With the tools created in the previous posts (chronologically speaking), we’re finally at a point to discuss our first serious machine learning tool starting from the foundational linear algebra all the way to complete python code. In fact, it is so easy that we will start with a 5×5 matrix to make it “clearer” when we get to the coding. matrix ( a )) >>> ainv matrix([[-2. , 1. If you don’t use Jupyter notebooks, there are complementary .py files of each notebook. With numpy.linalg.inv an example code would look like that: Let’s simply run these steps for the remaining columns now: That completes all the steps for our 5×5. dtype. I_{3} = If at some point, you have a big “Ah HA!” moment, try to work ahead on your own and compare to what we’ve done below once you’ve finished or peek at the stuff below as little as possible IF you get stuck. ], [ 1.5, -0.5]]) Inverses of several matrices can be computed at … \end{bmatrix} Yes! Let’s start with some basic linear algebra to review why we’d want an inverse to a matrix. Subtract 3.0 * row 1 of A_M from row 2 of A_M, and Subtract 3.0 * row 1 of I_M from row 2 of I_M, 3. Then, code wise, we make copies of the matrices to preserve these original A and I matrices, calling the copies A_M and I_M. Python Matrix. It is imported and implemented by LinearAlgebraPractice.py. This blog is about tools that add efficiency AND clarity. I_{2} = Inverse of a Matrix is important for matrix operations. , ... 1 & 0 & 0\\ Subtract -0.083 * row 3 of A_M from row 1 of A_M Subtract -0.083 * row 3 of I_M from row 1 of I_M, 9. To find out the solution you have to first find the inverse of the left-hand side matrix and multiply with the right side. A=\begin{bmatrix}5&3&1\\3&9&4\\1&3&5\end{bmatrix}\hspace{5em} I=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}. Data Scientist, PhD multi-physics engineer, and python loving geek living in the United States. I hope that you will make full use of the code in the repo and will refactor the code as you wish to write it in your own style, AND I especially hope that this was helpful and insightful. Now, we can use that first row, that now has a 1 in the first diagonal position, to drive the other elements in the first column to 0. Be sure to learn about Python lists before proceed this article. 1 The following line of code is used to create the Matrix. Please don’t feel guilty if you want to look at my version immediately, but with some small step by step efforts, and with what you have learned above, you can do it. \end{bmatrix} Returns the (multiplicative) inverse of invertible self. In case you’ve come here not knowing, or being rusty in, your linear algebra, the identity matrix is a square matrix (the number of rows equals the number of columns) with 1’s on the diagonal and 0’s everywhere else such as the following 3×3 identity matrix. For example: A = [[1, 4, 5], [-5, 8, 9]] We can treat this list of a list as a matrix having 2 rows and 3 columns. Subtract 2.4 * row 2 of A_M from row 3 of A_M Subtract 2.4 * row 2 of I_M from row 3 of I_M, 7. Plus, tomorrow… GitHub Gist: instantly share code, notes, and snippets. Write a NumPy program compute the inverse of a given matrix. We will also go over how to use numpy /scipy to invert a matrix at the end of this post. When you are ready to look at my code, go to the Jupyter notebook called MatrixInversion.ipynb, which can be obtained from the github repo for this project. 0 & 1 & 0 & 0\\ A_M and I_M , are initially the same, as A and I, respectively: A_M=\begin{bmatrix}5&3&1\\3&9&4\\1&3&5\end{bmatrix}\hspace{4em} I_M=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}, 1. Now I need to calculate its inverse. If our set of linear equations has constraints that are deterministic, we can represent the problem as matrices and apply matrix algebra. $$. If at this point you see enough to muscle through, go for it! All those python modules mentioned above are lightening fast, so, usually, no. 1 & 2 & 3 \\ We will be walking thru a brute force procedural method for inverting a matrix with pure Python. The main thing to learn to master is that once you understand mathematical principles as a series of small repetitive steps, you can code it from scratch and TRULY understand those mathematical principles deeply. Executing the above script, we get the matrix. Why wouldn’t we just use numpy or scipy? 