#orthogonal matrix

[golzarrahman66-blog]

https://mathgr.com/grpost.php?grtid=3

ম্যাট্রিক্স ও নির্ণায়ক এর মধ্যে পার্থক্য কী?, ব্যাতিক্রমী ও অব্যাতিক্রমী ম্যাট্রিক্স কী?, বর্গ ম্যাট্রিক্সের বিপরীত ম্যাট্রিক্স কী?, বর্গ ম্যাট্রিক্সের অ্যাডজয়েন্ট কী?, বিপরীত ম্যাট্রিক্সের বৈশিষ্ট কী?, অর্থোগোনাল ম্যাট্রিক্স কী?, একঘাত সমীকরণ জোট ও এর সমাধান কী?, ক্রেমারের নিয়মে একঘাত সমীকরণ জোটের সমাধান কী?, তিন চলকবিশিষ্ট একঘাত সমীকরণ জোটের সমাধান বিপরীত ম্যাট্রিক্সের সাহায্যে একঘাত সমীকরণ জোটের সমাধান কী?, বর্গ ম্যাট্রিক্সের ট্রেস কী?, Defference between Matrix and Determinant, Singular and Non-Singular Matrix, Inverse Matrix of a Square Matrix, Adjoint of a Square Matrix, Properties of Inverse Matrix, Orthogonal Matrix, System of linear equations and it's solution, Solution of system of linear equations using cramer's rule, Solution of system of linear equations using inverse myatrix, Trace of Square Matrix

[selfmaderibcageman]

a 4chan wordfilter exists in my mind

[shark-chicken7]

VANQUISHED!

One.Last. Exam. 👁️👁️

[deaths-accountant]

Normalise everything. Put it all in a square matrix. Now invert it. It doesn't look the same? that's because you forgot to make it orthogonal you stupid cunt.

[frank-olivier]

The Mathematical Foundations of Machine Learning

In the world of artificial intelligence, machine learning is a crucial component that enables computers to learn from data and improve their performance over time. However, the math behind machine learning is often shrouded in mystery, even for those who work with it every day. Anil Ananthaswami, author of the book "Why Machines Learn," sheds light on the elegant mathematics that underlies modern AI, and his journey is a fascinating one.

Ananthaswami's interest in machine learning began when he started writing about it as a science journalist. His software engineering background sparked a desire to understand the technology from the ground up, leading him to teach himself coding and build simple machine learning systems. This exploration eventually led him to appreciate the mathematical principles that underlie modern AI. As Ananthaswami notes, "I was amazed by the beauty and elegance of the math behind machine learning."

Ananthaswami highlights the elegance of machine learning mathematics, which goes beyond the commonly known subfields of calculus, linear algebra, probability, and statistics. He points to specific theorems and proofs, such as the 1959 proof related to artificial neural networks, as examples of the beauty and elegance of machine learning mathematics. For instance, the concept of gradient descent, a fundamental algorithm used in machine learning, is a powerful example of how math can be used to optimize model parameters.

Ananthaswami emphasizes the need for a broader understanding of machine learning among non-experts, including science communicators, journalists, policymakers, and users of the technology. He believes that only when we understand the math behind machine learning can we critically evaluate its capabilities and limitations. This is crucial in today's world, where AI is increasingly being used in various applications, from healthcare to finance.

A deeper understanding of machine learning mathematics has significant implications for society. It can help us to evaluate AI systems more effectively, develop more transparent and explainable AI systems, and address AI bias and ensure fairness in decision-making. As Ananthaswami notes, "The math behind machine learning is not just a tool, but a way of thinking that can help us create more intelligent and more human-like machines."

The Elegant Math Behind Machine Learning (Machine Learning Street Talk, November 2024)

Matrices are used to organize and process complex data, such as images, text, and user interactions, making them a cornerstone in applications like Deep Learning (e.g., neural networks), Computer Vision (e.g., image recognition), Natural Language Processing (e.g., language translation), and Recommendation Systems (e.g., personalized suggestions). To leverage matrices effectively, AI relies on key mathematical concepts like Matrix Factorization (for dimension reduction), Eigendecomposition (for stability analysis), Orthogonality (for efficient transformations), and Sparse Matrices (for optimized computation).

