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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

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I'll try to explain some of the History, which is unusual because it started at the highest level of abstraction and worked its way down.

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.

One of the reasons this seems so foreign at the level of matrices is that it's coming from a much more abstract infinite-dimensional setting where symmetry arises naturally in the study of differential equations of Physics and Engineering. But because of this natural connection, there are many applications of symmetric matrix theory as well.

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