1. If you go about it the way that you would program it, it is MUCH easier in my opinion. \begin{bmatrix} $$ Scale row 3 of both matrices by 1/3.667, 8. NOTE: The last print statement in print_matrix uses a trick of adding +0 to round(x,3) to get rid of -0.0’s. What is NumPy and when to use it? $$ The way that I was taught to inverse matrices, in the dark ages that is, was pure torture and hard to remember! This is the last function in LinearAlgebraPurePython.py in the repo. Note there are other functions in LinearAlgebraPurePython.py being called inside this invert_matrix function. Consider a typical linear algebra problem, such as: We want to solve for X, so we obtain the inverse of A and do the following: Thus, we have a motive to find A^{-1}. $$ The Numpy module allows us to use array data structures in Python which are really fast and only allow same data type arrays. \begin{bmatrix} Let’s start with the logo for the github repo that stores all this work, because it really says it all: We frequently make clever use of “multiplying by 1” to make algebra easier. We’ll do a detailed overview with numbers soon after this. You don’t need to use Jupyter to follow along. Think of the inversion method as a set of steps for each column from left to right and for each element in the current column, and each column has one of the diagonal elements in it, which are represented as the S_{k1} diagonal elements where k=1\, to\, n. We’ll start with the left most column and work right. I_{4} = 1 & 0 \\ Python buffer object pointing to the start of the array’s data. There are 7 different types of sparse matrices available. See the code below. which clearly indicate that writing one column of inverse matrix to hdf5 takes 16 minutes. In Python, the … It’s important to note that A must be a square matrix to be inverted. Great question. To work with Python Matrix, we need to import Python numpy module. An identity matrix of size $n$ is denoted by $I_{n}$. 0 & 1 & 0\\ Creating a Matrix in NumPy; Matrix operations and examples; Slicing of Matrices; BONUS: Putting It All Together – Python Code to Solve a System of Linear Equations. This blog is about tools that add efficiency AND clarity. I’ve also saved the cells as MatrixInversion.py in the same repo. The NumPy code is as follows. Doing the math to determine the determinant of the matrix, we get, (8) (3)- … Why wouldn’t we just use numpy or scipy? A^{-1}). , It all looks good, but let’s perform a check of A \cdot IM = I. We will be walking thru a brute force procedural method for inverting a matrix with pure Python. After you’ve read the brief documentation and tried it yourself, compare to what I’ve done below: Notice the round method applied to the matrix class. Let’s first define some helper functions that will help with our work. A_M has morphed into an Identity matrix, and I_M has become the inverse of A. You can verify the result using the numpy.allclose() function. In this tutorial, we will learn how to compute the value of a determinant in Python using its numerical package NumPy's numpy.linalg.det() function. Also, once an efficient method of matrix inversion is understood, you are ~ 80% of the way to having your own Least Squares Solver and a component to many other personal analysis modules to help you better understand how many of our great machine learning tools are built. I don’t recommend using this. Now we pick an example matrix from a Schaum's Outline Series book Theory and Problems of Matrices by Frank Aryes, Jr1. Let’s first introduce some helper functions to use in our notebook work. Python’s SciPy library has a lot of options for creating, storing, and operating with Sparse matrices. The first step (S_{k1}) for each column is to multiply the row that has the fd in it by 1/fd. An inverse of a square matrix $A$ of order $n$ is the matrix $A^{-1}$ of the same order, such that, their product results in an identity matrix $I_{n}$. Note that all the real inversion work happens in section 3, which is remarkably short. But it is remarkable that python can do such a task in so few lines of code. In this tutorial, we will make use of NumPy's numpy.linalg.inv() function to find the inverse of a square matrix. AA^{-1} = A^{-1}A = I_{n} This means that the number of rows of