$$

Vieta's formula

Problem

Let

$$A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$$

and

${\lambda }_1,{\lambda }_2,{\lambda }_3$ its eigenvalues. Find the values of

$$\begin{array}{rl}a& ={\lambda }_1+{\lambda }_2+{\lambda }_3,\\ b& ={\lambda }_1{\lambda }_2+{\lambda }_1{\lambda }_3+{\lambda }_2{\lambda }_3,\\ c& ={\lambda }_1{\lambda }_2{\lambda }_3.\end{array}$$

Thought

For any

$n\times n$ matrix

$A$, its eigenvalues

${\lambda }_1,\dots ,{\lambda }_n$ are the roots of its characteristic polynomial

$p_A(x)=det(A-xI)$. Therefore, we have

$$\begin{array}{rl}{p}_A(x)& =(-x{)}^n+s_1(-x{)}^{n-1}+\cdots +s_n\\ & =({\lambda }_1-x)({\lambda }_2-x)\cdots ({\lambda }_n-x),\end{array}$$

where

$$s_k=\sum _{\begin{array}{c}\alpha \subseteq [n]\\ |\alpha |=k\end{array}}det(A[\alpha ]).$$

Note that it is

$({\lambda }_i-x)$ instead of

$(x-{\lambda }_i)$ since the leading term of

$p_A(x)$ is

$(-x{)}^n$.

By comparing the coefficients (also known as the Vieta's formula, we have

$$\begin{array}{rl}{s}_1& ={\lambda }_1+{\lambda }_2+\cdots +{\lambda }_n,\\ s_2& ={\lambda }_1{\lambda }_2+{\lambda }_1{\lambda }_3+\cdots +{\lambda }_{n-1}{\lambda }_n,\\ ⋮& =\\ s_n& ={\lambda }_1{\lambda }_2\cdot \cdots \cdot {\lambda }_n.\end{array}$$

Sample answer

By comparing the coefficients, we know that

$a={\lambda }_1+{\lambda }_2+{\lambda }_3=s_1$, which is the trace of the matrix. Thus,

$a=\mathrm{tr}(A)=1+5+9=15$.

Next,

$b={\lambda }_1{\lambda }_2+{\lambda }_1{\lambda }_3+{\lambda }_2{\lambda }_3=s_2$. All the

$\alpha \subseteq [3]$ of order

$2$ are

$\{1,2\}$,

$\{1,3\}$, and

$\{2,3\}$. The corresponding

$A[\alpha ]$ are

$$\left[\begin{array}{cc}1& 2\\ 4& 5\end{array}\right],\left[\begin{array}{cc}1& 3\\ 7& 9\end{array}\right],\left[\begin{array}{cc}5& 6\\ 8& 9\end{array}\right]$$

with their determinans

$-3$,

$-12$, and

$-3$. Thus,

$b=s_2=-3-12-3=-18$.

Lastly,

$c={\lambda }_1{\lambda }_2{\lambda }_3=s_3$, which is the determinant of

$A$. Since

$A$ is singular,

$c=det(A)=0$.

This note can be found at Course website > Learning resources.