answer:Introduction: TrueBlue MD is a healthtech startup that specializes in mitochondrial genetics and epigenomics, providing personalized and effective treatment plans through mitochondrial genome sequencing and epigenetic profiling. The company aims to generate 5 million in revenue by 2025 and currently has a revenue of 800,000 for 2024. In this proposal, we will develop a comprehensive commercial plan that will help the company expand its business to the rare fields of mitochondrial genetics and epigenomics. Target Audience: Our target audience includes medical professionals specializing in mitochondrial disorders, aging-related diseases, and regenerative medicine. Medical professionals sign up for our service and provide us with their patients' genetic testing needs. We charge a fee per test and offer a range of pricing options depending on the number and type of tests ordered. Objective: The objective of our commercial plan is to expand TrueBlue MD's business to the rare fields of mitochondrial genetics and epigenomics, generating 5 million in revenue by 2025. The plan should identify potential markets, develop a comprehensive action plan, determine the staffing required to execute the plan, and provide recommendations on which states to target based on the number of medical professionals, acceptance of mitochondrial testing and epigenetics, and ease of entry. It should also include a list of possible keywords to use and examples of ad content. Action Plan: 1. Market Analysis: The first step is to analyze the market to identify potential markets for TrueBlue MD's services. We will analyze the number of medical professionals in each state, acceptance of mitochondrial testing and epigenetics, and ease of entry. 2. Action Plan: Based on the market analysis, we will develop a comprehensive action plan to expand TrueBlue MD's business to the rare fields of mitochondrial genetics and epigenomics. The plan should include the services to be provided, pricing strategies, marketing activities, and channels to be used. 3. Staffing: We will determine the staffing required to execute the plan. This will include identifying the roles required and the associated costs to run the commercial function. 4. State Targeting: We will provide recommendations on which states to target based on the number of medical professionals, acceptance of mitochondrial testing and epigenetics, and ease of entry. 5. Keywords and Ad Content: We will provide a list of possible keywords to use and examples of ad content to be used in promoting TrueBlue MD's services. Channels: We will use various channels to promote TrueBlue MD's services to our target audience. These will include: 1. Email Campaigns: TrueBlue MD currently engages in email campaigns to promote its services to medical professionals' offices. We will continue to use this channel to promote our services. 2. Telemarketing: We will also engage in telemarketing activities to reach out to medical professionals who have not yet signed up for our services. 3. Conference and Events: We will participate in mitochondrial medicine and epigenetics conferences and events to network with potential clients and partners. Conclusion: In conclusion, developing a comprehensive commercial plan is critical to TrueBlue MD's success in expanding its business to the rare fields of mitochondrial genetics and epigenomics. The plan should identify potential markets, develop a comprehensive action plan, determine the staffing required to execute the plan, and provide recommendations on which states to target based on the number of medical professionals, acceptance of mitochondrial testing and epigenetics, and ease of entry. It should also include a list of possible keywords to use and examples of ad content. By following this plan, TrueBlue MD can achieve its revenue goal of 5 million by 2025., Here's a comprehensive commercial plan for TrueBlue MD to expand its business in the field of mitochondrial genetics and epigenomics: **Objectives:** 1. Increase revenue to 5 million by 2025. 2. Expand the customer base by targeting medical professionals in specific states. 3. Increase brand awareness and establish TrueBlue MD as a leader in mitochondrial genetics and epigenomics. **Action Plan:** 1. **Market Research:** Conduct market research to identify the top 10 states with the highest number of medical professionals specializing in mitochondrial disorders, aging-related diseases, and regenerative medicine. Additionally, research acceptance of mitochondrial testing and epigenetics in those states and the ease of entry. 