A and number of columns of A must be equal. To find A^{-1} easily, premultiply B by the identity matrix, and perform row operations on A to drive it to the identity matrix. Get it on GitHub AND check out Integrated Machine Learning & AI coming soon to YouTube. Those previous posts were essential for this post and the upcoming posts. DON’T PANIC. If you do not have any idea about numpy module you can read python numpy tutorial.Python matrix is used to do operations regarding matrix, which may be used for scientific purpose, image processing etc. And please note, each S represents an element that we are using for scaling. NumPy Linear Algebra Exercises, Practice and Solution: Write a NumPy program to compute the inverse of a given matrix. See if you can code it up using our matrix (or matrices) and compare your answer to our brute force effort answer. PLEASE NOTE: The below gists may take some time to load. When we are on a certain step, S_{ij}, where i \, and \, j = 1 \, to \, n independently depending on where we are at in the matrix, we are performing that step on the entire row and using the row with the diagonal S_{k1} in it as part of that operation. 1 & 3 & 3 \\ data. \begin{bmatrix} I_{1} = If the generated inverse matrix is correct, the output of the below line will be True. We will be using NumPy (a good tutorial here) and SciPy (a reference guide here). We will see at the end of this chapter that we can solve systems of linear equations by using the inverse matrix. I would even think it’s easier doing the method that we will use when doing it by hand than the ancient teaching of how to do it. Try it with and without the “+0” to see what I mean. You can verify the result using the numpy.allclose() function. One way to “multiply by 1” in linear algebra is to use the identity matrix. 0 & 0 & 1 \begin{bmatrix} It’s interesting to note that, with these methods, a function definition can be completed in as little as 10 to 12 lines of python code. An inverse of a matrix is also known as a reciprocal matrix. left_hand_side_inverse = left_hand_side.I left_hand_side_inverse solution = left_hand_side_inverse*right_hand_side solution Plus, tomorrows machine learning tools will be developed by those that understand the principles of the math and coding of today’s tools. Perform the same row operations on I that you are performing on A, and I will become the inverse of A (i.e. Yes! Now, this is all fine when we are solving a system one time, for one outcome \(b\) . Python | Numpy matrix.sum() Last Updated: 20-05-2019 With the help of matrix.sum() method, we are able to find the sum of values in a matrix by using the same method. I would not recommend that you use your own such tools UNLESS you are working with smaller problems, OR you are investigating some new approach that requires slight changes to your personal tool suite. 0 & 0 & 0 & 1 In this article we will present a NumPy/SciPy listing, as well as a pure Python listing, for the LU Decomposition method, which is used in certain quantitative finance algorithms.. One of the key methods for solving the Black-Scholes Partial Differential Equation (PDE) model of options pricing is using Finite Difference Methods (FDM) to discretise the PDE and evaluate the solution numerically. I want to invert a matrix without using numpy.linalg.inv. The reason is that I am using Numba to speed up the code, but numpy.linalg.inv is not supported, so I am wondering if I can invert a matrix with 'classic' Python code. 0 & 1 \\ $$. Published by Thom Ives on November 1, 2018November 1, 2018. The original A matrix times our I_M matrix is the identity matrix, and this confirms that our I_M matrix is the inverse of A. I want to encourage you one last time to try to code this on your own. Use the “inv” method of numpy’s linalg module to calculate inverse of a Matrix. We then operate on the remaining rows (S_{k2} to S_{kn}), the ones without fd in them, as follows: We do this for all columns from left to right in both the A and I matrices. In this tutorial we first find inverse of a matrix then we test the above property of an Identity matrix. , If you didn’t, don’t feel bad. I_M should now be the inverse of A. Let’s check that A \cdot I_M = I . Using flip() Method. Since the resulting inverse matrix is a $3 \times 3$ matrix, we use the