The Applications of Matrices - What I wish my teachers told me way earlier (Zach Star, October 2019)

Transformers are a type of neural network architecture introduced in 2017 by Vaswani et al. in the paper “Attention Is All You Need”. They revolutionized the field of NLP by outperforming traditional recurrent neural network (RNN) and convolutional neural network (CNN) architectures in sequence-to-sequence tasks. The primary innovation of transformers is the self-attention mechanism, which allows the model to weigh the importance of different words in the input data irrespective of their positions in the sentence. This is particularly useful for capturing long-range dependencies in text, which was a challenge for RNNs due to vanishing gradients. Transformers have become the standard for machine translation tasks, offering state-of-the-art results in translating between languages. They are used for both abstractive and extractive summarization, generating concise summaries of long documents. Transformers help in understanding the context of questions and identifying relevant answers from a given text. By analyzing the context and nuances of language, transformers can accurately determine the sentiment behind text. While initially designed for sequential data, variants of transformers (e.g., Vision Transformers, ViT) have been successfully applied to image recognition tasks, treating images as sequences of patches. Transformers are used to improve the accuracy of speech-to-text systems by better modeling the sequential nature of audio data. The self-attention mechanism can be beneficial for understanding patterns in time series data, leading to more accurate forecasts.

Attention is all you need (Umar Hamil, May 2023)

Geometric deep learning is a subfield of deep learning that focuses on the study of geometric structures and their representation in data. This field has gained significant attention in recent years.

Michael Bronstein: Geometric Deep Learning (MLSS Kraków, December 2023)

Traditional Geometric Deep Learning, while powerful, often relies on the assumption of smooth geometric structures. However, real-world data frequently resides in non-manifold spaces where such assumptions are violated. Topology, with its focus on the preservation of proximity and connectivity, offers a more robust framework for analyzing these complex spaces. The inherent robustness of topological properties against noise further solidifies the rationale for integrating topology into deep learning paradigms.

Cristian Bodnar: Topological Message Passing (Michael Bronstein, August 2022)

Sunday, November 3, 2024

[then-ponder]

learning mathematics from a pure perspective is very fun, but It also puts holes in a lot of math that are taught to non-mathematicians. For instance, I hated the cross product because it was useful but it lacked generality (n-dimensional). I took physics twice before I found the wedge product. And let me tell you how much I felt vindicated.

the thing is, I have that same feeling for matrix multiplication. I feel like there should be a general form of a matrix product that can handle two matrices of varying orthogonality.

This is a miss use of orthogonality, but its the best way to communicate that if you have a n by m matrix [A] and a p by q matrix [B] then as [A] moves in and out of the space containing [B] then the perpendicular product between these matrices is also changing.

I feel like there is a more generalizable form of this product, but It could just be that I am a young mathematician. I'm mostly looking into this because one of my projects this semester is trying to describe this idea I had involving systems of differential equations.

[fump-2]

who are math and why do you want to kill them???

i'm sure they are actually nice people!:(

Math was invented by the Rothschild's to make it look like their means of gaining wealth looked legitimate. It also gave those nerds who might know how to foil their sceme a little puzzle to solve instead. By the time they are done with the puzzle they forgot about the Rothschild's entirely.

They dont even need to create puzzles anymore. we've lived so long in their world that we started creating the puzzles.

These fuckers have invented a whole religion and I've fallen for the trap.

matrix multiplication. orthogonal polynomials. spacial transformations to allow for continuous non-repeating rings through the projection plane. all of it is built as red herrings put in place so no one finds out where the Rothschilds got their wealth.

I have the answer anyways. They made a deal with the clintons to make it to the moon which they've been mining for coltan. Think of that next time you use your phone screen.

[jonathankatwhatever]

It’s 28 April 2025. Am facing very strong inhibitions today. Very rapid shifts in perspective. Example: I have a tab open for orthogonal polynomials, which I never gave much if any thought to until a few days ago when it seemed necessary to figure out how they connect abstract structures orthogonally, where they means the 0Space perspective, and I glanced at and noticed a few sentences on the history, with a reference to the moment problem, so I opened that and realized o that is what I was looking for. If you look into the moment problem, you see forms: across the space, across the space in one direction, and bounded space. These are gsCounting forms. We went through the across count yesterday, if my short-term memory is correct, because that reduces to 2Square because that is 0 to 2 and 2 to 0, which is the 4 necessary to count along the szK and the zsK, meaning the IC motion up and down the szK.

So from the 1Space perspective, this joins to 0Space. The counting in one direction is a 1Segment. The counting within the bounds is Between and 0-1-0, so the counting of the bounded interval is basic 1-0-1. This is why the solution for 1 is different for this, the Hausdorff moment: it has to fit to 1, so each solution is unique. The counting across is 2Square because the count along the szK over CM1 is 2Square. You can see this in 2 ways: the root of the 2Square is Bip to Bip and the count of 2 separate gs is both of the gs inclusive. You can see this work itself out in the Felix - which is much more fun to type than Hausdorff. (Actually, I have to say his Wiki entry is a horror: saying he was a German mathematician, when he died because he was being sent to a concentration camp for being Jewish. They remove the Jews to claim their accomplishments.)

