Importance of Decomposing Linear Operators

Question

I was just wondering what could be the real importance of decomposing linear operators into simpler forms. My instincts say it is important beacause we can easily calculate eigenvalues. I don't know exactly, kindly help

thanx and regards

Answer 1

Decomposing operators has its origin in Fourier analysis. The earliest example is the vibrating string fastened at points $x=0$, $x=1$ and stretched in between. This led to the wave equation $$ \frac{\partial^2 u}{\partial t^2} = c\frac{\partial^2 u}{\partial x^2} $$ where $c$ is a physical constant associated with the string, and $u$ is the displacement of the string from rest for $0 \le x \le 1$ and $t \ge 0$. This is an old problem from the 1700's, and they found that there were solutions of the form $$ u_n(x,t) = \left(A_n\cos(n\pi c t)+B_n\sin(n\pi ct)\right)\sin(n\pi x). $$ These effectively "diagonalize" $\frac{\partial^2}{\partial x^2}$ with $0$ endpoint conditions to find function $\sin(n\pi x)$. Then, using a sum, they wrote a more general solution as a sun of the $u_n$. These standing way solutions were called harmonics (which is where the term Harmonic Analysis came from,) and the need to find coefficients $A_n,B_n$ to match initial conditions $$ u(x,0)=f(x),\;\; u_t(x,0)=g(x), $$ led to what is now called Fourier analysis: $$ \sum_{n=1}^{\infty}A_n\sin(n\pi x)=f(x),\;\; \sum_{n=0}^{\infty}B_nn\pi\sin(n\pi x) = g(x). $$ For a long time it was believed that these conditions placed restrictions on the possible $f$, $g$ that could be allowed. Fourier conjectured the opposite by claiming that basically every mechanical function could be written in a trigonometric series. Eventually this led to general Fourier Analysis, diagonalization of operators and Spectral Theory. And, oddly enough, the infinite dimensional problem here came well before diagonalization of matrices. Symmetric operators also arose out of problems like this. Fourier's separation of variables introduce a parameter, and gave us what is now referred to as an eigenvalue/eigenvector problem.

Answer 2

Linear operators have proven to be applicable to such a large variety of problems it's hard to give all the examples, but for something less computational Orthographic projections are one example of decomposing a space into subspaces that has an entirely geometric purpose, technical drawings for engineering. Several two dimensional subspaces are used to give you the view from the top, bottom, left, right, front and back sides of the object so you can figure out what it should look like as a whole.These type of drawings are very common in architecture, engineering and construction.