2. **Keyword Research:** Identify relevant keywords such as "mitochondrial genetics," "epigenomics," "mitochondrial testing," "epigenetic markers," and "personalized treatment plans" to optimize website content and AdWords campaigns. 3. **Website Optimization:** Optimize the website for search engines and user experience, ensuring content is up-to-date and informative. Additionally, include clear calls-to-action and pricing information. 4. **Ad Content:** Develop ad content for Google AdWords campaigns targeting the identified keywords and medical professionals in the top 10 states. Additionally, create social media content and email campaigns to promote the services and establish thought leadership. 5. **Partnerships:** Partner with medical institutions, laboratories, and other healthcare organizations to offer joint services and expand the reach of TrueBlue MD's offerings. 6. **Sales Strategy:** Develop a sales strategy targeting medical professionals and institutions, ensuring a clear value proposition and competitive pricing. 7. **Customer Service:** Establish excellent customer service practices, ensuring medical professionals are provided with prompt and accurate test results and outstanding customer support. **Channels:** 1. Google AdWords campaigns targeting identified keywords and medical professionals in the top 10 states. 2. Social media campaigns on platforms such as LinkedIn and Twitter. 3. Email campaigns targeting medical professionals and institutions. 4. Participation in mitochondrial medicine and epigenetics conferences and events. 5. Partnerships with medical institutions, laboratories, and other healthcare organizations. **Staffing:** 1. A dedicated sales team to target medical professionals and institutions. 2. A customer service team to provide prompt and accurate test results and support. 3. A marketing team to create and execute AdWords campaigns, social media content, and email campaigns. 4. A partnership manager to establish and maintain relationships with medical institutions, laboratories, and other healthcare organizations. **Budget:** 1. AdWords campaigns: 200,000 per year. 2. Social media and email campaigns: 50,000 per year. 3. Conference and event participation: 75,000 per year. 4. Staffing: 500,000 per year. 5. Partnership management: 75,000 per year. **Recommendations for States to Target:** 1. California: High number of medical professionals and acceptance of mitochondrial testing and epigenetics. 2. Texas: High number of medical professionals and potential for growth in the field. 3. New York: High number of medical professionals and potential for partnerships with laboratories and research institutions. 4. Florida: High number of medical professionals and potential for growth in the field. 5. Pennsylvania: High number of medical professionals and potential for partnerships with medical institutions. 6. Massachusetts: High number of medical professionals and potential for growth in the field. 7. Illinois: High number of medical professionals and potential for partnerships with laboratories and research institutions. 8. Michigan: High number of medical professionals and potential for growth in the field. 9. Ohio: High number of medical professionals and potential for partnerships with medical institutions. 10. Georgia: High number of medical professionals and potential for growth in the field. By following this comprehensive commercial plan, TrueBlue MD can achieve its revenue goals and establish itself as a leader in the rare fields of mitochondrial genetics and epigenomics.,It seems that TrueBlue MD already has a well-established service and goal, but requires assistance in developing a comprehensive commercial plan. Here is a framework to help you develop this plan: 1. Objective setting: Start by setting clear and specific objectives for revenue, target market growth, and customer satisfaction. These objectives will guide your commercial efforts. 2. Analysis and research: Analyze your target market and competition to identify key gaps and opportunities within the field of mitochondrial genetics and epigenomics. You could leverage existing data and market research to gain insights into your target audience, the size of your market, and the competition. 3. Action plan: Develop an action plan that outlines the steps you need to take to achieve your objectives. This could include marketing strategies, sales tactics, and customer acquisition plan, among others. 