Matrix Inverse and Transpose Explained: Essential Linear Algebra for Machine Learning
Machine learning (Part 15)
📚Chapter: 3 -Linear Algebra
Matrix-vector multiplication is a fundamental operation in linear algebra and plays a crucial role in machine learning and deep learning. In this tutorial, you'll learn how to perform matrix multiplication in Python using NumPy, understand the dot product, and explore its applications in AI model optimization.
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Sections
Description
Matrix inverse
Matrix Transpose
Conclusion
Section 1- Description (Dr Andrew)
In machine learning, matrices are used to represent data and perform operations on it. Two important matrix operations are the inverse and transpose. Here’s what you need to know about them. In this Tutorial, I want to tell you about a couple of special matrix operations, called the matrix inverse and the matrix transpose operation.
Section 2-Matrix inverse
Let’s start by talking about matrix inverse, and as usual we’ll start by thinking about how it relates to real numbers. In the last Tutorial, I said that the number one plays the role of the identity in the space of real numbers because one times anything is equal to itself. It turns out that real numbers have this property that very number have an, that each number has an inverse, for example, given the number three, there exists some number, which happens to be three inverse so that that number times gives you back the identity element one. And so to me, inverse of course this is just one third. And given some other number, maybe twelve there is some number which is the inverse of twelve written as twelve to the minus one, or really this is just one twelve
So that when you multiply these two things together. the product is equal to the identity element one again. Now it turns out that in the space of real numbers, not everything has an inverse. For example the number zero does not have an inverse, right? Because zero’s a zero inverse, one over zero that’s undefined. Like this one over zero is not well defined. And what we want to do, in the rest of this slide, is figure out what does it mean to compute the inverse of a matrix.
Key points
The inverse of a matrix A is denoted A^-1.
Multiplying a matrix by its inverse results in the identity matrix.
A non-square matrix does not have an inverse matrix.
We can compute inverses of matrices in Octave with the pinv(A) function and in Matlab with the inv(A) function.
Matrices that don’t have an inverse are singular or degenerate.
Here’s the idea: If A is a n by n matrix, and it has an inverse, I will say a bit more about that later, then the inverse is going to be written A to the minus one and A times this inverse, A to the minus one, is going to equal to A inverse times A, is going to give us back the identity matrix.
Okay? Only matrices that are m by m for some the idea of M having inverse. So, a matrix is M by M, this is also called a square matrix and it’s called square because the number of rows is equal to the number of columns. Right and it turns out only square matrices have inverses, so A is a square matrix, is m by m, on inverse this equation over here.
Let’s look at a concrete example, so let’s say I have a matrix, three, four, two, sixteen. So this is a two by two matrix, so it’s a square matrix and so this may just could have an and it turns out that I happen to know the inverse of this matrix is zero point four, minus zero point one, minus zero point zero five, zero zero seven five. And if I take this matrix and multiply these together it turns out what I get is the two by two identity matrix, I, this is I two by two. Okay? And so on this slide, you know this matrix is the matrix A, and this matrix is the matrix A-inverse. And it turns out if that you are computing A times A-inverse, it turns out if you compute A-inverse times A you also get back the identity matrix. So how did I find this inverse or how did I come up with this inverse over here? It turns out that sometimes you can compute inverses by hand but almost no one does that these days. And it turns out there is very good numerical software for taking a matrix and computing its inverse. So again, this is one of those things where there are lots of open source libraries that you can link to from any of the popular programming languages to compute inverses of matrices.
