In the particular example in the question, the properties of a symmetric matrix have been confused with those of a positive definite one, which explains the discrepancies noted.
A brief tour of symmetry and positive semidefiniteness
Symmetric positive (semi)definite matrices play an important role in
statistical theory and applications, making it useful to briefly
explore some of their properties and from whence they arise. The proofs can be made reasonably short, making it feasible to do this exploration here.
A word on notation. Let $\newcommand{\reals}{\mathbb R}A$ be an $n \times n$ matrix
with real-valued entries. Often, this is denoted $A \in
M_n(\reals)$. In what follows, $\lambda$ will be an eigenvalue of $A$
and $v$ will be a corresponding unit eigenvector, i.e., $A v = \lambda
v$ and $\|v\|_2 = 1$. Even though $A$ is real-valued, $\lambda$ and $v$ both might be
complex-valued, which is an important point to keep in mind. The notation $v^T$ denotes the transpose of $v$ if it
is real-valued and $v^*$ denotes the conjugate transpose. The conjugate of $z \in \mathbb C$ is denoted $\bar z$.
Spectral decomposition and symmetry
For this part, we'll assume $A$ is symmetric, that is, $A = A^T$.
Theorem 1 The eigenvalues of $A$ are real.
Proof: We have $$ \lambda = \lambda v^* v = v^* A v = \sum_i a_{ii} v_i \bar v_i + \sum_{i < j} a_{ij} (v_i \bar v_j + \bar v_i v_j) \>,$$ where the last equality follows from the fact that $a_{ij} = a_{ji}$. Note that all of the terms in each sum on the right-hand
side are real. Hence, $\lambda$ is real.
The result above gives our first hint at what any potential eigendecomposition must look like. To get a further hint, consider the following.
Theorem 2 The eigenvectors corresponding to distinct eigenvalues are orthogonal.
Proof: Let $\lambda_1 \neq \lambda_2$ with corresponding $v_1$ and $v_2$. Then $$ \lambda_1 v_1^* v_2 = v_1^* A^* v_2 = v_1^* A v_2 = \lambda_2 v_1^* v_2 \>, $$ so, $(\lambda_1 - \lambda_2) v_1^* v_2 = 0$ and since $\lambda_1 \neq \lambda_2$, we must have $v_1^* v_2 = 0$.
So now we know that distinct eigenvalues yield orthogonal eigenvectors. But, we still don't know that the eigenvectors themselves are (i.e., can be chosen to be) real-valued.
Theorem 3 The eigenvectors of each eigenvalue can be chosen to be real-valued.
Proof: Decompose $v = u + i w$ where $u$ and $w$ are real-valued vectors. Then, $$ \lambda u + i \lambda w = \lambda v = A v = Au + i A w \>, $$ and $\lambda$ and $A$ are both real-valued, hence $A u = \lambda u$ and $A w = \lambda w$. This shows that
if $v$ is an eigenvector of $A$, so is $u$ (and $w$, for that matter).
So, we've got some real eigenvalues and some real eigenvectors that are orthogonal whenever the eigenvalues are distinct. Now we handle the case where a particular eigenvalue has multiple linearly independent eigenvectors.
Theorem 4 The (real) eigenvectors of a common eigenvalue form a vector subspace of $\reals^n$.
Proof: If $A v_1 = \lambda v_1$ and $A v_2 = \lambda v_2$, then for any $\alpha$ and $\beta$, we have $$ A (\alpha v_1 + \beta v_2) = \alpha A v_1 + \beta A v_2 = \lambda (\alpha v_1 + \beta v_2) \>, $$
and so the set of vectors $\mathcal V = \{v: A v = \lambda v\}$ is a linear subspace.
Since every finite-dimensional real-valued linear subspace of
$\reals^n$ has an orthogonal basis, this is enough to conclude that
any symmetric $A$ satisfies $A V = V \Lambda$ where $V$ is an
orthogonal matrix and $\Lambda$ is a real-valued diagonal matrix. Note
that if it happens that there are less than $n$ nonzero eigenvalues,
that we can "fill out" the columns of $V$ with an orthogonal basis
of the remaining subspace and place zeros along the diagonal of $\Lambda$ in the corresponding locations.
Hence, there is some orthogonal $V$ and some real-valued diagonal
$\Lambda$ such that
$$
A = V \Lambda V^T \>.
$$
We can also see that the converse is trivially true. If $A = V \Lambda
V^T$, then $A^T = (V \Lambda V^T)^T = V \Lambda V^T = A$ and so $A$ is
symmetric.
But, what about positive semidefiniteness?
So far, we've said nothing about positive definiteness; all of the properties we have derived deal purely with symmetry. Now, we'll try to briefly develop the properties of positive definite matrices and make some connections to symmetry.
A matrix $A$ (not necessarily symmetric!) is called positive semidefinite if for all $x \in \reals^n$, we
have $x^T A x \geq 0$. If $x \neq 0$ further implies that $x^T A x>
0$, then $A$ is called positive definite. This arises
frequently in statistics in the study of quadratic forms.
Theorem 5 Let $A \in M_n(\reals)$. Then, there exists a symmetric matrix $B$ such that $x^T A x = x^T B x$ for all $x \in \reals^n$.
Proof: $ x^T A x = (x^T A x)^T = x^T A^T x$ so choose $B = \frac{1}{2}(A + A^T)$.
Notice that there is nothing in the statement of the theorem about $A$ being positive semidefinite; it's completely general. What this says, though, is that when considering quadratic forms, we can always
implicitly assume that $A$ is symmetric. This motivates the fact that
oftentimes authors will assert that $A$ is symmetric immediately in the definition
of positive semidefiniteness.
Theorem 6 A symmetric positive semidefinite matrix $A$ has nonnegative eigenvalues.
Proof: If $v$ is a unit eigenvector corresponding to $\lambda$, then $$ \lambda = \lambda v^T v = v^T A v \geq 0 \>. $$
As an exercise, you should prove that if $A$ is symmetric positive definite,
then all eigenvalues must be strictly positive.
A counterexample
While the above shows that we can conclude that a symmetric $A$ having real positive eigenvalues is positive definite, this does not hold if we drop the symmetry requirement. Let
$$
A = \left(\begin{array}{rr}1 & -4 \\ 0 & 1 \end{array}\right) \>.
$$
Then $A$ has an eigenvalue of 1 (with algebraic multiplicity of 2), but by taking $x$ to be a vector of ones we see
that $A$ is not positive semidefinite. The motivation for this example can be found in this question and answer.