4. Channels and staffing: Identify the channels through which you will market your services such as digital advertising, content marketing, website, and email marketing. Define the skills and roles you will require to manage these channels and estimate the associated costs. 5. Budget: Based on your revenue goals and objectives, set a realistic budget for marketing, sales, and other commercial activities. Consider allocating a percentage of your revenue for continuous market research and improvement. 6. Keyword research: Use keyword research tools to identify the top-performing keywords in your industry. Based on your target audience and search trends, develop a list of relevant keywords and include them in your digital marketing efforts. 7. Ad content: Write clear, targeted ad copy that reflects your value proposition and communicates your unique selling points effectively. Use storytelling and thought leadership to differentiate yourself from your competitors. 8. Measuring and monitoring: As you implement your commercial plan, track your performance metrics regularly to gauge the effectiveness of your strategies. Use this information to make informed strategic decisions and continuously improve your approach. 9. State targeting: Utilize data and research to identify states with a high concentration of medical professionals and a high acceptance rate for mitochondrial testing and epigenetics. These states will be ideal for your expansion. Remember, each component of your commercial plan should be driven by insights and data. Conduct thorough research and continually listen to your customers to stay on top of market trends and make data-driven decisions.

answer:The output of the "writeln" function in PHP can be affected by changing the configuration settings related to buffering output. Buffering output in PHP enables control over when data is sent to the web browser. Buffering output can significantly improve performance by allowing PHP to store the output in memory rather than sending it to the browser immediately, which can be useful when dealing with large amounts of data. When the output buffering configuration settings are changed in PHP, this affects the behavior of both "writeln" and "echo". By default, both functions are affected by output buffering. This means that the output will be buffered until some event (such as a call to an "ob_*" function or a direct output with a "flush()" call) triggers the content to be sent to the browser. If you disable output buffering in the PHP configuration, any change you make to the "writeln" function or "echo" will immediately display on the web page without the intervention of output buffering. The behavior of "writeln" and "echo" when the output buffering configuration settings are changed is quite the same. Both functions are affected by the changes in output buffering, and their behavior will be synchronized., In PHP, the output buffering mechanisms can be controlled using several configuration settings, such as `output_buffering`, `zlib.output_compression`, and `implicit_flush`. These settings can have an impact on the behavior of the `writeln()` function, which is typically a custom function that combines the `echo` statement with a newline character (`n`). 1. output_buffering: When this configuration setting is enabled (e.g., `output_buffering = 4096` in php.ini), PHP automatically buffers output for performance reasons. This means that the output of `writeln()` (or any other output function) will not be immediately sent to the browser. Instead, it will be stored in an internal buffer until the buffer is full, or until you explicitly flush the buffer using `flush()` or `ob_flush()`. For example, consider the following code: ```php ini_set('output_buffering', 1024); function writeln(line) { echo line . "n"; } writeln("Hello, World!"); ``` In this case, the output won't be displayed until the buffer is full (1024 bytes) or flushed manually. To address this, you can either disable output buffering or manually flush the buffer. 2. zlib.output_compression: This setting enables output compression using zlib. When enabled (e.g., `zlib.output_compression = On` in php.ini), the output is compressed before being sent to the browser. This can lead to a slight delay in the display of the output since compression can take some time. The `writeln()` function's output will be compressed along with other output generated by the script. The `echo` function will behave similarly to `writeln()` in this scenario, as it is affected by the same PHP configuration settings. 