(23 replies) I guess this is a question to folks with some numpy background (but not necessarily). Executing the above script, we get the matrix. I encourage you to check them out and experiment with them. This is just a high level overview. Since the resulting inverse matrix is a $3 \times 3$ matrix, we use the numpy.eye() function to create an identity matrix. To calculate the inverse of a matrix in python, a solution is to use the linear … 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. A_M has morphed into an Identity matrix, and I_M has become the inverse of A. What is NumPy and when to use it? Python buffer object pointing to the start of the array’s data. I don’t recommend using this. Using this library, we can perform complex matrix operations like multiplication, dot product, multiplicative inverse, etc. I do love Jupyter notebooks, but I want to use this in scripts now too. In this post, we will be learning about different types of matrix multiplication in the numpy … This blog is about tools that add efficiency AND clarity. \begin{bmatrix} If you get stuck, take a peek, but it will be very rewarding for you if you figure out how to code this yourself. Great question. $$ Code faster with the Kite plugin for your code editor, featuring Line-of-Code Completions and cloudless processing. All those python modules mentioned above are lightening fast, so, usually, no. If our set of linear equations has constraints that are deterministic, we can represent the problem as matrices and apply matrix algebra. 0 & 1 & 0 & 0\\ 1 & 2 & 3 \\ bsr_matrix: Block Sparse Row matrix If the generated inverse matrix is correct, the output of the below line will be True. We start with the A and I matrices shown below. Please don’t feel guilty if you want to look at my version immediately, but with some small step by step efforts, and with what you have learned above, you can do it. matrix ( a )) >>> ainv matrix([[-2. , 1. So hang on! \begin{bmatrix} $$. Yes! \begin{bmatrix} 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. Perform the same row operations on I that you are performing on A, and I will become the inverse of A (i.e. The python matrix makes use of arrays, and the same can be implemented. I want to be part of, or at least foster, those that will make the next generation tools. 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. Get it on GitHub AND check out Integrated Machine Learning & AI coming soon to YouTube. We will also go over how to use numpy /scipy to invert a matrix at the end of this post. Using flip() Method. 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. If you don’t use Jupyter notebooks, there are complementary .py files of each notebook. Let’s get started with Matrices in Python. We will be using NumPy (a good tutorial here) and SciPy (a reference guide here). 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. Then come back and compare to what we’ve done here. \begin{bmatrix} \end{bmatrix} I_{4} = \begin{bmatrix} 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. 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. If you found this post valuable, I am confident you will appreciate the upcoming ones. There will be many more exercises like this to come. right_hand_side = np.matrix([[4], [-6], [7]]) right_hand_side Solution. If you did most of this on your own and compared to what I did, congratulations! 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. base. The larger square matrices are considered to be a combination of 2x2 matrices. 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. Python’s SciPy library has a lot of options for creating, storing, and operating with Sparse matrices. >>> import numpy as np #load the Library And please note, each S represents an element that we are using for scaling. The first step (S_{k1}) for each column is to multiply the row that has the fd in it by 1/fd. The flip() method in the NumPy module reverses the order of a NumPy array and returns the NumPy array object. In this tutorial, we will make use of NumPy's numpy.linalg.inv() function to find the inverse of a square matrix. An inverse of a matrix is also known as a reciprocal matrix. AA^{-1} = A^{-1}A = I_{n} 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. We’ll do a detailed overview with numbers soon after this. 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. We will see at the end of this chapter that we can solve systems of linear equations by using the inverse matrix. 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. 0 & 0 & 1 & 0\\ There are also some interesting Jupyter notebooks and .py files in the repo. 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. 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. Learning to work with Sparse matrix, a large matrix or 2d-array with a lot elements being zero, can be extremely handy. One way to “multiply by 1” in linear algebra is to use the identity matrix. 1 & 2 & 4 data. 