In the previous article, I shared my understanding of matrices from a purely algebraic perspective. We saw that by assigning corresponding operational rules to matrices, they demonstrate immense utility in fields such as algebra and analysis. However, pure algebra is not enough. To achieve a more perfect understanding, it is best to find corresponding geometric objects. Only in this way can we intuitively grasp the concept and achieve mastery.
Geometric Understanding
I assume the reader has already read Meng Yan’s three articles on "Understanding Matrices," so I will not repeat many of the details. We know that a matrix A: \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} is actually composed of two vectors [a_{11}, a_{21}]^T and [a_{12}, a_{22}]^T (where vectors here are column vectors), which describe a planar (affine) coordinate system. In other words, these two vectors are actually the two basis vectors of this coordinate system. The operation y = Ax tells us what a vector x, defined in the coordinate system A, looks like in the coordinate system I. Here, the I coordinate system is the standard Cartesian coordinate system we most commonly use. That is to say, any vector (including the vectors within a matrix), as long as it does not have a matrix acting upon it, is measured within the Cartesian coordinate system.
(In fact, the statement that the identity matrix I is the default Cartesian coordinate system is not always true, but since we are seeking an intuitive way to understand it, we will use the simplest approach.)
Words alone may not clarify the issue; we need a diagram to explain:
The matrix A used in the diagram is: \begin{pmatrix} 3 & 2 \\ 1 & 3 \end{pmatrix} This forms an affine coordinate system. In this coordinate system, there is a vector x = [2, 2]^T. Its coordinates measured in the Cartesian coordinate system are [10, 8]^T. Now, it is easy to find that by calculating directly using matrix multiplication: Ax = [3 \cdot 2 + 2 \cdot 2, 1 \cdot 2 + 3 \cdot 2]^T = [10, 8]^T This is exactly what we expected!
Why does it have such characteristics? In fact, this stems from our definition of matrix multiplication; conversely, if we define matrix multiplication in this geometric way, we will also obtain the matrix multiplication calculation formulas found in textbooks. Higher-order matrices can be understood through the same analogy. The derivation process is a very simple exercise; perhaps the reader would like to try it themselves?
Now we return to the perspective in Meng Yan’s articles. Regarding a matrix acting on a vector (corresponding to a point), we can either view it as the point remaining stationary while the coordinate system transforms from Cartesian to affine; or, we can view it as the matrix "moving" (transforming) a point A' in the Cartesian system to a point A. Both interpretations are valid, as Meng Yan says, "motion is relative." More accurately, both interpretations should be mentioned simultaneously for the best understanding. A matrix is a transformation from one point to another, and the method of transformation is the transformation of the coordinate system.
Of course, the above only discusses the multiplication of a matrix by a vector. What about matrix-matrix multiplication? For example, AB can be viewed as matrix B providing a coordinate system, but each component of this coordinate system is measured within the A coordinate system, while A is measured in the Cartesian coordinate system. Therefore, each component (column vector) of B must be multiplied by matrix A to obtain the "image" of this affine coordinate system in the Cartesian coordinate system. This directly leads to the calculation formula for matrix multiplication and clearly answers why "multiplication of n-th order square matrices is only meaningful with other n-th order square matrices"—because both must be measured in the same space to completely and uniquely determine the measured values. Just as it is meaningless to discuss n n-dimensional vectors in an (n+1)-dimensional space (because to an observer in (n+1)-dimensional space, they are merely a "surface" and the extra dimension can change arbitrarily), it is even more meaningless to discuss (n+1)-dimensional vectors in an n-dimensional space, as there are simply not enough dimensions.
With this intuitive geometric meaning, many problems become almost obvious, such as determinant issues, similar matrices, and so on, which will be discussed next time.
Introduction to Tensors
We have already roughly understood that an ordered combination of numbers produces a vector, and an ordered combination of vectors produces a matrix. These two newly constructed objects become increasingly powerful in their utility. Thus, one might wonder: could an ordered combination of matrices produce a "cubic array" with even greater power? And more generally, what about n-dimensional cubic arrays? This line of thought is reasonable, and such research objects exist in mathematics: they are called tensors.
In the most common terms, an n-th order tensor is an n-dimensional cubic array. Thus, a 0-th order tensor corresponds to a scalar, while vectors and matrices correspond to 1-st and 2-nd order tensors, respectively. The three-dimensional cubic array we mentioned is a 3-rd order tensor. Of course, tensors belong to a very advanced branch of mathematical theory, and their properties and roles cannot be explained so simply. Recall that Einstein used tensor analysis as a tool to establish his great General Theory of Relativity. If given the opportunity, we will certainly revisit them.
Next, we shall return to matrix problems and discuss the determinant of a matrix.
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