3. implicit_flush: When this configuration setting is enabled (e.g., `implicit_flush = On` in php.ini), PHP automatically calls `flush()` after each output statement, effectively disabling output buffering. This ensures that the output from `writeln()` (or any other output function) is displayed immediately, without waiting for a buffer to fill up or for an explicit `flush()` call. This setting is useful when you want to see the output generated by a script in real-time, such as in a long-running script or a progress indicator. However, it is important to note that using `implicit_flush` can have a negative impact on performance, as the output is not buffered and must be processed and transmitted to the browser after each output statement. When comparing the `writeln()` function to the `echo` function in the context of PHP configuration settings affecting output buffering, both functions behave similarly. The difference between these functions lies in their implementations: `echo` is a built-in language construct, while `writeln()` is often a custom function that combines `echo` with a newline character. However, their output will be subject to the same PHP configuration settings.,The "writeln" function in PHP outputs a string to the browser and appends a newline character at the end. The output of the "writeln" function is affected by changing the PHP configuration settings related to buffering output in the following ways: 1. Output buffering: PHP provides the ability to buffer the output of the script until the script completes its execution. If the output buffering is turned on, the "writeln" function will not immediately output the string to the browser. Instead, it will be stored in the buffer until the script completes its execution or the buffer is flushed manually. Once the buffer is flushed, the entire string will be output to the browser at once. 2. Implicit flush: PHP also provides an option to enable implicit flushing, which means that PHP will automatically flush the output buffer after a certain amount of output is accumulated. If this option is enabled, the "writeln" function will immediately output the string to the browser and flush the buffer. On the other hand, the "echo" function in PHP always outputs the string immediately to the browser regardless of the buffering settings. Therefore, changing the buffering settings does not affect the behavior of the "echo" function.

answer:The professional journey of a data analyst who has transitioned from academia to the tech industry is quite dynamic and challenging. After completing their advanced degree in a field such as statistics, mathematics, or economics, they may have started their career in academia, conducting research and teaching classes. However, they might have felt drawn to the tech industry due to the increasing demand for data analysis skills and the potential to make a more direct impact on business outcomes. Once they made the transition to the tech industry, they likely started working on complex projects such as designing and implementing data pipelines for large-scale e-commerce platforms. This involved working with large and complex datasets to extract, transform, and load data into data warehouses or data lakes, where it could be used for analysis and reporting. They may have also worked on developing predictive models using machine learning algorithms to optimize marketing campaigns and reduce customer churn. This required a strong understanding of machine learning techniques, as well as the ability to work with large datasets and implement models in a production environment. In addition to their technical expertise, they also developed a strong ability to collaborate with cross-functional teams, including product managers, engineers, and business stakeholders. This involved communicating complex data insights in a clear and concise way, as well as working closely with teams to ensure that data analysis was aligned with business goals and strategies. They also developed expertise in extracting meaningful insights from unstructured data, such as text data from customer reviews or social media posts, using tools like natural language processing and sentiment analysis. Their technical expertise includes proficiency in tools such as SQL, Python, and Tableau. They have likely used SQL to extract and manipulate data from databases, Python to build predictive models and perform statistical analysis, and Tableau to create visualizations and dashboards. Their ability to apply statistical methods to derive actionable insights from data is also a critical part of their skillset. They