Singular Value Decomposition is a matrix factorization that expresses... a matrix M as UΣVᵀ, where U and V are orthogonal matrices, and Σ is diagonal. It can interpreted as the action of a rotation, a scaling and a rotation/reflection.show more

Visualization of 2D rotation matrix. As θ sweeps from... 0 to 2π, shapes rotate rigidly; no distortion or scaling. Orthogonal transformations preserve distances, angles & inner products. That’s why we diagonalize symmetric matrices with orthogonal.show more

HIP MOBILITY - THREE moves to explore rotation of... the hips. First one is seated where we can pry a little more having leverage to dive deep into internal rotation. Second is exploring IR rotation from a squat and third is a combo of the figure 4 contour and elevated pigeon.show more

I built MatmulFlow ( — an interactive tool that... makes matrix multiplication dimensions visual, part of my AI by Hand ✍️ series. Matrix multiplication dimensions are confusing. Which is the inner dimension? Columns of the first or rows of the second? And when you chain five multiplications together, it gets worse. The idea: represent matrices as rectangles. Shift the second matrix up and to the right. The edges that must align become obvious. The result fills in the remaining space. No memorization. You can see it. It extends to chains. Stack vertically for left-multiplication. Stack horizontally for right-multiplication. Resize any matrix and watch the dimensions "flow" through the entire chain. Give it a try!show more

An eigenvector of a matrix is a direction it... does not turn, only scales: Av = λv. Here A is blended in from the identity, (1 − s)I + sA. The box on (−1, 2, 2), (2, −1, 2) and (2, 2, −1) never turns; its edges only scale, by 1.8, 1.2 and 0.6, the roots of det(A − λI) = 0, and a point inside rides along to Ax. A is symmetric, so the three directions are perpendicular, and A = λ₁v₁v₁ᵀ + λ₂v₂v₂ᵀ + λ₃v₃v₃ᵀ for the unit vᵢ.show more

Three days ago, the Tate brothers criticised Israel for... their disgusting crimes and exposed the Matrix is controlled by Jews. 🚨“THEY control the Matrix!“ This morning, DIICOT raided their home in a new investigation. This is not a coincidence, this is a planned Matrix attack.show more

Gustnado in Trussville earlier when the storm passed through.... A gustnado (a blend of the words “gust front” and “tornado”) is a brief, shallow, surface-based whirlwind that forms along the leading edge of a thunderstorm. They are not associated with storm scale rotation, and are classified as thunderstorm wind events. Video from Kayden Russellshow more

A gyroscope is a device that measures orientation and... angular velocity, using the principle of angular momentum to resist changes in its rotation. It helps maintain balance and stability, commonly, airplanes, and shipsshow more

One line of code is the difference between a... working shader and breaking out of the Matrix. 😎show more

U.S. company Powerus has unveiled the Matrix Fold, a... modular and foldable FPV drone reportedly selected for the Pentagon’s Drone Dominance program. The 12-inch platform is designed to fold into a compact form that can reportedly fit into a backpack and be deployed within seconds. Its 3D-printed construction is intended to allow individual components to be replaced in the field, with printing files reportedly supplied for manufacturing replacement parts. According to Powerus, the Matrix Fold has a maximum speed of 242 km/h, a payload capacity of up to 6.8 kg and an endurance of up to 35 minutes. The system features a modular architecture compatible with the company’s Matrix 5, 7, 10 and F platforms. Tool-free maintenance and rapid payload changes are also reportedly supported. Powerus states that the Matrix Fold is NDAA-compliant and Blue UAS certified, with its supply chain under company control. In May 2026, the Matrix Fold was reportedly selected for the second phase of the Pentagon’s Drone Dominance program, described as having a value of approximately $1 billion. The program is intended to support the procurement of tens of thousands of low-cost, expendable drones and expand U.S. domestic production capacity. Serial production is reportedly planned to follow the program’s next phase. #USA #Powerus #MatrixFold #FPV #FPVDrone #DroneDominance #Pentagon #UAV #DroneWarfare #3DPrinting #BlueUAS #NDAAshow more

Look at Ousmane Dieng go He went from outside... of a rotation and looking in at OKC to being a starter for MIL, and he’s showcasing that he can thrive with added responsibilities After all, there’s a ton of value having a big man who can corral a rebound and instantly jumpstart a transition opportunity by himselfshow more

A full day after landfall, what's left of Isaias... did this. The red streak is the storm's rotation on radar, the blue dots are WeatherWise storm chasers, and a tornado was confirmed along it again and again for three and a half hours.show more

