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I am used to seeing most basic mathematical objects being visually represented (for instance, a curve in the plane divided by the xy axis; the same goes for complex numbers, vectors, and so on....), However, I never saw a visual representation of a matrix. I do not mean the disposition in rows and columns, of course. I would like to know whether they can be graphically represented in some intuitive way. If so, how? Could you illustrate it, say, with a 2x2 quadratic matrix?

Thanks a lot in advance

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  • $\begingroup$ What do you mean by a "quadratic matrix"? Do you mean a symmetric matrix, which would in turn represent a quadratic form? $\endgroup$ Jun 6, 2015 at 23:13
  • $\begingroup$ I mean a matrix with the same number of rows and columns...for instance 2x2 $\endgroup$ Jun 6, 2015 at 23:35
  • $\begingroup$ Stress is a tensor (a special kind of matrix). That most likely doesn't simplify the interpretation though. More along that route, matrices can be used to rotate or transform shapes. So you can think of matrices in terms of the regions they transform. In other words as an operator. $\endgroup$
    – Zach466920
    Jun 6, 2015 at 23:41
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    $\begingroup$ This might help. $\endgroup$ Jun 6, 2015 at 23:45
  • $\begingroup$ You can also think of a $m\times n$ matrix with all complex entries as the collection of $m$ number of vectors in the complex-affine space $\Bbb C^n$ (I hope this makes sense). :P $\endgroup$ Jun 6, 2015 at 23:46

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There is no one way to "visualize" matrices. In fact, visualizing abstract quantities can be useful but also lead to loss of information through the visualization.

As far as 2x2 matrices go, one way of visualizing them is to note that they have the same mathematical structure as complex numbers when it comes to addition and multiplication (See http://en.wikipedia.org/wiki/2_%C3%97_2_real_matrices#2.E2.80.89.C3.97.E2.80.892_real_matrices_as_complex_numbers)

Of course, this is highly specific as far as matrices go. A more useful approach to "visualizing" a matrix is to view it a linear transformation on a vector -- and observe its action on a standard set of vectors. Linear transformations and their properties in terms of dilating and rotating vectors are useful ways to understand matrices, although they are not the only useful information contained inside a matrix.

I would recommend Artin's text as a useful first resource to highlight some of these connections. Depending on what you want to do with matrices, you can expand in many different directions.

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  • $\begingroup$ Maybe instead of saying visualizing, which implies some sort of mental imagination which may remain as such....I should have said drawing......so..any way of, for example, darwing matrices on a plane or something like that? $\endgroup$ Jun 6, 2015 at 23:49
  • $\begingroup$ What want to do is the following. I want to visualize (or draw, if you like) some specific matrices which are right ideals within the nxn matrix ring......because that is the kind of representation (ideals) some inguist propose for some stuff I am working on.... $\endgroup$ Jun 7, 2015 at 0:03
  • $\begingroup$ Is there some kind of inter-relationship or order relation between the matrices? What do you seek to gain from "drawing" the matrices versus displaying them? What aspect (not necessarily mathematical) of these matrices are you trying to represent? $\endgroup$ Jun 7, 2015 at 0:11
  • $\begingroup$ What specific book by Artin do you refer to? Could you provide the entire reference (at best with reference to chapter and pages where he deals with the topic??? THanks in advance. $\endgroup$ Jul 31, 2015 at 17:21
  • $\begingroup$ I don't have a specific page or chapter but the book I was referring to is amazon.com/Algebra-2nd-Featured-Titles-Abstract-ebook/dp/… -- a great book to develop intuition if you have looked at linear algebra before. $\endgroup$ Aug 3, 2015 at 23:18
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You can represent a $2 \times 2$ matrices $A = \left[\begin{smallmatrix}a & b\\c & d\end{smallmatrix}\right]$ as a parallelogram $Q_A \subset \mathbb{R}^2$ with vertices $(0,0), (a,c), (b,d)$ and $(a+b,c+d)$. If one identify the plane $\mathbb{R}^2$ with $M^{2\times 1}(\mathbb{R})$, the space of $2 \times 1$ column matrices, then $Q_A$ is the image of the unit square $[0,1] \times [0,1]$ under linear transform

$$[0,1] \times [0,1] \ni \begin{bmatrix}x \\ y\end{bmatrix} \quad \mapsto \quad \begin{bmatrix}x' \\ y'\end{bmatrix} = \begin{bmatrix}a & b\\c & d\end{bmatrix} \begin{bmatrix}x \\ y\end{bmatrix} \in Q_A $$

Since a linear transformation is uniquely determined by its action on a basis, this provides a faithful represent of the $2 \times 2$ matrices. Under this representation, some geometry related operations now correspond to familiar geometric shapes. e.g.

  • The matrix $\left[\begin{smallmatrix}s & 0\\0 & s\end{smallmatrix}\right]$ represents a scaling of geometric objects. It corresponds to a square of side length $s$, axis aligned with the standard $x$- and $y$-axis.

  • The matrix $\left[\begin{smallmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \;\cos\theta\end{smallmatrix}\right]$ represents a counterclockwise rotation of angle $\theta$. It corresponds to the unit square rotated counterclockwisely for angle $\theta$.

  • The matrices $\left[\begin{smallmatrix}1 & m\\ 0& 1\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix}1 & 0\\ m& 1\end{smallmatrix}\right]$ represent sheer mappings in horizontal and vertical directions. They can visualized as parallelogram with one pair of its sides staying horizontal and vertical respectively.

This sort of shapes provides a useful visual mnemonics for what are the effects of those matrices (when viewed as a transformation of the plane).

Finally, one can also use this to introduce the concept of determinant to students.