have likely used techniques such as A/B testing and regression analysis to understand the impact of different variables on business outcomes and make data-driven recommendations. Their contribution to driving data-driven decision-making and their success in improving business outcomes, such as increasing revenue and reducing costs, is a key part of their professional journey. They have likely worked on projects that have resulted in significant cost savings, increased revenue, or improved customer satisfaction. They have also demonstrated a strong ability to communicate the impact of their work to stakeholders, using data visualization and storytelling techniques to make a compelling case for data-driven decision-making. In terms of industry certifications, they may have earned certifications in areas such as data science, machine learning, or data visualization. They may also have published research or articles in academic or industry publications, showcasing their expertise and contributing to the broader field of data analysis. Finally, their experience working in a fast-paced, agile environment is critical to their success in the tech industry. They have likely worked on projects with tight deadlines and shifting priorities, requiring them to be adaptable and flexible. They have also developed leadership skills in mentoring junior analysts and collaborating with stakeholders to drive strategic initiatives. Their ability to lead cross-functional teams, communicate effectively, and drive impactful results has made them a valuable asset to any organization.,The data analyst in question started their career in academia, where they honed their analytical skills and gained a deep understanding of statistical methods. They then made the transition to the tech industry, where they have taken on a variety of complex projects. Their work has included designing and implementing data pipelines for large-scale e-commerce platforms, developing predictive models using machine learning algorithms to optimize marketing campaigns and reduce customer churn, and collaborating with cross-functional teams to extract meaningful insights from unstructured data. The analyst is well-versed in a variety of technical tools, including SQL, Python, and Tableau. They are proficient in using these tools to manipulate data, conduct analyses, and visualize insights. They have a strong understanding of statistical methods, such as A/B testing and regression analysis, which they use to derive actionable insights from data. Their ability to apply these techniques has helped them to make data-driven decisions and improve business outcomes, such as increasing revenue and reducing costs. The analyst has also achieved notable industry certifications, such as the Tableau Desktop Specialist certification, which demonstrates their expertise in using Tableau to create data visualizations. Additionally, they have published several academic papers on statistical methods and their applications in the tech industry, showcasing their ability to bridge the gap between academia and industry. In a fast-paced, agile environment, the analyst has demonstrated their ability to thrive. They have a keen eye for detail and are able to work quickly and accurately to meet project deadlines. Their leadership skills have also been evident, as they have taken on the role of mentoring junior analysts and collaborating with stakeholders to drive strategic initiatives. Their ability to communicate complex insights in a clear and concise manner has helped to drive data-driven decision-making across the organization.,The data analyst in question embarked on their professional journey in academia, building a strong foundation in statistical methods and data analysis techniques. It was this foundation that enabled them to transition smoothly into the tech industry, where their expertise in tools like SQL, Python, and Tableau became invaluable. They began their tech career working on complex projects involving data pipelines for large-scale e-commerce platforms. These required a keen understanding of data processing and management, as well as an ability to optimize data flows for maximum efficiency. They quickly expanded their responsibilities to include developing predictive models using machine learning algorithms to optimize marketing campaigns and reduce customer churn. This required not only technical expertise but also a keen understanding of business goals and objectives, and how to align data-driven insights with those objectives. Their ability to collaborate