What is an individual? Can we accept the idea... that an individual is not an isolated being, but a totality of traits, ideas, and deeds that arise within humanity, are shaped by it, and return to it? Every human being is a temporary form through which humanity expresses a part of itself. : The author redefines the individual not as an isolated entity but as a transient expression of humanity itself: a unique constellation of traits, ideas, and deeds that emerge from the collective, are molded by it, and ultimately return to enrich it. Each person serves as a temporary vessel through which humanity manifests and explores its own nature. This dissolves the illusion of absolute separateness, revealing every life as an integral thread in the living fabric of humankind. It portrays individuality as both gift and responsibility: a fleeting opportunity to contribute distinct value to the whole. The concept elevates personal existence to a sacred role in humanity’s continuous self-creation. It leads to recognition that our thoughts, actions, and choices ripple beyond the self, shaping the shared human story. Ultimately, framing every individual as both creation and co-creator of humanity’s evolving essence, bound in profound interdependence.show more

The Helmholtz decomposition is one of the fundamental results... of vector calculus. It says any well-behaved vector field can be split into two parts, one capturing sources and sinks through divergence, and one capturing rotation through curl.show more

[Deep RNN] by Hand ✍️ A Deep Recurrent Neural... Network (RNN) extends a basic single-layer RNN into multiple layers of hidden states, effectively incorporating deep learning into the RNN architecture. How does a Deep RNN work? [1] Given ↳ A sequence of four inputs X1, X2, X3, X4 ⬛️ ↳ Recurrent weights and biases for hidden layers a 🟩, b 🟧, c 🟪, and the output layer y 🟦. [2] Initialize Hidden States ↳ Set a0, b0, c0 to zeros — Process X1 (t = 1)— [3] First Hidden Layer (a) 🟩: a0 → a1 ↳ The transformation matrix is horizontal concatenation of input weights, hidden state weights and biases, visualized as [⬛️ | 🟩 | ⬜️] . ↳ The state matrix is vertical concatenation of input X1, previous hidden state a0, and an extra 1, visualized as [⬛️ ; 🟩 ; 1]. ↳ Multiply the two matrices to obtain new hidden state a1 = [0 ; 1]. [4] Second Hidden Layer (b) 🟪: b0 → b1 ↳ First layer a1 🟩 becomes the input. ↳ The transformation matrix is visualized as [🟩 | 🟪 | ⬜️]. ↳ The state matrix is the combination of a1, b0, and 1, visualized as [🟩; 🟪 ; 1]. ↳ Multiply the two matrices to obtain new hidden state b1 = [1; -1]. [5] Third Hidden Layer (c) 🟧: c0 → c1 ↳ Second layer b 🟪 becomes the input. ↳ The transformation matrix is visualized as [🟪 | 🟧 | ⬜️]. ↳ The state matrix is the combination of a1, b0, and 1, visualized as [🟪; 🟧; 1]. ↳ Multiply the two matrices to obtain new hidden state b1 = [1; -1]. [6] Output Layer (Y) 🟦 ↳ The transformation matrix is visualized as [🟧 | ⬜️]. ↳ The state matrix is the combination of c0 and , visualized as [🟧; 1]. ↳ Multiply the two matrices to obtain output Y1 = [3; 0; 3]. — Process X2 (t = 2)— [7] Previous Hidden States ↳ Copy the values of a1, b1, c1. [8] Hidden 🟩🟪🟧 + Output 🟦 ↳ Repeat [3]-[6] to obtain output Y2 = [5; 0; 4] — Process X3 (t = 3)— [9] Previous Hidden States ↳ Copy the values of a2, b2, c2. [10] Hidden 🟩🟪🟧 + Output 🟦 ↳ Repeat [3]-[6] to obtain output Y3 = [13; -1; 9] — Process X4 (t = 4)— [11] Previous Hidden States ↳ Copy the values of a3, b3, c3. [12] Hidden 🟩🟪🟧 + Output 🟦 ↳ Repeat [3]-[6] to obtain output Y4 = [15; 7; 2]show more

The tiger is a monk. It won't bother you,... or be bothered by you. It tries to maintain its composure as much as it can. Even if you are around it, it will most likely be unfazed. And even when a tiger expresses its aggression, it is mock. It's a construct.show more