  • What is the the determinant of a matrix $A$? It is just the area of $Q_A$.
  • What does it mean $\det A < 0$? It just mean the $Q_A$ has been flipped.

I'm sure there are other ways to use this visualization as a teaching tool.

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  • $\begingroup$ How about a 2x2 matrix with zero entries in the bottom row? $\endgroup$ Jun 7, 2015 at 0:34
  • $\begingroup$ @JavierArias a degenerate parallelogram with both sides pointing along the $x$ axis. In fact, all non-invertible (non-zero) matrix looks like this except the direction of the sides is different. For the zero-matrix, it is just a dot. $\endgroup$ Jun 7, 2015 at 0:40
  • $\begingroup$ Thanks....I will think about it thorougly one or two days...see if I deeply understand it......Have a good night (or day, depending on where you are.....) $\endgroup$ Jun 7, 2015 at 0:44
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You probably want a visual or mental picture of a matrix. Now that's truly hard. Let me explain why with a simple example:

Imagine you have a door and you open it. You can prove to me that the door is in two different states because you did a certain transformation to it and you actually rotated the door around the hinge. You did that transformation of a 3D object in TIME. So a transformation is nothing else but a structure in 4D space-time for the door, and you're lucky here because you transform just 2 coordinates, x, y(you can't rise up the door when you open it). Math lovers tend to generalize stuff and say...ok...but I'm a math guru, so I make the time dimension a spatial one...and I add another one...n actually.

Going back now to 4D a 2x2 matrix could actually be seen in 4D space. But that would also mean that for you opening a door is actually meaningless in 4D space(it's important this, when it's time it has meaning), because you see a geometrical object that has all the transitory states between the two states(door opened in state 1 or state 0). In time, it would mean that you see events in space-time like single stationary objects. For example a human speaking to you wouldn't actually be speaking to you. You would see him before he spoke and as it spokes and for you is a single object, even though for mere mortals he passed thru an infinite number of transition.

Pretty nice stuff though...helps alot in physics.

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  • $\begingroup$ MY plan and aim goes as follows. In my field, the study of language, it has been established (though it is not a very popular subject in the discipline) that the so called sublanguages (for instance, the language of medicine, weather reports, science languages, and so on) are in a very particular relation to the language as a whole.....I will not go into details....just say that is is suggested that relation is that of a right ideal to a ring....A good example of that in maths are certain matrices......That is why I want to visualize them, to better grasp the relations in my field $\endgroup$ Jun 7, 2015 at 0:15
  • $\begingroup$ To be more specific, this is what is commonly assumed: "the sublanguage grammar contains rules which the language violates and the language grammar contains rules which the sublanguage never meets. It follows that while the sentences of such science object-languages are included in the language as a whole, the grammar of these sublanguages intersects (rather than is included in) the grammar of the language as a whole"....In some other places, that very same author describes the bulk of those relations as that of a ring ideal to a ring $\endgroup$ Jun 7, 2015 at 0:26
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I understand your frustration with wanting to "visualize" matrices (even simple $2 \times 2$s)!

So, let's start first with the complex $1 \times 1$ matrices $[z]$ with $z \in \mathbb{C}$. Let's also constrain this set of matrices by imposing a rule that each of these matrices must be unitary; namely, $z$'s conjugate transpose is also its inverse. So $\bar{z} = z^{-1}$ implies $|z| = 1$, which means the set of all such $1 \times 1$ matrices is in fact the 1-dimensional unit circle. If we replace $\mathbb{C}$ with $\mathbb{R}$, we get two points: $\lbrace -1, 1 \rbrace$ (the 0-dimensional "circle"). If we lose the unitary requirement, then the real $1 \times 1$ matrices just form an infinite line: $\mathbb{R}$.

Ok, now let's move up to $2 \times 2$ real matrices. We know such matrices belong in $\mathbb{R}^{4}$ (in fact, all together they are $\mathbb{R}^4$), but how do we make sense of them?

Let's consider all such matrices (real and $2 \times 2$) which are also invertible. They form an open subspace of $\mathbb{R}^4$ that is disconnected: one component represents the matrices of positive determinant; the other negative. Ok, so this isn't very helpful, but if you think of these matrices as what they really are (linear transformations), then it will begin to make sense.

Note that a linear transformation does two things to a vector: (1) it scales it, and (2) it rotates it. Then we get a new vector. So we are sending $\mathbb{R}^2 \longrightarrow \mathbb{R}^2$. Scaling can be parameterized by a single real number $\lambda$; similarly, rotation can be parameterized by a single real angle $\phi \in \mathbb{R} / \pi\mathbb{Z}$ (in fact, the set of all real $2 \times 2$ rotation matrices is the unit circle). These two parameters can be thought of as your extra $2$ dimensions; namely, to visualize an invertible real $2 \times 2$ matrix $A$, you could view it as a $4$-tuple $(x, y, \lambda, \phi)$ ($\in \mathbb{R}^4$), where $v = [x, y]^\top$ is a vector in $\mathbb{R}^2$, and $\lambda$ and $\phi$ are the scaling and rotating components that will send $v$ to $Av$.

Another kind of hand-wavy way is to say that a $2 \times 2$ real invertible matrix can be decomposed into it's scaling part and it's rotating part (each independent). The rotating part, as claimed above, is really just the circle $S^1$. The scaling part is really just the real line $\mathbb{R}$. So together they form sort of an infinite cylinder $\mathbb{R} \times S^1$. So the "total space" is $\mathbb{R}^2 \times \mathbb{R} \times S^1 = \mathbb{R}^3 \times \mathbb{R}/\mathbb{Z}$ which is "in" (hand-wavy) $\mathbb{R}^4$.

In line with the door comment, this is a space in which the vector $v$ under transformation by $A$ is "preserved" through all states until it reaches its final state $Av$.

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