with cross-functional teams to extract meaningful insights from unstructured data further contributed to their success as a data analyst. Their technical skills, combined with their ability to understand and communicate complex ideas, made them an invaluable asset to their teams. In terms of their technical expertise, they have demonstrated a thorough understanding of statistical methods, including A/B testing and regression analysis. They have also shown a deep understanding of Python and SQL, which they've used to design efficient data pipelines and build predictive models. They are proficient in Tableau, using its powerful visualizations to bring data insights to life. Throughout their career, they have contributed to driving data-driven decision-making, leading to significant improvements in business outcomes, such as increased revenue and reduced costs. Their achievements in these areas have been recognized by their peers, earning them notable industry certifications and publications. Finally, their experience working in a fast-paced, agile environment has honed their ability to juggle multiple, complex projects simultaneously. They have also demonstrated excellent leadership skills, mentoring junior analysts and collaborating with stakeholders to drive strategic initiatives.

answer:Creating a step-by-step visualization of the Singular Value Decomposition (SVD) process for matrices using the manim library is a complex task that requires a deep understanding of both linear algebra and computer graphics. I'll provide an outline and some code snippets to help you get started. However, due to the complexity and length of the code, I cannot provide the complete code in this response. First, let's define the SVD of a matrix A: A = U * S * V^T where U is an orthogonal matrix, S is a diagonal matrix with non-negative entries, and V^T is the transpose of an orthogonal matrix V. Here's an outline of how to create an interactive SVD visualization: 1. Define the input matrix, and create methods to change its values interactively. 2. Compute the SVD of the input matrix and store the results (U, S, V^T). 3. Create animations to visualize each step of the SVD process. 4. Combine all animations and interactions into a single manim scene. Now, let's look at some code snippets that demonstrate how to achieve each of these steps: 1. Define and interact with the input matrix: ```python class SVDScene(Scene): def __init__(self, matrix_size=3, **kwargs): self.matrix_size = matrix_size self.input_matrix = Matrix( random.random() for _ in range(matrix_size) for _ in range(matrix_size) ) self.matrix_input = MatrixInput( self.input_matrix, lambda m: self.input_matrix.set_entries(m.get_matrix()), width=FRAME_WIDTH - 2, height=FRAME_HEIGHT - 2, **kwargs ) super().__init__() def construct(self): self.add(self.matrix_input) ``` 2. Compute the SVD of the input matrix: ```python def compute_svd(self): self.U, self.S, self.V_T = svd(self.input_matrix.get_matrix()) ``` 3. Visualize each step of the SVD process: - Computing the QR decomposition of A: A = Q * R ```python def create_qr_decomposition(self, matrix): q, r = manim_linear_algebra.qr_decomposition(matrix) qr_decomposition = VGroup( Matrix(q), Matrix(r, buff=0.1) ).arrange(direction=DOWN, aligned_edge=LEFT) qr_label = Tex('A = QR').next_to(qr_decomposition, UP, buff=0.5) return VGroup(qr_decomposition, qr_label) ``` - Computing the QR decomposition of A^T: A^T = Q' * R' ```python def create_qr_decomposition_transpose(self, matrix): q, r = manim_linear_algebra.qr_decomposition(matrix.transpose()) qr_decomposition_transpose = VGroup( Matrix(q), Matrix(r, buff=0.1) ).arrange(direction=DOWN, aligned_edge=LEFT) qr_label_transpose = Tex('A^T = Q'R'').next_to( qr_decomposition_transpose, UP, buff=0.5) return VGroup(qr_decomposition_transpose, qr_label_transpose) ``` - Computing the SVD of R (or R' for even dimensions) using power iteration: ```python def create_svd_r(self, matrix, svd_r): u, s, v = svd_r svd_r_components = VGroup( Matrix(u), Matrix(s.to_numpy().reshape(-1, 1), buff=0.1), Matrix(v.transpose()) ).arrange(direction=DOWN, aligned_edge=LEFT) svd_r_label = Tex('R = USV^T').next_to(svd_r_components, UP, buff=0.5) return VGroup(svd_r_components, svd_r_label) ``` 4. Combine animations and interactions into a single manim scene: ```python class InteractiveSVDScene(SVDScene): def construct(self): self.compute_svd() # Add the steps here, e.g: qr_decomposition = self.create_qr_decomposition(self.input_matrix.get_matrix()) self.play(Create(qr_decomposition)) self.wait(1) # ... self.wait(1) ``` You