"New Understanding of Matrices, Part 4": On Similar Matrices

《新理解矩阵4》：相似矩阵的那些事儿 苏剑林 · 2012-11-11 · Mathematics algebra matrix linear

This post is probably going to be the last one in this series—I might come back to linear algebra later, but if so, it'll be as a standalone piece. This article is mainly about similar matrices. My views here are quite rough, and honestly even feel a bit hazy to me, so I'd ask readers to read carefully. In Meng Yan's article, there's a wonderful description of matrices and their similarity:

"A matrix is a description of a linear transformation on a linear space. Within a linear space, once we fix a basis, any linear transformation can be described by a determinate matrix."
Likewise, for a given linear transformation, once you fix a basis, you can find a matrix describing that transformation. Change the basis, and you get a different matrix. All these matrices are descriptions of one and the same linear transformation, yet none of them is the linear transformation itself.

All the matrices described above—descriptions of the same linear transformation—are similar to one another. Meng Yan also mentioned that the formula for similar matrices can be proven in a very intuitive way, but he never followed up on that. I never got in touch with him about it, but I've long been looking for an intuitive understanding of this myself. After going through many books, I finally arrived at an answer I'm fairly satisfied with. What might surprise readers is that what got me there wasn't a mathematics text, but a book leaning more toward physics: Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (on the symmetry connections between group theory and quantum mechanics).

Part 1

First, let's take a more physics-flavored view. Matrix A describes a motion from vector x to vector y, i.e., $y=Ax$. But this is only as measured in the standard (Cartesian) coordinate system. In a new coordinate system P, suppose the measured result is $y'=Bx'$.

Based on the geometric understanding of matrices we gave earlier, a vector measured as $x'$ in coordinate system P is measured as $x$ in the standard system, which can be written as $Px'=x$; similarly we have $Py'=y$. Substituting these in gives $Py'=APx'$, which can be rearranged slightly into $Py'=P(P^{-1}AP)x'$. In other words, in coordinate system P, the motion from $x'$ to $y'$ is represented by the matrix $B=P^{-1}AP$—and this is precisely a similar matrix of A! So a family of similar matrices is nothing more than the measured result of one and the same linear transformation under different coordinate systems.

Part 2

Actually, there's a somewhat more intuitive solid geometric model for similar matrices as well. We know a matrix A consisting of n column vectors actually defines an n-dimensional parallelepiped in n-dimensional space (analogous to the 2D parallelogram and 3D parallelepiped). The identity matrix I, meanwhile, defines an n-dimensional unit cube. Suppose there's some correspondence between the two.

Now, the measured result of matrix A under the new coordinate system P is $P^{-1} A$, i.e., $A=P(P^{-1}A)$; and the measured result of I under P is $I=P(P^{-1})$. In other words, under the new coordinate system, $P^{-1}$ corresponds to $P^{-1}A$. So what does the unit cube correspond to under the new coordinate system? It's this:

$$\begin{aligned}P^{-1} \to P^{-1}P=I \\ P^{-1}A \to P^{-1}AP\end{aligned}$$

In other words, the unit cube under the new coordinate system corresponds exactly to the n-dimensional solid described by the similar matrix!

This is just the pairing principle at work, plain and simple!

This also makes it easy to understand why similar matrices always have the same determinant. The geometric meaning of the determinant is volume. Although the solid represented by matrix A does change in volume after the coordinate transformation, the volume of the unit cube changes too—that is, under the new coordinate system, all the scales have changed. But from the "counting grid cells" point of view, the number of grid cells hasn't changed, so the volume, in that sense, hasn't changed either.

The Great Matrix

In physics, matrices are used broadly in almost every field, but the discipline most tightly bound up with matrices has to be quantum mechanics. Many people know that quantum mechanics has three equivalent formulations: Schrödinger's wave equation (which is what I'm currently studying), Heisenberg's matrix mechanics, and the ingenious Feynman's path integral. As it happens, when Heisenberg was working out quantum mechanics, linear algebra as a subject had already been developed quite thoroughly—but he himself had never studied it. Instead, he "invented" his own thing, which he called an "energy table," as a tool to describe his ideas. When he finally submitted his paper to his advisor, Born, Born told him bluntly: "This new energy table of yours is exactly what mathematicians call a matrix, and they've already studied it extensively." Funny enough—the founder of matrix mechanics didn't even know what a matrix was! Heisenberg later caught up on matrix theory and, together with his advisor, published the results of matrix mechanics.

Recently, while reading about quantum mechanics and special relativity, I've noticed that the two are, to a large extent, presented in a unified way: both start with some foundational material, then move on to linear algebra, group theory, and so on. In the end, both essentially boil down to using matrices and group theory to do the analysis. I suspect this reflects physics' underlying need for a unified mode of description. What's struck me as a bit unexpected is that this kind of synthesized, abstract approach actually feels easier for me to get into—perhaps precisely because I'm a math enthusiast at heart.

Finally, let me wrap up this series of New Understanding of Matrices posts.

These articles have been rough, unrestrained, and rather unpolished, and some of the views expressed may well not be correct—because intuitive understanding tends to give a one-sided impression, glossing over the enormous power of abstraction. But I believe it's only once we've built up an intuitive grasp that we can wield the more abstract tools with real facility; a fuller understanding is something that gradually dawns on us, and accumulates, on the back of these intuitions. I suspect the historical development of linear algebra followed a similar path—so why shouldn't we learn it the way history unfolded it?