This article is expected to be the final installment of this series. While I may continue to discuss linear algebra in the future, those discussions will be presented independently. This post primarily focuses on the topic of similar matrices. The perspectives shared here are somewhat rough and even a bit fuzzy to me, so I invite readers to read carefully. In Meng Yan’s articles, he provided a brilliant description of matrices and their similarity:
"A matrix is a description of a linear transformation in a linear space. In a linear space, as long as we select a basis, any linear transformation can be described by a specific matrix."
Similarly, for a linear transformation, as long as you choose a basis, you can find a matrix to describe it. Change the basis, and you get a different matrix. All these matrices are descriptions of the same linear transformation, yet none of them are the linear transformation itself.
All these matrices that describe the same linear transformation are called similar matrices. Meng Yan also mentioned that the formula for similar matrices could be proven in a very intuitive way, but he did not elaborate further. I have not been in contact with him, but I have been seeking an intuitive understanding of this myself. After consulting many books, I finally found an answer that satisfies me. What might surprise readers is that the source of this understanding was not a mathematical work, but a physics-oriented book titled Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra.
Part 1
First, let’s look at a more physical understanding: A matrix A describes a motion from vector x to vector y, i.e., y = Ax. However, this is only the measurement in a Cartesian coordinate system. In a new coordinate system P, suppose the measurement result is y' = Bx'.
According to the geometric understanding of matrices we established previously, the vector x' measured in coordinate system P is represented as x in the Cartesian system, which can be expressed as Px' = x. Similarly, we have Py' = y. Substituting these into the original equation, we get: Py' = APx' This can be slightly rewritten as: 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. This is exactly a similar matrix of A! Therefore, a family of similar matrices is simply the measurement results of the same linear transformation under different coordinate systems.
Part 2
In fact, similar matrices also have a relatively intuitive geometric 3D model. We know that a matrix A consists of n column vectors, which actually define an n-dimensional parallelepiped in n-dimensional space (analogous to a parallelogram in 2D or a parallelepiped in 3D). The identity matrix I defines an n-dimensional unit cube. Suppose there is a certain correspondence between them.
The measurement result of matrix A in the new coordinate system P is P^{-1}A, meaning A = P(P^{-1}A). The measurement of I in P is I = P(P^{-1}). This means that in the new coordinate system, P^{-1} corresponds to P^{-1}A. So, what does the unit cube in the new coordinate system correspond to? It follows the mapping: \begin{aligned} P^{-1} &\to P^{-1}P = I \\ P^{-1}A &\to P^{-1}AP \end{aligned}
That is to say, the unit cube in the new coordinate system corresponds to the n-dimensional parallelepiped described by the similar matrix!
This is essentially the "pairing principle"!
From this, it is not difficult to understand why the determinants of similar matrices are identical. The geometric meaning of a determinant is volume. Although the volume of the parallelepiped represented by matrix A changes after a coordinate transformation, the volume of the unit cube also changes. In other words, while all scales change in the new coordinate system, from the perspective of "counting the grids," the number of grids remains unchanged. Therefore, the relative volume remains the same.
The Greatness of Matrices
In physics, matrices are widely used in almost every field. However, the discipline most closely linked to matrices is quantum mechanics. Many people know that quantum mechanics has three equivalent forms of expression: Schrödinger’s wave equation (which I am currently studying), Heisenberg’s matrix mechanics, and finally, the path integral formulation by the genius Feynman. It is said that when Heisenberg was conceiving quantum mechanics, the field of linear algebra was already well-developed, but he had not studied it. Instead, he "invented" something he called "energy tables" to serve as a tool for his conceptualization. When he finally submitted his paper to his advisor, Max Born, Born told him bluntly: "Your new energy table is exactly the matrix that mathematicians have long studied." It is fascinating that the founder of matrix mechanics initially did not know what a matrix was. Later, Heisenberg studied matrix theory and collaborated with his advisor to publish the results of matrix mechanics.
Recently, while looking at quantum mechanics and special relativity, I found that the descriptive methods of both have been largely unified. Both usually start with basic knowledge, followed by linear algebra, group theory, and so on. Ultimately, they both rely on matrices and group theory for analysis. I believe this is necessary for a unified description of physics. What surprised me is that this integrated abstract model actually feels easier to grasp. Perhaps it is because I am a mathematics enthusiast.
Finally, to summarize these few posts in the "New Understanding of Matrices" series:
These articles are quite rough, bold, and immature. Some viewpoints may not even be entirely correct, as intuitive understanding can sometimes lead to overgeneralization and ignore the powerful role of abstraction. However, I believe that only after gaining an intuitive understanding can one use these tools more proficiently. A more comprehensive understanding is gradually realized and accumulated through such intuitive effects. I believe the development of linear algebra in mathematical history followed a similar path. If so, why shouldn’t we learn it in a way that mirrors its historical development?
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