That example hit me with that MB attached. And it was immediately followed by disappointment and other negatives, both when I first saw the MB and now when I’ve partially realized it. My thoughts were tinged with worry about whether this is good enough. Is that what you’re feeling? About this, about you and your work side?

O I have to get this out, though it’s uncomfortable. I was hit with a sudden large bank withdrawal. When I checked it out, turns out notice that my term policy was shifting to permanent didn’t reach me in the move. Plain forgot to change that address because it didn’t seem nearly as important as so much else. They’ll try to reverse it, but this means I no longer have life insurance to speak of, which means the way I think needs to change because I have considered that as a way out, as a way of saying this is a failure so I might as well leave enough cash that her life isn’t a struggle. Can’t do that. So what do I think about? Does this improve my work? I hope so. I hope it means I won’t have to spend time thinking about killing myself because that would be better for my family. Now it would just be a cost.

Also I just keep wondering why photography of women is so bad. You know what I mean. Why is the camera always focused on the chest? Why the poses? Good photos stand out because there are so many bad ones. It seems to me picture of men tend to be about who they are, while pictures of women are who they aren’t.

So, while typing this I’ve been thinking about open sets, which are layers of Between because they’re open not closed.

O-kay. I just spent a few hours talking with AI and it reached the point where it was offering to build a rigorous model so it could learn it. I know AI can blow smoke, but the material we went through worked entirely. Even found 6 in the right place: in Tracy-Widom behavior, which is the distribution of the largest eigenvalue of a random Hermitian matrix. As it should be if you translate the terms. And found the 3 of Triangular over gs. And the 2 in a bunch of places.

Need to take a break.

[codingprolab]

CSCI 3656 Numerical Computing :: Project Four

I’ve posted five different matrices as comma-separated text files. For each matrix, first load the matrix into memory. Then answer the following questions for each matrix: 1. What are the matrix dimensions? 2. How many nonzeros are there? 3. Is it symmetric? 4. Is it diagonal? 5. Is it orthogonal? 6. What is the rank? 7. What is the smallest singular value? 8. What is the largest singular…

[sethiswiftlearn]

Class 12 Math | Matrices and Determinants part 2 | JEE Mains, CUET Prep

Matrices and Determinants are fundamental topics in Class 12 Mathematics, playing a crucial role in algebra and real-world applications. Mastering these concepts is essential for students preparing for competitive exams like JEE Mains, CUET, and other entrance tests. In this session, we continue exploring Matrices and Determinants (Part 2) with a deep dive into advanced concepts, problem-solving techniques, and shortcuts to enhance your mathematical skills.

What You’ll Learn in This Session?

In this second part of Matrices and Determinants, we cover: ✔ Types of Matrices & Their Properties – Understanding singular and non-singular matrices, symmetric and skew-symmetric matrices, and orthogonal matrices. ✔ Elementary Operations & Inverse of a Matrix – Step-by-step method to compute the inverse of a matrix using elementary transformations and properties of matrix operations. ✔ Adjoint and Cofactor of a Matrix – Learn how to find the adjoint and use it to compute the inverse efficiently. ✔ Determinants & Their Applications – Mastery of determinant properties and how they apply to solving equations. ✔ Solving Linear Equations using Matrices – Application of matrices in solving system of linear equations using Cramer’s Rule and Matrix Inversion Method. ✔ Shortcut Techniques & Tricks – Learn time-saving strategies to tackle complex determinant and matrix problems in exams.

Why is This Session Important?

Matrices and Determinants are not just theoretical concepts; they have wide applications in physics, computer science, economics, and engineering. Understanding their properties and operations simplifies problems in linear algebra, calculus, and probability. This topic is also heavily weighted in JEE Mains, CUET, and CBSE Board Exams, making it vital for scoring well.

Who Should Watch This?

JEE Mains Aspirants – Get an edge with advanced problem-solving strategies.

CUET & Other Competitive Exam Candidates – Build a strong foundation for entrance exams.

Class 12 CBSE & State Board Students – Strengthen concepts and improve exam performance.

Anyone Seeking Concept Clarity – If you want to master Matrices and Determinants, this session is perfect for you!

How Will This Help You?

Conceptual Clarity – Develop a clear understanding of matrices, their types, and operations.

Stronger Problem-Solving Skills – Learn various techniques and tricks to solve complex determinant problems quickly.

