left\{\array{\lambda_0&=&a+b+c+d\\\lambda_1&=&(a-c)+i(b-d)\\\lambda_2&=&a-b+c-d\\\lambda_3&=&(a-c)+i(d-b)&=&\bar{\lambda_1}}\right.$$

Note 1: You indeed get $a+b+c+d$ as an eigenvalue, and $4a$ as the sum of the eigenvalues, as you mentioned in your question ;)
Note 2: This method can be generalised to any circulant matrix of size $n$, for which you would then use the n-th roots of unity.
