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Matrix multiplication seems complex until you observe it step by step. Watch how each of the nine entries is formed: one row from matrix A, one column from matrix B, three products, and one sum.

The Math Flow

62,009 subscribers

37,848 次观看 • 2 天前 •via X (Twitter)

艺术教育科学技术

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Max Stozek 的头像

Max Stozek2 天前

I can understand the meaning of what is a sum, a multiplication, even the different meanings of what is a matrix. I can repeat the process of multiplying one matrix by another, but understanding what this means is still something obscure to me

G Labs 的头像

G Labs1 天前

Mathematician here. Many wonder what matrix multiplication means, physically. I'm here to explain that. A matrix represents a phenomena that could change a vector. For example, wind conditions that could change a ship's velocity. Matrix multiplication is one matrix acting on another, resulting in a different matrix as an output. It represents one phenomena acting on another, generating a new vector altering phenomena as a result. For example, temperature conditions acting on wind conditions, resulting in different wind conditions. To those interested, see my lecture on Matrix Multiplication:

ESteiner001 的头像

ESteiner0011 天前

Oh DAMMIT, Mr. Wing🫠

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You start with a plain old complex polynomial p(z) = a₀ + a₁z + … + aₙzⁿ. Instead of fixing the coefficients, you treat them as variables in a controlled experiment and modulate them. Maybe you draw each aₖ from some distribution, or you let them drift along a parameter path. Do that 50,000 times, say, and you no longer have a polynomial. You have an ensemble. For each realization, the goal is the same Find its roots. Instead of attacking p(z)=0 directly with something like fsolve, you rewrite the polynomial as a matrix problem. Specifically, you build its Companion Matrix. For a monic polynomial zⁿ + bₙ₋₁zⁿ⁻¹ + … + b₀ the companion matrix is an n×n matrix whose last column contains the coefficients (with minus signs), and whose subdiagonal is filled with ones. Everything else is zero. It’s sparse, structured and deterministic. The theorem that makes this worth doing is simple and beautiful The eigenvalues of the companion matrix are exactly the roots of the polynomial. So each polynomial sample becomes a matrix. Each matrix becomes an eigenvalue problem. And eigenvalues are something numerical linear algebra is very, very good at computing.

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This is a great lecture at MIT by David Shirokoff on Markov Chains. He covers the fundamentals of Markov Chains using a simple particle movement example. He starts by explaining how a particle moves between two positions, A & B, with different probabilities. From there, the talk converts the problem into matrix form using a Markov matrix. The main topics covered are: - Transition probabilities - Markov matrices - Probability vectors - Matrix multiplication in Markov Chains - Finding probabilities after n steps - Eigenvalues and eigenvectors - Matrix diagonalization - Long-term steady state distribution

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A deleted scene from Scary Movie 3 where Cindy enters the matrix to battle hundreds of Samara’s and realize she’s the one

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𝚃𝙷𝙺

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15,182 次观看 • 5 个月前

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Discrete Fourier Transform by hand ✍️ ~ 12 steps walkthrough below Here is a little-known secret about the DFT and the inverse DFT: it is just matrix multiplication in both directions, one the transpose of the other, exactly like the forward pass and backpropagation I drew in other examples. Goal: recover which cosine waves a signal is made of, using nothing but multiplication and addition. = 1. Given = Three signals written as sums of cosines, and a fourth, X, that we do not know yet. = 2. Frequency matrix F = Let us write the coefficients as a matrix. Each signal is a row, each frequency a column, so A = cos(w) + 2cos(2w) becomes [1, 2, 0, 0]. = 3. Sample the waves = We read the four cosine waves at ten discrete time points. That word "discrete" is the whole difference between this and the continuous transform. = 4. Cosine matrix W = Let us write those samples as a matrix: each frequency a row, each time point a column. = 5. Frequency to time = We multiply F by W. That combines the four cosine waves in the proportions F specifies, and the result T is the three signals as they would look in time. = 6. Transpose = Let us stand each signal up as a column. = 7. Time to frequency = We multiply W by that transpose. Every cell is the dot product of one signal with one cosine wave, which measures how much of that wave the signal contains. Zero means none of it. = 8. Scale = Let us multiply by 2/n, with n = 10. The projections come out five times too large, and this is the correction. = 9. Transpose back = We turn it back around, and it is F again, exactly. That is the check: the transform recovered the coefficients we started from. = 10. Now solve for X = Let us run the same multiplication on the one signal whose recipe we never knew. = 11. Scale = We divide by 5 again. = 12. Transpose back = And X reads [0, 0, 3, 2], which says X = 3cos(3w) + 2cos(4w). Note: I originally drew this to show that the DFT is a special case of a convolution layer, its filters fixed to sine and cosine waves rather than learned. No wonder, then, that a convolution layer free to learn its own filters can be trained to process signals. 💾 Save this post!

Tom Yeh

25,684 次观看 • 1 个月前

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Discrete Fourier Transform by hand ✍️ ~ 12 steps walkthrough below Here is a little-known secret about the DFT and the inverse DFT: it is just matrix multiplication in both directions, one the transpose of the other, exactly like the forward pass and backpropagation I drew in other examples. Goal: recover which cosine waves a signal is made of, using nothing but multiplication and addition. = 1. Given = Three signals written as sums of cosines, and a fourth, X, that we do not know yet. = 2. Frequency matrix F = Let us write the coefficients as a matrix. Each signal is a row, each frequency a column, so A = cos(w) + 2cos(2w) becomes [1, 2, 0, 0]. = 3. Sample the waves = We read the four cosine waves at ten discrete time points. That word "discrete" is the whole difference between this and the continuous transform. = 4. Cosine matrix W = Let us write those samples as a matrix: each frequency a row, each time point a column. = 5. Frequency to time = We multiply F by W. That combines the four cosine waves in the proportions F specifies, and the result T is the three signals as they would look in time. = 6. Transpose = Let us stand each signal up as a column. = 7. Time to frequency = We multiply W by that transpose. Every cell is the dot product of one signal with one cosine wave, which measures how much of that wave the signal contains. Zero means none of it. = 8. Scale = Let us multiply by 2/n, with n = 10. The projections come out five times too large, and this is the correction. = 9. Transpose back = We turn it back around, and it is F again, exactly. That is the check: the transform recovered the coefficients we started from. = 10. Now solve for X = Let us run the same multiplication on the one signal whose recipe we never knew. = 11. Scale = We divide by 5 again. = 12. Transpose back = And X reads [0, 0, 3, 2], which says X = 3cos(3w) + 2cos(4w). Note: I originally drew this to show that the DFT is a special case of a convolution layer, its filters fixed to sine and cosine waves rather than learned. No wonder, then, that a convolution layer free to learn its own filters can be trained to process signals. 💾 Save this post!

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