Adds the eigenvalues lecture and, separately, the lake lecture #94
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| @jstac have these lectures been added to |
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@jstac looks like this empty line needs to be removed for jupytext to understand the metadata
No, sorry, I forgot. All help is appreciated. |
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| Let's first see examples of a sequence of iterates $(A^kx)_{k \geq 0}$ under different maps $A$. | ||
| Let's first see examples of a sequence of iterates $(A^k x)_{k \geq 0}$ under | ||
| different maps $A$. |
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The methodmatrix_power in the python code cell after the sentence above needs to be defined or imported.
Useful resources:
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Thanks for writing such a beautiful lecture @jstac. The idea of using functions grid_transform and circle_transform to visualise the matrix transformation looks pretty cool to me.
Please find my comments above. Feel free to let me know if you like some of them and I can update them in a commit.
lectures/eigen.md Outdated
| \lambda I$ are linearly dependent. | ||
| ### Complex Values | ||
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| So far our definition of eigenvalues and eigenvector seems straightforward. |
lectures/eigen.md Outdated
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| So far our definition of eigenvalues and eigenvector seems straightforward. | ||
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| There is, however, one complication we haven't metioned yet: |
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| Although we truncate the infite sum at $k = 50$, | ||
| both methods give us the same result which illustrates the result of the Neumann Series lemma. | ||
| Although we truncate the infite sum at $k = 50$, both methods give us the same |
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| Consider the following equation where we multiply a $n \times m$ matrix $A$ with a $m \times 1$ | ||
| column vector $x$ to obtain a $n \times 1$ column vector $y$. | ||
| To understand the second point of view, suppose we multiply a $n \times m$ |
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a $n \times m$ -->> an $n \times m$
lectures/eigen.md Outdated
| Consider the following equation where we multiply a $n \times m$ matrix $A$ with a $m \times 1$ | ||
| column vector $x$ to obtain a $n \times 1$ column vector $y$. | ||
| To understand the second point of view, suppose we multiply a $n \times m$ | ||
| matrix $A$ with a $m \times 1$ column vector $x$ to obtain a $n \times 1$ |
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a $n \times 1$ -->> an
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| 2. $A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ is irreducible since $a_{12},a_{21} >0$ and $a^{2}_{11},a^{2}_{22} >0$. | ||
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| 3. $A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ is reducible since $A^k = A$ for all $k \geq 0$ and thus |
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Here
lectures/eigen.md Outdated
| If a matrix $A \geq 0$ then, | ||
| 1. the dominant eigenvalue of A, r(A), is real-valued and nonnegative. |
lectures/eigen.md Outdated
| 2. for any other eigenvalue (possibly complex) $\lambda$ of $A$, $|\lambda| \leq r(A)$. | ||
| 3. we can find a nonnegative and nonzero eigenvector $v$ such that $Av = r(A)v$. | ||
| Moreover if A is also irreducible then, |
| r = max(abs(λ) for λ in evals) # compute spectral radius | ||
| print(r) | ||
| ``` | ||
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$(I-A)^{1}$ should be $(I-A)^{-1}$ in line 999.
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| and 4 units is the external demand by consumers. | ||
| The first row depicts how agriculture's total output $x_1$ is distributed | ||
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| * $30\%$ is used as inputs within agriculture itself, |
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Should it be $0.3 x_1$ to be consistent with the following notations?
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| * $30\%$ is used as inputs within agriculture itself, | ||
| * $0.2x_2$ is used as inputs by the industry sector to produce $x_2$ units | ||
| * $0.2x_3$ is used as inputs by the service sector to produce $x_3$ units and |
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The $0.2x_3$ in line 1052 should be $0.3x_3$.
| @mmcky I've added eigen to toc but the build is still failing with a cache error. Can you please tell me what I'm doing wrong? |
| @jstac looking at the build the warnings indicate that Does this run locally for you? I will write up a note on the Also should
I should be able to dive in and take a closer look later today. |
| Thanks @jstac I'll take a look at the issues right now. |
| BTW -- I have set the I will fix those warnings now |
| @jstac this is now building with generated previews. |
| @maanasee this lecture looks really nice -- thank you. I was wondering if this Could be fun to put together some math animations / transforms etc. |
| I'm going to go ahead and merge this. Then we'll iterate. |
Note that, since this lecture is quite long, I pulled out the discussion of the lake model of employment and unemployment and put it in file
lake_model.md.