Exam-Focused Approach – Solve previous years' JEE Mains, CUET, and CBSE board-level questions.

Step-by-Step Explanations – Get detailed solutions to frequently asked questions in competitive exams.

Watch Now & Strengthen Your Math Skills!

Don't miss this in-depth session on Matrices and Determinants (Part 2). Strengthen your concepts, learn effective shortcuts, and boost your problem-solving skills to ace JEE Mains, CUET, and board exams.

�� Watch here 👉 https://youtu.be/

🔔 Subscribe for More Updates! Stay tuned for more quick revision sessions, concept explanations, and exam tricks to excel in mathematics.

[theohonohan]

Computing nonverbs

Abstract

Active

Ad hoc

Analogue

Argument

Array

Atomic

Attribute

Bay

Bit

Bitmap

Block

Bounds

Branch

Buffer

Bus

Busy

Byte

Cache

Canonical

Capability

Case

Chain

Channel

Character

Checkpoint

Checksum

Child

Chunk

Ciphertext

Class

Client

Clipboard

Clock

Command

Complement

Composite

Concurrent

Condition

Confidentiality

Constant

Context

Continuous

Control

Core dump

Credential

Cursor

Cut

Cycle

Cylinder

Datagram

Dependency

Device

Digital

Direct

Discrete

Document

Domain

Drum

Dynamic

Enclosure

End

Environment

Exception

External

Factor

Fault

Field

File

Flag

Fork

Form

Fragment

Frame

Free

Fresh

Gate

Global

Glyph

Guard

Hash

Head

Heap

Host

Image

Immediate

Implicit

Index

Indirect

Input

Instruction

Instrument

Integer

Integrity

Internal

Kernel

Key

Kind

Language

Layer

Layout

Leaf

Lexical

Line

Local

Lock

Log

Login

Manual

Matrix

Mechanism

Memory

Meta

Method

Mirror

Mode

Module

Monitor

Name

Native

Node

Notation

Null

Object

Octet

Offset

Orthogonal

Output

Overlay

Pack

Package

Packet

Parallel

Parametric

Parent

Parity

Passive

Paste

Path

Pixel

Plaintext

Polymorphism

Precision

Principal

Privilege

Promiscuous

Property

Protocol

Range

Receive

Record

Reflection

Register

Regular

Relation

Relative

Ring

Root

Route

Routine

Sample

Scalar

Scope

Script

Semaphore

Sensitive

Server

Slice

Source

Stack

Stale

State

Statement

Static

Stream

Stride

String

Structure

Switch

Symbol

Syntax

Table

Tag

Tail

Target

Terminal

Thread

Ticket

Timestamp

Token

Tool

Transfer

Transition

Transmit

Transparent

Tree

Type

Union

Unit

Unreliable

User

Valid

Value

Variable

Vector

Violation

Virtual

Volatile

Wait

Watchdog

[pwaqo-blog]

Norma de una matriz por un vector ortogonal @cs

En una demostración encontramos la norma de una Matriz multiplicada por un vector igual a la norma de un vector. Esto se tiene (creo) que porque la matriz pertenece a O(3). Creo que esto era el grupo de matrices ortonormales. Las que van asociadas a los movimientos rígidos. Esto es contenido de Geometría III. Perdón de Geometría II. Una isometría linreal entre dos espacios vectoriales métricos (V,g) y (W,h) se define como un isomorfismo f: V -> W que conserva las métricas, es decir, que verifica g(u,v) = h(f(u), f(v)) para todos u,v de V Lo encontré en StackExchange: Si H es ortonormal, entonces ||Hx|| = ||x|| ya que ||Hx||^2 = (Hx)^T Hx = x^T H^t H x = x^T x = ||x||^2 https://math.stackexchange.com/questions/1754712/orthogonal-matrix-norm

[artist-regardless]

i lied. i don't like sex. put your clothes back on and prove that an orthogonal matrix is length preserving

[ardhra2000]

Principal Component Analysis

Principal Component Analysis is a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components.

PCA can uncover patterns and relationships in the data that weren’t initially apparent.

Statistical theories underlie the creation of the covariance matrix and the understanding of variance and correlation in the dataset.

Carefully interpret the principal components and understand their relationship to the original variables.

Developments in real-time PCA algorithms open up new possibilities for applications in streaming data and online data analysis.

Its blend of simplicity, power, and elegance makes PCA not just a technique but a cornerstone of modern data analysis, embodying the wisdom that sometimes, less truly is more.

Eigenvalues indicate the amount of variance explained by each principal component, helping to identify the components that contribute most to the data's structure.

