Decompose a real symmetric matrix Prove that, without using induction, A real symmetric matrix $A$ can be decomposed as $A = Q^T \Lambda Q$, where $Q$ is an orthogonal matrix and $\Lambda$ is a diagonal matrix with eigenvalues of $A$ as its diagonal elements.
I can see that all eigenvalues of $A$ are real, and the corresponding eigenvectors are orthogonal, but I failed to see that when putting all (interesting) eigenvectors together, they form a basis of $\mathbb{R}^n$.
Edit
The reason I asked this question is to show that a real symmetric matrix is diagonalizable, so let's not use that fact for a while. Other than that, any undergraduate level linear algebra can be used. 
Edit 2
After reading Algebraic Pavel's answer, I feel like ruling out Schur Decomposition as well, but I can't keep ruling out theorems, so...if a proof is too obvious, that's probably not what I am looking for, though, it maybe a technically correct answer.
Thanks.
 A: Provided that the Schur decomposition is an allowed tool:
Using the Schur decomposition, we have that there exists an orthogonal $Q$ and an upper triangular $R$ such that $A=QRQ^T$. Since $A$ is symmetric, $Q^TAQ=R$ is symmetric as well. Therefore $R$ is symmetric. A symmetric triangular matrix is necessarily diagonal.

There is also a neat theory behind tridiagonal matrices, which can help:
It is easy to show that for any real $A$ there is an orthogonal matrix $Q$ such that $Q^TAQ=H$, where $H$ is upper Hessenberg. If $A$ is symmetric it then follows that $H$ is symmetric as well and hence tridiagonal. 
Now if the tridiagonal matrix $H$ is unreduced (none of the upper and lower diagonal entries are zero), then the eigenvalues of $H$ (and therefore of $A$) are distinct. Equivalently, if $A$ has repeated eigenvalues then $H$ is reduced (some upper and lower (symmetrically) diagonal entries are zero and hence $H$ is a block diagonal matrix). Consequently, one must have that each repeated eigenvalue must be in different unreduced diagonal blocks of $H$. In each of these blocks we must find at least one eigenvector (actually, only one) and it is almost trivial to show that these eigenvectors must be linearly independent (therefore, we can orthogonalize them).
A: A matrix $Q$ is orthogonal if and only if its columns forms a orthonormal basis, if and only if $Q^{-1}=Q^T$. 
Therefore, if there exists an orthornomal basis of eigenvectors of $A$, we have
that the matrix of change of basis if ortogonal. That is to say, there is $Q$ orthogonal so that
$Q^{-1}AQ=\Lambda$ 
But then $A=Q^{-1}\Lambda Q=Q^T\Lambda Q$
It remains to show that an orthonormal basis of eigenvectors exists.
Eigenspaces corresponding to different eigenvalues are orthogonal. As A is diagonalizable we have $\mathbb R^n=V_1\oplus V_2\dots\oplus V_k$ where $V_i$ is the eigenspace corresonding to the eignevalue $\lambda_i$.
Each $V_i$ has a orthonormal basis $v_{i,1},\dots,v_{i,n_i}$ where $n_i=\dim V_i$. (from the comments we know that we can use this fact.)
Thus, putting toghether these basis we got an orthonormal basis $v_{i,j}$ of $\mathbb R^n$ consisting of eigenvectors of $A$. 
Edit
If you cannot use that $A$ is diagonalizable, then use the following workaround to shot that in fact $A$ is diagonalizable.
Write $\mathbb R^n=V_1\oplus\dots\oplus V_k\oplus W$. Where the $V_i$ are the eigenspaces of $A$ and $W$ is the orthogonal complement of $V_1\oplus\dots\oplus V_k$.
Then, the resctriction of $A$ to $W$ is symmetric and has no eigenvectors.
Now, consider the function
$\max \langle Aw,w\rangle$ when $w$ runs on the spaces of unitary vectors of $W$ (that is to say, $||w||=1$). 
Any max point is an eigenvector, contradicting the fact that $A|_W$ has no eigen vectors. (See below.) Therefore $W$ contains no unitary vectors, hence $W=\{0\}$ and 
$V_1\oplus\dots\oplus V_k=\mathbb R^n$.
Let's see that max points are eigenvectore. We restrict to $W$ a and
Let $F(w)=\langle Aw,w\rangle$ then the derivative of $F$ at point $w$, in the direction $v$ is $dF_w[v]=2\langle Aw,v\rangle$ (because $A$ is symmetric). The tangent space of $\{||w||=1\}$ at the point $w$ is $w^\perp$. Thus, $w$ is a critical value if and only if $dF_w[v]=0\forall v\in w^\perp$. That is to say, $\langle Aw,v\rangle=0$ for all $v$ such that $\langle w,v\rangle=0$. Therefore $Aw$ must be a multiple of $w$ as the orthogonal ov $w$ is contained in the orthogonal of $Aw$. 
Edit another way to see that $A|_W$ has an eigenvector is to consider its characteristic polynomial. It has roots in $\mathbb C$ because of the fundamental theorem of algebra. They are reals because $A|_W$ is symmetric. So $A|_W$ has at least one eigenvalue, whence eigenvector.