can extend this outline and code snippets by adding more steps and animations to visualize the full SVD process. Note that this response does not provide caching, parallel processing, or optimized algorithms, as this would add complexity to the example and depend on use-cases. However, these techniques can be implemented using libraries like `functools` (for caching) in combination with your preferred concurrency and parallelism methods. For a deeper understanding of linear algebra, matrix decompositions, and computer graphics, consider studying resources like 3Blue1Brown's Essence of Linear Algebra series (<https://www.3blue1brown.com/topics/linear-algebra>).,Unfortunately, as an AI, I am not capable of coding or running code, but I will guide you through the steps needed to create an animation using the 3b1b/manim library to illustrate the Singular Value Decomposition (SVD) for matrices. 1. First, you need to install the 3b1b/manim library. You can install it using pip: ``` pip install manim ``` 2. To create an animation using manim, you need to write a Python script that defines the objects, colors, and animations you want to include. You can create a separate file for this or use an existing SVD Python script as a starting point. 3. To visualize the key steps and components of the decomposition, you can use various manim's functionalities, such as drawing colors, circles, polygons, etc. You can also use these objects to represent the operations in the SVD process, such as calculating the row sums, pivoting the matrix, and sorting the rows in descending order. 4. Optimize the code for efficiency by using caching and parallel processing. To cache results, you can store them in global variables, and to parallelize the computations, you can use libraries like Numba or Dask. These techniques can significantly reduce the execution time of the SVD algorithm, especially for large matrices. 5. Make your code interactively change the input matrix, allowing the user to modify it and visualize the corresponding SVD decomposition. This can be achieved through interactive tools like Jupyter notebooks or by implementing a function that updates the input matrix and reruns the script automatically. 6. Finally, execute your script using the command `manim SVD_animation.py SVD_Scene --debug` to see the SVD animation. This command runs the "SVD_Scene" scene from the "SVD_animation.py" script, and debug mode allows you to see the intermediate steps of the animation.,To create a step-by-step visualization of the Singular Value Decomposition (SVD) process for matrices using manim library, we need to break down the steps involved in the decomposition process and create animations for each of those steps. The SVD process involves decomposing a given matrix A into three matrices U, Sigma, and V^T such that: A = U Sigma V^T where U is an orthogonal matrix, Sigma is a diagonal matrix with non-negative elements, and V is an orthogonal matrix. Here are the steps we will visualize: 1. Compute A^TA and AA^T matrices 2. Compute the eigenvalues and eigenvectors of A^TA and AA^T matrices 3. Sort the eigenvalues and corresponding eigenvectors in descending order 4. Construct the matrix Sigma using the sorted eigenvalues 5. Normalize the eigenvectors of A^TA and AA^T to obtain the columns of U and V matrices, respectively 6. Create the final SVD decomposition of the input matrix A using U, Sigma, and V^T matrices For the animations, we will use the `ManimCE` version of manim library, which is an open-source animation engine for explanatory math videos. We will create a class `SVDScene` that inherits from `Scene` class and contains the animation code. Here's a code snippet to get started with the `SVDScene` class: ```python from manim import * import numpy as np class SVDScene(Scene): def construct(self): # Define input matrix A A = np.array(3, 2, 2, 3) # Compute A^TA and AA^T matrices ATA = A.T @ A AAT = A @ A.T # Compute eigenvalues and eigenvectors of ATA and AAT matrices eigvals_ATA, eigvecs_ATA = np.linalg.eig(ATA) eigvals_AAT, eigvecs_AAT = np.linalg.eig(AAT) # Sort eigenvalues and corresponding eigenvectors in descending order idx_ATA = eigvals_ATA.argsort()::-1 eigvals_ATA = eigvals_ATAidx_ATA eigvecs_ATA = eigvecs_ATA:,idx_ATA idx_AAT = eigvals_AAT.argsort()::-1 eigvals_AAT = eigvals_AATidx_AAT eigvecs_AAT = eigvecs_AAT:,idx_AAT # Construct matrix Sigma using sorted eigenvalues Sigma = np.zeros(A.shape) Sigma:min(A.shape), :min(A.shape) = np.diag(np.sqrt(eigvals_ATA)) # Normalize