numpy.eye() function to create an identity matrix. Learning to work with Sparse matrix, a large matrix or 2d-array with a lot elements being zero, can be extremely handy. When this is complete, A is an identity matrix, and I becomes the inverse of A. Let’s go thru these steps in detail on a 3 x 3 matrix, with actual numbers. We’ll call the current diagonal element the focus diagonal element, or fd for short. Matrix Operations: Creation of Matrix. We then divide everything by, 1/determinant. $$. When what was A becomes an identity matrix, I will then be A^{-1}. Here are the steps, S, that we’d follow to do this for any size matrix. bsr_matrix: Block Sparse Row matrix Or, as one of my favorite mentors would commonly say, “It’s simple, it’s just not easy.” We’ll use python, to reduce the tedium, without losing any view to the insights of the method. which is its inverse. Can numpy help in this regard? ctypes. However, compared to the ancient method, it’s simple, and MUCH easier to remember. In this post, we will be learning about different types of matrix multiplication in the numpy … The first matrix in the above output is our input A matrix. in a single step. The only really painful thing about this method of inverting a matrix, is that, while it’s very simple, it’s a bit tedious and boring. However, we may be using a closely related post on “solving a system of equations” where we bypass finding the inverse of A and use these same basic techniques to go straight to a solution for X. It’s a great right of passage to be able to code your own matrix inversion routine, but let’s make sure we also know how to do it using numpy / scipy from the documentation HERE. There are also some interesting Jupyter notebooks and .py files in the repo. Doing such work will also grow your python skills rapidly. Python statistics and matrices without numpy. right_hand_side = np.matrix([[4], [-6], [7]]) right_hand_side Solution. However, we can treat list of a list as a matrix. print(np.allclose(np.dot(ainv, a), np.eye(3))) Notes Let’s get started with Matrices in Python. In other words, for a matrix [[a,b], [c,d]], the determinant is computed as ‘ad-bc’. This is just a high level overview. How to do gradient descent in python without numpy or scipy. The identity matrix or the inverse of a matrix are concepts that will be very useful in the next chapters. Python provides a very easy method to calculate the inverse of a matrix. Base object if memory is from some other object. Python matrix determinant without numpy. Would I recommend that you use what we are about to develop for a real project? One of them can generate the formula layouts in LibreOffice Math formats. The flip() method in the NumPy module reverses the order of a NumPy array and returns the NumPy array object. which is its inverse. Then come back and compare to what we’ve done here. >>> import numpy as np #load the Library I want to be part of, or at least foster, those that will make the next generation tools. In Linear Algebra, an identity matrix (or unit matrix) of size $n$ is an $n \times n$ square matrix with $1$'s along the main diagonal and $0$'s elsewhere. We will use NumPy's numpy.linalg.inv() function to find its inverse. Find the Determinant of a Matrix with Pure Python without Numpy or , Find the Determinant of a Matrix with Pure Python without Numpy or Scipy AND , understanding the math to coding steps for determinants IS In other words, for a matrix [[a,b], [c,d]], the determinant is computed as ‘ad-bc’. The numpy.linalg.det() function calculates the determinant of the input matrix. In this post, we create a clustering algorithm class that uses the same principles as scipy, or sklearn, but without using sklearn or numpy or scipy. 