# understanding the matrix transpose

Can you explain me, please, what does it mean the transpose of a matrix ? I know the definition in the context of matrix theory and its generalization to adjoint operators (transpose of a linear application). What is the fundamental idea behind transpose ? and why it is introduced and considered in today's mathematics ? Thank you

The study of symmetric differential operators predates the systematic study of symmetric matrices. While studying differential operators such as $$Lf = -\frac{d}{dx}p\frac{d}{dx}f + V f$$ on an interval $[a,b]$ with two endpoint conditions such as (a) $f(a)=f(b)=0$ or (b) $f(a)=f(g)$, $f'(a)=f'(b)$, it was found that $$\int_{a}^{b} (Lf)g\,dx = \int_{a}^{b}f(Lg)\,dx,$$ provided $f$, $g$ satisfy the endpoint conditions. And it was this property that was isolated as the reason for the integral 'orthogonality' of eigenfunction solutions of $Lf=\lambda f$. For example, the classical Fourier series functions are eigenfunctions of $L=-\frac{d^{2}}{dx^{2}}$ with periodic endpoint conditions on $[0,2\pi]$: $$1,\cos x,\sin x,\cos 2x,\sin 2x,\cos 3x,\sin 3x,\cdots\;.$$ This is an infinite-dimensional space of functions with infinitely many eigenvalues $0,1^{2}, 2^{2}, 3^{2},\cdots$. If you take any two of these which are different--call them $f$ and $g$--then $\int_{0}^{2\pi}fg\,dx=0$. And this happens for a broad class of eigenfunction solutions associated with such an operator $L$, including Bessel functions, Legendre Polynomials, Hermite Polynomials, etc..
The study of eigenfunction solutions came out of Fourier's separation of variables technique from ~1805. The trick they found to finally explain the general orthogonality of solutions of these ODEs was this: If $Lf=\lambda f$ and $Lg=\mu g$ with $\mu \ne \lambda$, then $$(\lambda-\mu)(f,g) = (Lf,g)-(f,Lg) = 0.$$ This abstract pairing eventually became a general inner product $(\cdot,\cdot)$ as shown above, and symmetric operators became a focus because of the properties similar to Fourier series expansion in orthogonal functions $1,\cos x,\sin x,\cos 2x,\sin 2x,\cdots$ on $[0,2\pi]$.
In the context of matrices, a matrix $A$ which is equal to its transpose is the natural symmetric operator, because $$y^{T}(Ax) = (A^{T}y)x.$$ Written as an inner product $(x,y) = y^{T}x$, this becomes $(Ax,y)=(x,A^{T}y)$. So symmetry in this context is $A=A^{T}$, which is the same as symmetry across the diagonal. Like a symmetric differential operator, a symmetric matrix also has a basis of eigenfunctions which are mutually orthogonal for different eigenvalues; this is easily shown using the same trick used to show orthogonality for eigenfunction solutions of symmetric ODEs coming from separation of variables--same notation, just different objects.