eigenvectors to obtain U and V matrices U = eigvecs_AAT / np.sqrt(eigvals_AAT) V = eigvecs_ATA / np.sqrt(eigvals_ATA) # Create final SVD decomposition of matrix A U_matrix = Matrix(U) Sigma_matrix = Matrix(Sigma) V_matrix = Matrix(V.T) A_matrix = Matrix(A) SVD_matrix = U_matrix @ Sigma_matrix @ V_matrix # Add animations for each step of SVD process ... ``` We start by defining the input matrix `A` and then computing the `A^TA` and `AA^T` matrices. We then use the `numpy.linalg.eig` function to compute the eigenvalues and eigenvectors of these matrices and sort them in descending order. We construct the diagonal matrix `Sigma` using the sorted eigenvalues and normalize the eigenvectors to obtain the `U` and `V` matrices. Finally, we create the final SVD decomposition matrix `SVD_matrix` using these matrices. Now, let's add the animations for each step of the SVD process. We will use the `Create`, `Transform`, and `Write` functions of manim library to create various mathematical objects and animate them. ```python from manim import * import numpy as np class SVDScene(Scene): def construct(self): # Define input matrix A A = np.array(3, 2, 2, 3) # Compute A^TA and AA^T matrices ATA = A.T @ A AAT = A @ A.T # Compute eigenvalues and eigenvectors of ATA and AAT matrices eigvals_ATA, eigvecs_ATA = np.linalg.eig(ATA) eigvals_AAT, eigvecs_AAT = np.linalg.eig(AAT) # Sort eigenvalues and corresponding eigenvectors in descending order idx_ATA = eigvals_ATA.argsort()::-1 eigvals_ATA = eigvals_ATAidx_ATA eigvecs_ATA = eigvecs_ATA:,idx_ATA idx_AAT = eigvals_AAT.argsort()::-1 eigvals_AAT = eigvals_AATidx_AAT eigvecs_AAT = eigvecs_AAT:,idx_AAT # Construct matrix Sigma using sorted eigenvalues Sigma = np.zeros(A.shape) Sigma:min(A.shape), :min(A.shape) = np.diag(np.sqrt(eigvals_ATA)) # Normalize eigenvectors to obtain U and V matrices U = eigvecs_AAT / np.sqrt(eigvals_AAT) V = eigvecs_ATA / np.sqrt(eigvals_AAT) # Create final SVD decomposition of matrix A U_matrix = Matrix(U) Sigma_matrix = Matrix(Sigma) V_matrix = Matrix(V.T) A_matrix = Matrix(A) SVD_matrix = U_matrix @ Sigma_matrix @ V_matrix # Add animations for each step of SVD process A_text = TextMobject("A=", A_matrix).scale(1.5).to_edge(UP) self.play(Write(A_text)) ATA_text = TextMobject("A^TA=", Matrix(ATA)).scale(1.5).next_to(A_text, DOWN) self.play(Create(ATA_text)) AAT_text = TextMobject("AA^T=", Matrix(AAT)).scale(1.5).next_to(ATA_text, DOWN) self.play(Create(AAT_text)) eigenvectors_ATA_text = TextMobject("Eigenvectors of A^TA").to_edge(LEFT) eigenvectors_AAT_text = TextMobject("Eigenvectors of AA^T").to_edge(LEFT) self.play(Create(eigenvectors_ATA_text)) eigenvectors_ATA = Matrix(eigvecs_ATA).scale(0.75).next_to(eigenvectors_ATA_text, DOWN) self.play(Create(eigenvectors_ATA)) eigenvectors_AAT = Matrix(eigvecs_AAT).scale(0.75).next_to(eigenvectors_AAT_text, DOWN) self.play(Transform(eigenvectors_ATA, eigenvectors_AAT)) Sigma_text = TextMobject("Sigma=", Sigma_matrix).scale(1.5).next_to(AAT_text, DOWN) self.play(Create(Sigma_text)) U_text = TextMobject("U=", U_matrix).scale(1.5).next_to(Sigma_text, DOWN) self.play(Create(U_text)) V_text = TextMobject("V^T=", V_matrix).scale(1.5).next_to(U_text, DOWN) self.play(Create(V_text)) SVD_text = TextMobject("SVD of A").to_edge(RIGHT) self.play(Create(SVD_text)) self.play(Transform(A_text, SVD_matrix)) ``` We start by creating a `TextMobject` with the input matrix `A` and use the `Write` function to animate it onto the scene. We then create `TextMobject`s with the `A^TA` and `AA^T` matrices and use the `Create` function to animate them onto the scene. We also create `TextMobject`s for the eigenvectors of these matrices and use the `Create` function to animate them onto the scene. We use the `Transform` function to animate the eigenvectors of `AA^T` matrix to the position of the eigenvectors of `A^TA` matrix. We create a `TextMobject` with the diagonal matrix `Sigma` and use the `Create` function to animate it onto the scene. We do the same for the `U` and `V` matrices. Finally, we create a `TextMobject` with the SVD decomposition of `A` and use the `Transform` function to animate it from the `A` matrix. This code creates a basic visualization of the SVD process. However, to make it more interactive, we need to add user input and a feature to update the SVD decomposition when the input matrix changes. We will do this by creating a custom `MatrixInput` class that allows the user to input a matrix and updating the animation when the matrix changes. ```python from manim import * import numpy as np class MatrixInput(VGroup