[Discrete Fourier Transform] by Hand ✍️ In signal processing,... the Discrete Fourier Transform (DFT) is no doubt the most important method. But the math involved is extremely complex, literally, involving a summation over a complex number term e^(-iwt). I developed this exercise to demonstrate that underneath such complexity, DFT is just a series of matrix multiplications you can calculate by hand. ✍️ Once you see that, it should not surprise you that a deep neural network, which is also a series of matrix multiplications, with activation functions in-between, can learn to perform DFT to process and analyze signals so effectively. How does DFT work? [1] Given ↳ Signals A, B, and C in the 🟧 frequency domain: ◦ A = cos(w) + 2cos(2w) ◦ B = cos(w) + cos(3w) + cos(4w) ◦ C = -cos(2w) + cos(3w) ◦ Each signal is a weighed sum of four cosine waves at frequencies 1w, 2w, 3w, and 4w. ◦ We will apply Inverse DFT to convert the signals to time domain representations, and then demonstrate DFT can convert back to their original frequency domain representations. ↳ Signal X in the 🟩 time domain. X is sampled at 10 time points 1t, 2t, …, 10t: ◦ X = [-2.5, -1.8, 3, -0.7, -1.0, -0.7, 3, -1.8, -2.5, 5] ◦ Suppose X is also a weighted sum of the same four cosine waves, but we don’t already know their weights. We will apply DFT to discover them. [2] 🟧 Frequency Matrix (F) ↳ Write the coefficients of A, B, C as a matrix F. Each signal is a row. Each frequency is a column. ↳ A → [1, 2, 0, 0] ↳ B → [1, 0, 1, 1] ↳ C → [0, 1-, 1, 0] [3] Cosine → Discrete ↳ Sample from the continuous cosine waves at discrete time points 1t, 2t, 3t, to 10t. [4] Cosine Matrix (W) ↳ Write the samples as a matrix, Each frequency is a row. Each time point is a column. [5] Inverse DFT: 🟧 Frequency → 🟩 Time ↳ Multiply the frequency matrix F and the cosine matrix W. ↳ The meaning of this multiplication is to linearly combine the four cosine waves (rows in W) into time-domain signals (rows in T) using the weights specified in F. ↳ The result is matrix T, which are signals A, B, C converted to the time domain. Each signal is a row. Each time point is a column. [6] Transpose ↳ Transpose T, converting each signal’s time domain representation from a row to a column. [7] DFT: 🟩 Time → 🟧 Frequency ↳ Multiply the cosine matrix W with the transpose of matrix T. ↳ The purpose of this multiplication is to take a dot-product between each time-domain signal (columns in the transpose of T) and each cosine wave (rows in W), which has the effect of projecting the signal onto a cosine wave to determine how much they are correlated. Zero means not correlated at all. ↳ The result is an intermediate version of the “recovered” frequency matrix where each column corresponds to a signal and each row corresponds to a frequency. ↳ Compared to the original frequency matrix F, this intermediate matrix has non-zero weights in the correct places, but scaled up by a factor of 5 (n/2, n=10). For example, signal A, originally [1,2,0,0], is recovered at [5,10,0,0]. [8] Scale ↳ Multiply each value by 2/n = 1/5 to scale down the intermediate matrix to match the magnitude of the original frequency matrix F. [9] Transpose ↳ Transpose the recovered frequency matrix back to the same orientation of the original frequency matrix F. ↳ Like magic 🪄, the result is identical to the original F, which means DFT successfully recovered the frequency components of signals A, B, C. [10] Apply DFT to X: 🟩 Time → 🟧 Frequency ↳ Now that we have some confidence in DFT’s ability to recover frequency components, we apply DFT to X’s time-domain representation by multiplying W with X. ↳ The result is the an intermediate matrix. [11] Scale ↳ Similarly, we scale down by a factor of 5 to obtain the recovered frequency components of X (a column). [12] Transpose ↳ Similarly, we transpose the recovered column to row to match the orientation of the frequency matrix. ↳ Using the coefficients [0,0,3,2], we can write the equation of X as 3cos(3w) + 2cos(4w). Notes: I hope this by hand exercise helps you understand the essence of DFT. But there is more technical details, such as: • Sine: The complete DFT math also includes sine waves that follow a similar calculation process. • Phase: Here, we assume all the cosine waves are aligned at the origin, namely, phase is 0. If a phase p is added, for example, cos(w+p), we will need to calculate the sine component and use their ratio to figure out what p is. • Magnitude: If phase is not zero, the magnitude will need to be calculated by combining both cosine and sine terms.show more

There's no need to speak. You must only -... concentrate and recall all your past life. When a man thinks of the past, he becomes kinder. ~ Stalker (1979) Tarkovsky A déjà vu is usually a glitch in the Matrix. It happens when they change something. ~ The Matrix (1999) Wachowskishow more

I am developing B3D, a minimal software 3D renderer... that supports triangle rasterization with depth buffering and clipping, perspective projection with a configurable FOV, model transformations (translation, rotation, scaling), and camera control.show more

The Matrix launch video lands on one line; in... Matrix, anyone can become a CEO. That’s the real claim. Not one assistant doing one task. A company that keeps running after the first prompt, routing intent through a CEO Office, into OKRs, down to departments that do the work and send back proof to review. Connect Codex, point your leftover credits at it, and the first agent company is live.show more