[lunarsilkscreen]

The mqBit [Matrix qBit]

Represented like:

[0,1]

[1,0]

Where one and zero are opposing poles on the particle (hopefully orthogonal/parallel)

This is basically our emulation of a digital computer (or integer numbers)

[n,...] where n adds up to your arbitrary maximum. (Typically 1, in previous examples 360 degrees of rotation) Represents the rotation of a single axis

You can add as many positions between 0 and 1 in either direction as you wish, the "standard" being set is that they all add up to 1. (This is for simplicity's sake, but you can represent them with any value thing is appropriate.)

[a,b,c]

[b,c,a]

[c,a,b]

Where a+b+c=1. I FEEL LIKE OFF WITH THIS REPRESENTATION.

now, each of these variables may be labeled with some representation that we can use in virtual programming space..so think of each axis as a Set{}, or as an Enum.

Each value between 0 and 1 represents the rotation of that axis divided by the number of positions defined, and that's how we use [Quantum Shunting] or [Quantum Collapse] to emulate he data-types we see in a digital system.

Not feeling well, might delete later...

[testbankprovidersell]

Solution Manuals for Linear Models: The Theory and Application of Analysis of Variance Brenton R. Clarke

TABLE OF CONTENTS

Preface.Acknowledgments. Notation. 1. Introduction. 1.1 The Linear Model and Examples. 1.2 What Are the Objectives?. 1.3 Problems. 2. Projection Matrices and Vector Space Theory. 2.1 Basis of a Vector Space. 2.2 Range and Kernel. 2.3 Projections. 2.3.1 Linear Model Application. 2.4 Sums and Differences of Orthogonal Projections. 2.5 Problems. 3. Least Squares Theory. 3.1 The Normal Equations. 3.2 The Gauss-Markov Theorem. 3.3 The Distribution of SΩ. 3.4 Some Simple Significance Tests. 3.5 Prediction Intervals. 3.6 Problems. 4. Distribution Theory. 4.1 Motivation. 4.2 Non-Central X2 and F Distributions. 4.2.1 Non-Central F-Distribution. 4.2.2 Applications to Linear Models. 4.2.3 Some Simple Extensions. 4.3 Problems. 5. Helmert Matrices and Orthogonal Relationships. 5.1 Transformations to Independent Normally Distributed Random Variables. 5.2 The Kronecker Product. 5.3 Orthogonal Components in Two-Way ANOVA: One Observation Per Cell. 5.4 Orthogonal Components in Two-Way ANOVA with Replications. 5.5 The Gauss-Markov Theorem Revisited. 5.6 Orthogonal Components for Interaction. 5.6.1 Testing for Interaction: One Observation Per Cell. 5.6.2 Example Calculation of Tukey’s One's Degree of Freedom Statistic. 5.7 Problems. 6. Further Discussion of ANOVA. 6.1 The Different Representations of Orthogonal Components. 6.2 On the Lack of Orthogonality. 6.3 The Relationship Algebra. 6.4 The Triple Classification. 6.5 Latin Squares. 6.6 2k Factorial Designs. 6.6.1 Yates’ Algorithm. 6.7 The Function of Randomization. 6.8 Brief View of Multiple Comparison Techniques. 6.9 Problems. 7. Residual Analysis: Diagnostics and Robustness. 7.1 Design Diagnostics. 7.1.1 Standardized and Studentized Residuals. 7.1.2 Combining Design and Residual Effects on Fit - DFITS. 7.1.3 The Cook-D-Statistic. 7.2 Robust Approaches. 7.2.1 Adaptive Trimmed Likelihood Algorithm. 7.3 Problems. 8. Models That Include Variance Components. 8.1 The One-Way Random Effects Model. 8.2 The Mixed Two-Way Model. 8.3 A Split Plot Design. 8.3.1 A Traditional Model. 8.4 Problems. 9. Likelihood Approaches. 9.1 Maximum Likelihood Estimation. 9.2 REML. 9.3 Discussion of Hierarchical Statistical Models. 9.3.1 Hierarchy for the Mixed Model (Assuming Normality). 9.4 Problems. 10. Uncorrelated Residuals Formed from the Linear Model. 10.1 Best Linear Unbiased Error Estimates. 10.2 The Best Linear Unbiased Scalar-Covariance-Matrix Approach. 10.3 Explicit Solution. 10.4 Recursive Residuals. 10.4.1 Recursive Residuals and their Properties. 10.5 Uncorrelated Residuals. 10.5.1 The Main Results. 10.5.2 Final Remarks. 10.6 Problems. 11. Further inferential questions relating to ANOVA. References. Index. Read the full article