Are Matrices Related to Space and Time?

In summary: ZwcZ9x0hW0QIn summary, matrices are not limited to applications in space and time. They can be used to solve linear systems in various fields such as circuits, weather modeling, and financial modeling. Additionally, a vector is a matrix with one column or one row, and there are also tensors which are more general concepts. In special relativity, matrices with 4 entries are used to represent the 3 dimensions of space and 1 dimension of time. For example, the metric tensor in general relativity is a 4x4 matrix that defines time, distance, volume, and curvature in spacetime.
  • #1
paulo84
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Hi,

I just have a question relative to matrices, mostly. Is the reason there are 4 values in a matrix because there are (at least in basic terms) 3 dimensions of space and one of time?

Like it seems kind of obvious, but for some weird reason in school they never state it explicitly in those terms.

Also, do matrices have any applications outside of space and time?

For future reference, can someone also please let me know if I've posted this in the right section, and if it's ok to ask questions like this, or should I be Googling/consulting a textbook/inferring...
 
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  • #2
Matrices could be used to help solve a variety of linear systems - not just spatial ones. Solving for the currents and voltages in a circuit, for example. Depending on the complexity of the circuit, the size can be much larger than 4. There are other multivariable systems, such as weather and financial modeling, which matrices can be useful. You may find this of interest - http://ulaff.net/
 
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  • #3
scottdave said:
Matrices could be used to help solve a variety of linear systems - not just spatial ones. Solving for the currents and voltages in a circuit, for example. Depending on the complexity of the circuit, the size can be much larger than 4. There are other multivariable systems, such as weather and financial modeling, which matrices can be useful.

Relative to scalars and vectors, is there another one one-up from a vector?
 
  • #4
paulo84 said:
Relative to scalars and vectors, is there another one one-up from a vector?
A vector is a matrix which has 1 column - or 1 row, depending on how you are working with it. A matrix is a combination of multiple column vectors. Then there are tensors. You may find this YouTube video by Dan Fleisch helpful -
 
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  • #5
scottdave said:
A vector is a matrix which has 1 column - or 1 row, depending on how you are working with it. A matrix is a combination of multiple column vectors. Then there are tensors. You may find this YouTube video by Dan Fleisch helpful -


Thanks, that was very helpful.
 
  • #6
paulo84 said:
I just have a question relative to matrices, mostly. Is the reason there are 4 values in a matrix because there are (at least in basic terms) 3 dimensions of space and one of time?
No, there is no connection.
A matrix can have any number of values, if you include one-dimensional matrices (AKA vectors). Generally speaking, a matrix is a rectangular array of numbers. A 2 x 2 matrix has two rows and two columns, but you can have 2 x 3 matrices, 3 x 2 matrices, 8 x 8 matrices, and on and on.

paulo84 said:
Like it seems kind of obvious, but for some weird reason in school they never state it explicitly in those terms.
For good reason.

paulo84 said:
Also, do matrices have any applications outside of space and time?
See the wiki article, https://en.wikipedia.org/wiki/Matrix_(mathematics), under "Applications".

paulo84 said:
For future reference, can someone also please let me know if I've posted this in the right section, and if it's ok to ask questions like this, or should I be Googling/consulting a textbook/inferring...
This should probably go in the math technical section. For basic questions like this, it's a good idea to start by a web search to get some basic information. For more information, there are lots of linear algebra books out there that discuss matrices in much greater detail.
 
  • #7
Ok, makes sense if you're dealing with multiple dimensions of space and/or time, as well as the other applications mentioned.
 
  • #8
Special relativity has many vectors with 4 entries and matrices with 4*4=16 entries because we have 3 space and 1 time dimension. The concepts of vectors and matrices are much more general.
 
  • #9
mfb said:
Special relativity has many vectors with 4 entries and matrices with 4*4=16 entries because we have 3 space and 1 time dimension. The concepts of vectors and matrices are much more general.

Could you please share a little more about the 16 entry matrices which relate to space and time?
 
  • #11
jedishrfu said:
One example would be the metric tensor of General Relativity

https://en.m.wikipedia.org/wiki/Metric_tensor_(general_relativity)

It’s a 4x4 matrix of values that's used to define time, distance, volume, curvature... in a spacetime.

That is extremely impressive. I'm going to have to read it a number of times.
 
  • #12
Here's a video of on the Einstein Field equations that show applications of the metric tensor and other 4x4 marices:

 

1. What is the difference between a vector and a matrix?

A vector is a one-dimensional array of numbers, while a matrix is a two-dimensional array of numbers. Vectors are represented as a single column or row, while matrices are represented as multiple rows and columns.

2. How are vectors and matrices used in scientific research?

Vectors and matrices are used in a variety of scientific fields, such as physics, biology, and economics, to represent and manipulate data and equations. They are particularly useful in linear algebra, which is used in many areas of science for modeling and analyzing complex systems.

3. Can vectors and matrices be multiplied together?

Yes, vectors and matrices can be multiplied together. However, the dimensions of the two must be compatible for multiplication to be possible. For example, a vector with n elements can only be multiplied by a matrix with n rows and n columns.

4. What is the difference between a scalar and a vector?

A scalar is a single numerical value, while a vector is a collection of multiple values. Scalars can be represented by a single number, while vectors are represented by multiple numbers arranged in a specific order.

5. How are vectors and matrices used in machine learning and data analysis?

Vectors and matrices are essential in machine learning and data analysis, as they are used to store and manipulate large datasets. They are used for tasks such as data preprocessing, feature extraction, and model training in machine learning algorithms.

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