implied eigenvalue equations for an arbitrary, symmetric and positive definite matrix

Question

I have a matrix $M$, where $M$ = $\begin{bmatrix}a & b\\c & d\end{bmatrix}$. It is known that $M$ is symmetric and positive definite. Also, it is known that $x^TMy$ is a valid dot product in $R^2$. I know that to characterize $M$ by its eigenvectors, it can be shown as $Mv_i$ = λ$v_i$.

I was told in my class that based on this context, we can also gather the following information regarding the eigenvalues of this matrix:

$a - c = λ_1$ , $c - d = -λ_1$ , $a + c = λ_2$ and $c + d= λ_2$.

Why is this true? Is this always the case?

Answer 1

Let $M=\begin{bmatrix} a & c \\ c & d\end{bmatrix}$ be a symmetric matrix. The eigenvalues of this matrix are given by $$(a-\lambda)(d-\lambda)-c^2=0$$ $$\lambda^2-(a+d)\lambda +ad-c^2$$ Using the quadratic formula we obtain $$\frac{(a+d)\pm \sqrt{(a+d)^2-4(ad-c^2)}}{2}$$ $$\frac{(a+d)\pm \sqrt{a^2+d^2-2ad+4c^2}}{2}$$ Note that for the symmetric and positive definite matrix $M=\begin{bmatrix}2 & 1 \\ 2 & 3 \end{bmatrix}$ the quadratic formula will give us two irrational values, while the stated formula only gives whole values. So the stated formulae are not true in general but in just a few cases.

Case 1: $c=0$ $$\frac{(a+d)\pm \sqrt{a^2+d^2-2ad}}{2}$$ $$\frac{(a+d)\pm (a-d)}{2}$$ $$\lambda = a,d$$

Case 2: $a=d$ $$\frac{2a\pm \sqrt{4c^2}}{2}$$ $$a\pm c$$ $$\lambda = a-c, a+c$$

Answer 2

This is patently untrue for a general positive-definite matrix: Combining the first and third equations produces $\lambda_1+\lambda_2 = \operatorname{tr}(A)=2a$, while the second and fourth give $\operatorname{tr}(A)=2d$. Any diagonal matrix with different positive entires on the diagonal is a counterexample.

Note, too, that the statements that $A$ is symmetric and positive-definite, and that $x^TAy$ is a valid inner (not “dot”) product are equivalent.