Python
import numpy as np
# Create a matrix A = np.array([[1, 2], [3, 4]])
# Calculate the inverse of the matrix
A_inv = np.linalg.inv(A)
# Print the inverse matrix print(A_inv)Let me show you a quick example. How I actually computed this inverse, and what I did was I used software called Optive. So let me bring that up. We will see a lot about Optive later. Let me just quickly show you an example. Set my matrix A to be equal to that matrix on the left, type three four two sixteen, so that’s my matrix A right. This is matrix 34, 216 that I have down here on the left. And, the software lets me compute the inverse of A very easily. It’s like P over A equals this. And so, this is right, this matrix here on my four minus, on my one, and so on. This given the numerical solution to what is the inverse of A. So let me just write, inverse of A equals P inverse of A over that I can now just verify that A times A inverse the identity is, type A times the inverse of A and the result of that is this matrix and this is one one on the diagonal and essentially ten to the minus seventeen, ten to the minus sixteen, so Up to numerical precision, up to a little bit of round off error that my computer had in finding optimal matrices and these numbers off the diagonals are essentially zero so A times the inverse is essentially the identity matrix. Can also verify the inverse of A times A is also equal to the identity, ones on the diagonals and values that are essentially zero except for a little bit of round dot error on the off diagonals. If a definition that the inverse of a matrix is, I had this caveat first it must always be a square matrix, it had this caveat, that if A has an inverse, exactly what matrices have an inverse is beyond the scope of this linear algebra for review that one intuition you might take away that just as the number zero doesn’t have an inverse, it turns out that if A is say the matrix of all zeros, then this matrix A also does not have an inverse because there’s no matrix there’s no A inverse matrix so that this matrix times some other matrix will give you the identity matrix so this matrix of all zeros, and there are a few other matrices with properties similar to this. That also don’t have an inverse. But it turns out that in this review I don’t want to go too deeply into what it means matrix have an inverse but it turns out for our machine learning application this shouldn’t be an issue or more precisely for the learning algorithms where this may be an to namely whether or not an inverse matrix appears and I will tell when we get to those learning algorithms just what it means for an algorithm to have or not have an inverse and how to fix it in case. Working with matrices that don’t have inverses. But the intuition if you want is that you can think of matrices as not have an inverse that is somehow too close to zero in some sense. So, just to wrap up the terminology, matrix that don’t have an inverse Sometimes called a singular matrix or degenerate matrix and so this matrix over here is an example zero zero zero matrix. is an example of a matrix that is singular, or a matrix that is degenerate.
Section 3-Matrix Transpose
Finally, the last special matrix operation I want to tell you about is to do matrix transpose. So suppose I have matrix A, if I compute the transpose of A, that’s what I get here on the right. This is a transpose which is written and A superscript T, and the way you compute the transpose of a matrix is as follows.
To get a transpose I am going to first take the first row of A one to zero. That becomes this first column of this transpose. And then I’m going to take the second row of A, 3 5 9, and that becomes the second column. of the matrix A transpose. And another way of thinking about how the computer transposes is as if you’re taking this sort of 45 degree axis and you are mirroring or you are flipping the matrix along that 45 degree axis. so here’s the more formal definition of a matrix transpose.
Key points
The transposition of a matrix is like rotating the matrix 90° in the clockwise direction and then reversing it.
We can compute transposition of matrices in MatLab with the transpose(A) function or A’.
The transpose operation provides a short notation used as an element in many matrix operations.
The matrix inversion operation is not computed directly, but rather the inverted matrix is discovered through a numerical operation, where a suite of efficient methods may be used, often involving forms of matrix decomposition. However, A^-1 is primarily useful as a theoretical tool, and should not actually be used in practice for most software applications
Let’s say A is a m by n matrix. And let’s let B equal A transpose and so BA transpose like so. Then B is going to be a n by m matrix with the dimensions reversed so here we have a 2x3 matrix. And so the transpose becomes a 3x2 matrix, and moreover, the BIJ is equal to AJI. So the IJ element of this matrix B is going to be the JI element of that earlier matrix A. So for example, B 1 2 is going to be equal to, look at this matrix, B 1 2 is going to be equal to this element 3 1st row, 2nd column. And that equal to this, which is a two one, second row first column, right, which is equal to two and some of the example B 3 2, right, that’s B 3 2 is this element 9, and that’s equal to a two three which is this element up here, nine. And so that wraps up the definition of what it means to take the transpose of a matrix and that in fact concludes our linear algebra review.
matrix = [[1, 2], [3, 4], [5, 6]]
transposed_matrix = transpose_matrix(matrix)
print(transposed_matrix)Section 4- Conclusion
So by now hopefully you know how to add and subtract matrices as well as multiply them and you also know how, what are the definitions of the inverses and transposes of a matrix and these are the main operations used in linear algebra for this course. In case this is the first time you are seeing this material. I know this was a lot of linear algebra material all presented very quickly and it’s a lot to absorb but if you there’s no need to memorize all the definitions we just went through and if you download the copy of either these slides or of the lecture notes from the course website. and use either the slides or the lecture notes as a reference then you can always refer back to the definitions and to figure out what are these matrix multiplications, transposes and so on definitions. And the lecture notes on the course website also has pointers to additional resources linear algebra which you can use to learn more about linear algebra by yourself. And next with these new tools. We’ll be able in the next few tutorialto develop more powerful forms of linear regression that can view of a lot more data, a lot more features, a lot more training examples and later on after the new regression we’ll actually continue using these linear algebra tools to derive more powerful learning algorithims as well.
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References
1- Machine Learning — Andrew
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