Question

The first problem April 2014: http://math.unt.edu/problem-of-the-month

Squared Matrices and Characteristic Polynomials. Let A be a real 4×4-matrix with characteristic polynomial p(λ)=det(λI−A)=λ4−s1λ3+s2λ2−s3λ+s4. Show that if A admits real square roots, in the sense that there is a real 4×4-matrix S such that S2=A, then s1+s2+s3+s4≥−1.

Answer 1

@ikram002p @dan815 @Somy

Answer 2

The thing I'm thinking of is \[S=\left[\begin{matrix}a & b&c&d \\ e&f&g&h \\ i&j&k&l \\ m&n&o&p\end{matrix}\right]\]

Take this and square it:
\[\LARGE S^2=A\]

\[\Large \det( \lambda I - A)=\det(\lambda I - S^2)\]

We can compute this and compare the coefficients on both polynomials.

Answer 3

but that will be a hassle.

Answer 4

( ˘ ³˘)❤