1 & 0 & 0 & 0\\ I'm using fractions.Fraction as entries in a matrix because I need to have very high precision and fractions.Fraction provides infinite precision (as I've learned from advice from this list). We start with the A and I matrices shown below. B: The solution matrix Inverse of a Matrix using NumPy. Python Matrix. \end{bmatrix} Using the steps and methods that we just described, scale row 1 of both matrices by 1/5.0, 2. Code faster with the Kite plugin for your code editor, featuring Line-of-Code Completions and cloudless processing. There will be many more exercises like this to come. \end{bmatrix} NumPy: Determinant of a Matrix. Inverse of an identity [I] matrix is an identity matrix [I]. \begin{bmatrix} The other sections perform preparations and checks. We will see two types of matrices in this chapter. To calculate the inverse of a matrix in python, a solution is to use the linear … So hang on! Thus, a statement above bears repeating: tomorrows machine learning tools will be developed by those that understand the principles of the math and coding of today’s tools. Here, we are going to reverse an array in Python built with the NumPy module. Subtract 0.472 * row 3 of A_M from row 2 of A_M Subtract 0.472 * row 3 of I_M from row 2 of I_M. Kite is a free autocomplete for Python developers. The function numpy.linalg.inv() which is available in the python NumPy module is used to c ompute the inverse of a matrix.. Syntax: numpy… In future posts, we will start from here to see first hand how this can be applied to basic machine learning and how it applies to other techniques beyond basic linear least squares linear regression. This type of effort is shown in the ShortImplementation.py file. base. The shortest possible code is rarely the best code. Since the resulting inverse matrix is a $3 \times 3$ matrix, we use the numpy.eye() function to create an identity matrix. , If the generated inverse matrix is correct, the output of the below line will be True. The 2-D array in NumPy is called as Matrix. Create a Python Matrix using the nested list data type; Create Python Matrix using Arrays from Python Numpy package; Create Python Matrix using a nested list data type. My encouragement to you is to make the key mathematical points your prime takeaways. I love numpy, pandas, sklearn, and all the great tools that the python data science community brings to us, but I have learned that the better I understand the “principles” of a thing, the better I know how to apply it. So how do we easily find A^{-1} in a way that’s ready for coding? I do love Jupyter notebooks, but I want to use this in scripts now too. Success! An object to simplify the interaction of the array with the ctypes module. Great question. It should be mentioned that we may obtain the inverse of a matrix using ge, by reducing the matrix \(A\) to the identity, with the identity matrix as the augmented portion. You want to do this one element at a time for each column from left to right. \end{bmatrix} If a is a matrix object, then the return value is a matrix as well: >>> ainv = inv ( np . Using this library, we can perform complex matrix operations like multiplication, dot product, multiplicative inverse, etc. Subtract 0.6 * row 2 of A_M from row 1 of A_M Subtract 0.6 * row 2 of I_M from row 1 of I_M, 6. (23 replies) I guess this is a question to folks with some numpy background (but not necessarily). This blog’s work of exploring how to make the tools ourselves IS insightful for sure, BUT it also makes one appreciate all of those great open source machine learning tools out there for Python (and spark, and there’s ones fo… Plus, if you are a geek, knowing how to code the inversion of a matrix is a great right of passage! Matrix methods represent multiple linear equations in a compact manner while using the existing matrix library functions. As previously stated, we make copies of the original matrices: Let’s run just the first step described above where we scale the first row of each matrix by the first diagonal element in the A_M matrix. Applying Polynomial Features to Least Squares Regression using Pure Python without Numpy or Scipy, AX=B,\hspace{5em}\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}\begin{bmatrix}x_{11}\\x_{21}\\x_{31}\end{bmatrix}=\begin{bmatrix}b_{11}\\b_{21}\\b_{31}\end{bmatrix}, X=A^{-1}B,\hspace{5em} \begin{bmatrix}x_{11}\\x_{21}\\x_{31}\end{bmatrix} =\begin{bmatrix}ai_{11}&ai_{12}&ai_{13}\\ai_{21}&ai_{22}&ai_{23}\\ai_{31}&ai_{32}&ai_{33}\end{bmatrix}\begin{bmatrix}b_{11}\\b_{21}\\b_{31}\end{bmatrix}, I= \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}, AX=IB,\hspace{5em}\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}\begin{bmatrix}x_{11}\\x_{21}\\x_{31}\end{bmatrix}= \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix} \begin{bmatrix}b_{11}\\b_{21}\\b_{31}\end{bmatrix}, IX=A^{-1}B,\hspace{5em} \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix} \begin{bmatrix}x_{11}\\x_{21}\\x_{31}\end{bmatrix} =\begin{bmatrix}ai_{11}&ai_{12}&ai_{13}\\ai_{21}&ai_{22}&ai_{23}\\ai_{31}&ai_{32}&ai_{33}\end{bmatrix}\begin{bmatrix}b_{11}\\b_{21}\\b_{31}\end{bmatrix}, S = \begin{bmatrix}S_{11}&\dots&\dots&S_{k2} &\dots&\dots&S_{n2}\\S_{12}&\dots&\dots&S_{k3} &\dots&\dots &S_{n3}\\\vdots& & &\vdots & & &\vdots\\ S_{1k}&\dots&\dots&S_{k1} &\dots&\dots &S_{nk}\\ \vdots& & &\vdots & & &\vdots\\S_{1 n-1}&\dots&\dots&S_{k n-1} &\dots&\dots &S_{n n-1}\\ S_{1n}&\dots&\dots&S_{kn} &\dots&\dots &S_{n1}\\\end{bmatrix}, A_M=\begin{bmatrix}1&0.6&0.2\\3&9&4\\1&3&5\end{bmatrix}\hspace{5em} I_M=\begin{bmatrix}0.2&0&0\\0&1&0\\0&0&1\end{bmatrix}, A_M=\begin{bmatrix}1&0.6&0.2\\0&7.2&3.4\\1&3&5\end{bmatrix}\hspace{5em} I_M=\begin{bmatrix}0.2&0&0\\-0.6&1&0\\0&0&1\end{bmatrix}, A_M=\begin{bmatrix}1&0.6&0.2\\0&7.2&3.4\\0&2.4&4.8\end{bmatrix}\hspace{5em} I_M=\begin{bmatrix}0.2&0&0\\-0.6&1&0\\-0.2&0&1\end{bmatrix}, A_M=\begin{bmatrix}1&0.6&0.2\\0&1&0.472\\0&2.4&4.8\end{bmatrix}\hspace{5em} I_M=\begin{bmatrix}0.2&0&0\\-0.083&0.139&0\\-0.2&0&1\end{bmatrix}, A_M=\begin{bmatrix}1&0&-0.083\\0&1&0.472\\0&2.4&4.8\end{bmatrix}\hspace{5em} I_M=\begin{bmatrix}0.25&-0.083&0\\-0.083&0.139&0\\-0.2&0&1\end{bmatrix}, A_M=\begin{bmatrix}1&0&-0.083\\0&1&0.472\\0&0&3.667\end{bmatrix}\hspace{5em} I_M=\begin{bmatrix}0.25&-0.083&0\\-0.083&0.139&0\\0&-0.333&1\end{bmatrix}, A_M=\begin{bmatrix}1&0&-0.083\\0&1&0.472\\0&0&1\end{bmatrix}\hspace{5em} I_M=\begin{bmatrix}0.25&-0.083&0\\-0.083&0.139&0\\0&-0.091&0.273\end{bmatrix}, A_M=\begin{bmatrix}1&0&0\\0&1&0.472\\0&0&1\end{bmatrix}\hspace{5em} I_M=\begin{bmatrix}0.25&-0.091&0.023\\-0.083&0.139&0\\0&-0.091&0.273\end{bmatrix}, A_M=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\hspace{5em} I_M=\begin{bmatrix}0.25&-0.091&0.023\\-0.083&0.182&-0.129\\0&-0.091&0.273\end{bmatrix}, A \cdot IM=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}, Gradient Descent Using Pure Python without Numpy or Scipy, Clustering using Pure Python without Numpy or Scipy, Least Squares with Polynomial Features Fit using Pure Python without Numpy or Scipy, use the element that’s in the same column as, replace the row with the result of … [current row] – multiplier * [row that has, this will leave a zero in the column shared by. Subtract 1.0 * row 1 of A_M from row 3 of A_M, and Subtract 1.0 * row 1 of I_M from row 3 of I_M, 5. T. Returns the transpose of the matrix. I encourage you to check them out and experiment with them. The second matrix is of course our inverse of A. Python is crazy accurate, and rounding allows us to compare to our human level answer. The python matrix makes use of arrays, and the same can be implemented. I know that feeling you’re having, and it’s great! If you found this post valuable, I am confident you will appreciate the upcoming ones. If you get stuck, take a peek, but it will be very rewarding for you if you figure out how to code this yourself. Matrix Multiplication in NumPy is a python library used for scientific computing. Below is the output of the above script. When dealing with a 2x2 matrix, how we obtain the inverse of this matrix is swapping the 8 and 3 value and placing a negative sign (-) in front of the 2 and 7. As per this if i need to calculate the entire matrix inverse it will take me 1779 days. 1 & 2 & 4 0 & 0 & 1 & 0\\ If you did most of this on your own and compared to what I did, congratulations! When we multiply the original A matrix on our Inverse matrix we do get the identity matrix. I love numpy, pandas, sklearn, and all the great tools that the python data science community brings to us, but I have learned that the better I understand the “principles” of a thing, the better I know how to apply it. Python doesn't have a built-in type for matrices. Following the main rule of algebra (whatever we do to one side of the equal sign, we will do to the other side of the equal sign, in order to “stay true” to the equal sign), we will perform row operations to A in order to methodically turn it into an identity matrix while applying those same steps to what is “initially” the identity matrix. The larger square matrices are considered to be a combination of 2x2 matrices. My approach using numpy / scipy is below.

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