I think of xx as x x. It's the "function x" working on the "vector x". This helps compute the covariance matrix , a measure of self-similarity in the data. How does this help us?
When we see an equation like this from the Machine Learning class :. I now have an instant feel of what's happening. This should give us a single value. More complex derivations like this:. In some cases it gets tricky because we store the data as rows not columns in the matrix, but now I have much better tools to follow along.
You can start estimating when you'll get a single value, or when you'll get a "permutation grid" as a result. Geometric scaling and linear composition have their place, but here I want to think about information. Long story short, don't get locked into a single intuition. Multiplication evolved from repeated addition, to scaling decimals , to rotations imaginary numbers , to "applying" one number to another integrals , and so on.
Why not the same for matrix multiplication? You may be curious why we can't use the other combinations, like x x or x' x'. Simply put, the parameters don't line up: we'd have functions expecting 3 inputs only being passed a single parameter, or functions expecting single inputs getting passed 3. We define an anonymous function of 3 arguments, and immediately pass it 3 parameters.
Remember that [3 4 5] is the function and [3; 4; 5] or [3 4 5]' is how we'd write the data vector. I wanted to explain to myself — in plain English — why we wanted x' x and not the reverse. Now, in plain English: We're treating the information as a function, and passing the same info as the parameter. Learn Right, Not Rote. Home Articles Popular Calculus. Feedback Contact About Newsletter.
We will find the transformations for the unit vectors and separately as vectors and then combine them into a matrix just like S. So, the transformation for the unit vector is, and the transformation for the unit vector is, Combining the above results in a matrix, The next transformation matrix A is now applied on the product BS Again, we will find the transformations for the unit vectors and separately as vectors and then combine them into a matrix.
So, the transformation for the unit vector is, and the transformation for the unit vector is, Combining the above results in a matrix,. Share this post: WhatsApp. Intuition behind Matrix Multiplication Ask Question. Asked 10 years, 7 months ago. Active 1 year, 3 months ago. Viewed 61k times. What is the intuitive way of thinking about multiplication of matrices? Jeel Shah 8, 17 17 gold badges 69 69 silver badges bronze badges.
Happy Mittal Happy Mittal 2, 4 4 gold badges 22 22 silver badges 30 30 bronze badges. Show 8 more comments. Active Oldest Votes. Searke Searke 1, 1 1 gold badge 10 10 silver badges 8 8 bronze badges. Textbooks which only give the definition in terms of coordinates, without at least mentioning the connection with composition of linear maps, such as my first textbook on linear algebra!
It may save you a good amount of time. If there isn't an identity, you get a representation as composition, but it might not be faithful. Add a comment. Under elementwise multiplication, we have commutativity. This operation is independent of how you define or "redefine" multiplication. The second multiplication is componentwise but the inverse and first multiplication is still the usual one.
The following is covered in a text on linear algebra such as Hoffman-Kunze : This makes most sense in the context of vector spaces over a field. Eivind Dahl Eivind Dahl 1, 9 9 silver badges 15 15 bronze badges. You could also watch the matrices work on Mona step by step too, to help your intuition. Glorfindel 3, 10 10 gold badges 22 22 silver badges 36 36 bronze badges. Then, think on a Matrix, multiplicated by a vector. The Matrix is a "vector of vectors". Finally, Matrix X Matrix extends the former concept.
Herman Junge Herman Junge 1 1 silver badge 4 4 bronze badges. Now explained enough though. It is a similar way to how Gilbert Strang 'builds' up matrix multiplication in his video lectures. Martin Sleziak Martin Sleziak But you already said it all. Michael Hardy Michael Hardy 1. This meta question , and those cited therein, is a good discussion of the concerns.
As linear algebra is really about linear systems so this answer fits in with that definition of linear algebra. If you wish to explain this I'd really appreciate. The same is true of the numbers you mention. Show 1 more comment. So that is one way of getting the 'meaning' of matrix multiplication. Mitch Mitch 8, 2 2 gold badges 33 33 silver badges 67 67 bronze badges. John B 15k 9 9 gold badges 21 21 silver badges 49 49 bronze badges.
MichaelChirico 3, 13 13 silver badges 28 28 bronze badges. That is how matrices and matrix multiplication can be discovered. Nathan Petrangelo Nathan Petrangelo 11 1 1 bronze badge.
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