Context
I am studying normal modes oscillations and normal modes [1,2].
I am trying to complete a proof to show that the eigenvalues (i.e., the eigenvalues are positive). There is a proof in [2], but I find it verbose and likely incorrect. There is a short proof of this in [4]. Nonetheless, I proceed with my questions on the topic.
Questions
Question 1
In my understanding an eigenvalue equation [3], I write the following definition.
Definition
If $T$ is a linear transformation from a finite-dimensional vector space $\mathbb{C}^n$ over the field of complex number into itself and $\mathbf{v}$ is a nonzero vector in $\mathbb{C}^n$, then $\mathbf{v}$ is an eigenvector of $T$ if $T\,\mathbf{v}$ is a scalar multiple of $\mathbf{v}$. This can be written as the eigenvalue equation
$$T(\mathbf{v}) = \lambda \mathbf{v},$$
where $\lambda$ is a scalar in $\mathbb{C}$, known as the eigenvalue associated with $\mathbf{v}$.
However, in the analysis of normal mode of dynamical systems, I have $n$-dimensional real symmetric matrices $\mathbf{V}$ and $\mathbf{T} $, and the "eigenvalue equation"
$$\mathbf{V}\,\mathbf{a} = \lambda\,\mathbf{T}\,\mathbf{a}.\tag{1}$$
In my mind, if $T$ is invertible, I can make an actual eigenvalue equation
$$\left( \mathbf{V}\,\mathbf{T}^{-1}\right)\left(\mathbf{T}\,\mathbf{a}\right) = \lambda\,\left(\mathbf{T}\,\mathbf{a}\right).\tag{2}$$
Now, I would have that $\mathbf{T}\mathbf{a}$ is an eigenvector of $\mathbf{V}\mathbf{T}^{-1}$ since $\mathbf{V} \mathbf{a}$ is a scalar multiple of $\mathbf{T}\mathbf{a}$.
So my first series of questions:
(1) Can we agree that, as written, Equation 1 is not actually an eigenvalue equation?
(2) Does this type of equation have a name? If so what is it?
Question 2
In [2] there is a claim that $\lambda$ is always finite and positive. It comes down to faith. I really like [2], but when it comes to such matters as the proofs of mathematical claim, I just have no faith in [2]. For example, there is even an example later in the book when one of the eigenvalues is zero valued.
(3) How can we prove or disprove the following propositions? [Please note that both propositions cannot be proven to be true. An amended proposition is given and proven in OP's answer below.]
Proposition. Given two $n$ dimensional real symmetric matrices $\mathbf{T}$ and $\mathbf{V}$ show that any solution of the equation
$$\mathbf{V}\,\mathbf{a} = \lambda\,\mathbf{T}\,\mathbf{a}$$
has a scalar lambda that is positive.
Proposition. Given two $n$ dimensional real symmetric matrices $\mathbf{T}$ and $\mathbf{V}$ show that any solution of the equation
$$\mathbf{V}\,\mathbf{a} = \lambda\,\mathbf{T}\,\mathbf{a}$$
has a scalar lambda that is non-negative.
These two propositions are closely related. Thus, I assume that some other predicates need to be set in order to distinguish the proofs.
Bibliography
[1] https://en.wikipedia.org/wiki/Normal_mode
[2] Goldstein, "Classical Mechanics," 3rd edition, page 242-3.
[3] https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors#Formal_definition
[4] Prove that the eigenvalues for $A\vec{w} = \lambda B\vec{w}$ are real and positive
Best Answer
I believe it's called a generalized eigenvalue equation.
If $T$ is positive definite, you can write the equation as $$T^{-1/2} V T^{-1/2} w = \lambda w $$ where $w = T^{1/2} v$. The advantage of this over the form $T^{-1} V v = \lambda v$ is that $T^{-1/2} V T^{-1/2}$ is symmetric (and positive definite if $V$ is).
Your propositions are not true. Try $V = I$ (the identity matrix) and $T = -I$, or vice versa.