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. Inverse of a Matrix is important for matrix operations. Matrix Operations: Creation of Matrix. Now I need to calculate 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’. Python Matrix. Now, this is all fine when we are solving a system one time, for one outcome \(b\) . You can verify the result using the numpy.allclose() function. There are 7 different types of sparse matrices available. I_M should now be the inverse of A. Let’s check that A \cdot I_M = I . When we multiply the original A matrix on our Inverse matrix we do get the identity matrix. Let’s first define some helper functions that will help with our work. 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. 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. \end{bmatrix} Python provides a very easy method to calculate the inverse of a matrix. Doing such work will also grow your python skills rapidly. Why wouldn’t we just use numpy or scipy? 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. 1 which is its inverse. 0 & 1 \\ Great question. in a single step. 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. 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. Data Scientist, PhD multi-physics engineer, and python loving geek living in the United States. The following line of code is used to create the Matrix. Using the steps and methods that we just described, scale row 1 of both matrices by 1/5.0, 2. As per this if i need to calculate the entire matrix inverse it will take me 1779 days. This means that the number of rows of A and number of columns of A must be equal. Note there are other functions in LinearAlgebraPurePython.py being called inside this invert_matrix function. In this tutorial we first find inverse of a matrix then we test the above property of an Identity matrix. Try it with and without the “+0” to see what I mean. Now we pick an example matrix from a Schaum's Outline Series book Theory and Problems of Matrices by Frank Aryes, Jr1. B: The solution matrix Inverse of a Matrix using NumPy. 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 $I_{n}$. 0 & 0 & 0 & 1 Base object if memory is from some other object. 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. If you didn’t, don’t feel bad. Returns the (multiplicative) inverse of invertible self. Those previous posts were essential for this post and the upcoming posts. We will use NumPy's numpy.linalg.inv() function to find its inverse. Inverse of an identity [I] matrix is an identity matrix [I]. A^{-1}). Write a NumPy program compute the inverse of a given matrix. 1. 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}. 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. 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. Here, we are going to reverse an array in Python built with the NumPy module. NumPy: Determinant of a Matrix. Yes! 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. I want to invert a matrix without using numpy.linalg.inv. 1 & 0 & 0 & 0\\ 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. This is the last function in LinearAlgebraPurePython.py in the repo. To find out the solution you have to first find the inverse of the left-hand side matrix and multiply with the right side. 0 & 0 & 1 My encouragement to you is to make the key mathematical points your prime takeaways. 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. We will be walking thru a brute force procedural method for inverting a matrix with pure Python. $$. With numpy.linalg.inv an example code would look like that: We then operate on the remaining rows (S_{k2} to S_{kn}), the ones without fd in them, as follows: We do this for all columns from left to right in both the A and I matrices. You want to do this one element at a time for each column from left to right. 1 & 0 & 0\\ The other sections perform preparations and checks. The shortest possible code is rarely the best code. Plus, if you are a geek, knowing how to code the inversion of a matrix is a great right of passage! However, we can treat list of a list as a matrix. I_{3} = $$ Python statistics and matrices without numpy. If you go about it the way that you would program it, it is MUCH easier in my opinion. Would I recommend that you use what we are about to develop for a real project? 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. An object to simplify the interaction of the array with the ctypes module. 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… Since the resulting inverse matrix is a $3 \times 3$ matrix, we use the numpy.eye() function to create an identity matrix. If at this point you see enough to muscle through, go for it! 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. GitHub Gist: instantly share code, notes, and snippets. 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. This blog is about tools that add efficiency AND clarity. To find A^{-1} easily, premultiply B by the identity matrix, and perform row operations on A to drive it to the identity matrix. Python doesn't have a built-in type for matrices. print(np.allclose(np.dot(ainv, a), np.eye(3))) Notes In Python, the … , 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. left_hand_side_inverse = left_hand_side.I left_hand_side_inverse solution = left_hand_side_inverse*right_hand_side solution This type of effort is shown in the ShortImplementation.py file. Scale row 3 of both matrices by 1/3.667, 8. , Kite is a free autocomplete for Python developers. Be sure to learn about Python lists before proceed this article. So how do we easily find A^{-1} in a way that’s ready for coding? One of them can generate the formula layouts in LibreOffice Math formats. 