Assuming without loss of generality that a real symmetric matrix is in fact diagonal Since we have that all real symmetric matrices are diagonalizable, is it safe to write when solving a problem that we assume without loss of generality that the matrix is in fact diagonal?
 A: This will strongly depend on the property that you want to prove. For example, if you wanted to prove that all symmetric matrices have only zeroes outside of the diagonal (which is obviously false), you can't assume that the matrix is diagonal.
A: It would be safe if you bare in mind what similarity transformation preserves.
To repeat the list in Wikipedia:

Similar matrices share any properties that are really properties of the represented linear operator:
  
  
*
  
*Rank
  
*Characteristic polynomial, and attributes that can be derived from it:
  
*Determinant
  
*Trace
  
*Eigenvalues, and their algebraic multiplicities
  
*Geometric multiplicities of eigenvalues (but not the eigenspaces, which are transformed according to the base change matrix $P$ used).
  
*Minimal polynomial
  
*Elementary divisors, which form a complete set of invariants for similarity
  
*Rational canonical form
  

A: I would be very careful to stick to formalism. 
"Since $M$ is a real symmetric matrix, it is diagonalizable. As such, there exists an invertible matrix $P$ such that $PAP^{-1}=D$, where $D$ is a diagonal matrix."
That should be fine, but keep in mind Dominik's excellent answer, and also that the process of diagonalizing can destroy some of your nice properties, since it essentially amounts to a change of basis, and most advanced proofs make use of properties such as orthonormal bases of eigenvectors, and such.