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). I know that feeling you’re having, and it’s great! 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. Let’s simply run these steps for the remaining columns now: That completes all the steps for our 5×5. , Can numpy help in this regard? 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. Note that all the real inversion work happens in section 3, which is remarkably short. Success! 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. The 2-D array in NumPy is called as 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. But it is remarkable that python can do such a task in so few lines of code. Python Matrix. Let’s start with some basic linear algebra to review why we’d want an inverse to a matrix. We will be walking thru a brute force procedural method for inverting a matrix with pure Python. It’s important to note that A must be a square matrix to be inverted. T. Returns the transpose of the matrix. NumPy Linear Algebra Exercises, Practice and Solution: Write a NumPy program to compute the inverse of a given matrix. 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. Doing the math to determine the determinant of the matrix, we get, (8) (3)- … Matrix Multiplication in NumPy is a python library used for scientific computing. ], [ 1.5, -0.5]]) Inverses of several matrices can be computed at … We then divide everything by, 1/determinant. 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. 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. DON’T PANIC. How to do gradient descent in python without numpy or scipy. An identity matrix of size $n$ is denoted by $I_{n}$. Plus, tomorrow… We will see two types of matrices in this chapter. To work with Python Matrix, we need to import Python numpy module. 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. 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. My approach using numpy / scipy is below. Matrix methods represent multiple linear equations in a compact manner while using the existing matrix library functions. The identity matrix or the inverse of a matrix are concepts that will be very useful in the next chapters. The second matrix is of course our inverse of A. Published by Thom Ives on November 1, 2018November 1, 2018. It is imported and implemented by LinearAlgebraPractice.py. which clearly indicate that writing one column of inverse matrix to hdf5 takes 16 minutes. The way that I was taught to inverse matrices, in the dark ages that is, was pure torture and hard to remember! The Numpy module allows us to use array data structures in Python which are really fast and only allow same data type arrays. 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. Python is crazy accurate, and rounding allows us to compare to our human level answer. I_{1} = Why wouldn’t we just use numpy or scipy? 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. I’ve also saved the cells as MatrixInversion.py in the same repo. You don’t need to use Jupyter to follow along. 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. 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… \end{bmatrix} $$. \end{bmatrix} The numpy.linalg.det() function calculates the determinant of the input matrix. The NumPy code is as follows. 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. It all looks good, but let’s perform a check of A \cdot IM = I. In other words, for a matrix [[a,b], [c,d]], the determinant is computed as ‘ad-bc’. However, compared to the ancient method, it’s simple, and MUCH easier to remember. The first matrix in the above output is our input A 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. Let’s first introduce some helper functions to use in our notebook work. , ... dtype. 1 & 3 & 3 \\ 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. When what was A becomes an identity matrix, I will then be A^{-1}. ctypes. which is its inverse. You can verify the result using the numpy.allclose() function. See the code below. 1 & 0 \\ 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. Here are the steps, S, that we’d follow to do this for any size matrix. Python matrix determinant without numpy. 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. If the generated inverse matrix is correct, the output of the below line will be True. I_{2} = See if you can code it up using our matrix (or matrices) and compare your answer to our brute force effort answer. Below is the output of the above script. $$ 0 & 1 & 0\\ If a is a matrix object, then the return value is a matrix as well: >>> ainv = inv ( np . 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}. 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. 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. Plus, tomorrows machine learning tools will be developed by those that understand the principles of the math and coding of today’s tools. Use the “inv” method of numpy’s linalg module to calculate inverse of a Matrix. \end{bmatrix} We’ll call the current diagonal element the focus diagonal element, or fd for short.

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