Matrix Difference Equation Puns

I would like to find the inverse of a hollow unstructured square sparse matrix. You can find below an example code. This code works on my computer, and the mldivide function in the line "L_ml = IminA \ I;" takes a couple of seconds to execute. However, I would like to increase the matrix order from 10.000 to 100.000, and in this case the code doesn't execute, it takes forever. I was wondering if there is an alternative, more efficient way to solve A * X = I, given the matrix attributes I have. I hope I have clearly expressed myself, and thanks in advance for you help!

% Enter matrix order and density.
n = 10000
d = 0.0001

% Generate random sparse matrix.
A = sprand(n,n,d);
clear n d

% Replace the diagonal with 0s to make it a hollow matrix.
v = zeros(10000);
A = A - diag(diag(A)) + diag(v);
clear v

% Generate the identity matrix.
I = speye(n);

% Construct I-A matrix.
IminA = I-A;
clear A

% Invert I-A matrix
disp("Using the 'mldivide' function:")
tic
L_ml = IminA \ I;
toc

We are currently learning Matrix equations in pre cal and I have come to realize that it takes a good 5 minutes to solve one equation. There has to got be a quicker and more efficient way of solving these problems.

So like how Ax =b has the same solution set as x1a1 + x2a2... + xn an = b which has the same solution set as [a1 a2 ... an b] like what is significant about this at all, how are these different? Like I fail to see what's cool about this result, they are the same thing right? I geuss my question is how are they different, and why is that significant? Like I can see how an integral can be looked at with two different perspectives: 1 as an anti derivative, and one as a way to find area under a curve, but I fail to see how these 3 relations are the same thing but from a different perspective. To me they just look like the same thing but different notations, like of like the differnce between dy/dx and y'. I see these all as saying that if you distribute the x1 x2 ...xn to their corresponding column matrices, ie a1 , a2 ... an, then it should equal some other vector b.

Equation is Ay=b, size of this matrix is 50 and b = [5,5,5,5,5...... 5]^T

https://preview.redd.it/7vfo1y9dxl181.png?width=616&format=png&auto=webp&s=1ef2897dd4e3cadc9a716751fc4555a3d8a78cc0

I missed one class on computer algebra because of important family matter and I'm pretty much stuck with this excercise. I know that if there were not 1's but 0's I could easily save the two diagonals in two lists of size 50 and solve using back substitution but I can't come up with any other ideas other than bruteforcing it with LU decomposition.

The reccurence relation of the naive approach of matrix multiplication taken from this site is T(n) = 8T(n/2) + O(n^2).

I want to check my understanding of how this was formulated and ask a few questions about it.

So the 8(T/2) part where we split the matrix until we reach the base case which is a matrix of size 1 aka the individual components.

The article says that the cost of adding the multiplied components is O(n^2) and I'm kind of shaky on how they got this. Is it because in the worst case, you'd have to navigate a matrix to get the entry in the last column of the last row thus iterating through all of the entries in an matrix? In that case since we have two matrices that we're working with, does that mean the time complexity of navigating both of them is O(2n^2) which simplifies to O(n^2)?

They mention the cost of adding but what about the cost of multiplying the individual components such as ae, bg, af etc.?

https://preview.redd.it/f6yznwvbbky71.png?width=1417&format=png&auto=webp&s=b026bbcc012d72a14521eef2ddceec2a85b016bd

I am trying to make up some m x n matrix A, some ? x ? vector x and some 1 x n vector b for which their combination works out but I can't find any. Can someone help me? My teacher says that the total number of parameters we'd need to work out would be mn+n but i don't get why.

I have a linear system of 21 equations and variables in MATLAB but I don't want to write a coefficient matrix to solve it since it's very large. Is it possible to solve the system with just the equations as input ?

Hi guys. I've been banging my head against a wall trying to figure out how to convert this equation to a matrix with form A. This is for a programming class I'm taking, and a nudge in the right direction would be appreciated as I never took linear algebra. t and f are column vectors, phi is a set of basis functions, and xi is a constant. Thanks in advance!

https://preview.redd.it/k3bqt9keguv71.png?width=162&format=png&auto=webp&s=195d72d38c95a7551cb4ad7ddcb5c3e30912fef2

Hi, I got this equation yesterday, which I need to solve for the 2-by-2 matrix X.

Although this problem is really easy if we name every entry of X, respectively, a b c d and then solve for each variable. I wonder if we can solve for X without naming every entry like that.

I've been diving into this problem with this way for like over an hour, and I ended up something like this: http://imgur.com/a/yi895MN

So, thanks for reading and please tell me if I missed anything or it's actually difficult to approach this way.

https://preview.redd.it/sf4dfc2x5gs71.png?width=681&format=png&auto=webp&s=e7a80fe1de5a2a123f736e5e0faf1214d7c836b9

I'm just wondering if there is a special name for a matrix after you divide the coefficients through so that each equation is equal to 1.

For example if the original equations were:

4x + 8y = 2
21x - 9y = 3

After dividing:

2x + 4y = 1
7x - 3y = 1

Sorry if this is a stupid question, the only linear algebra I had was in Algebra 2 in HS a very long time ago.

I am trying to solve a hyperbolic equation using finite difference as shown below.

My main confusion is that to calculate for U_i,2 (i.e. the first iteration), where do I get U_i,1 from? Because the only given initial condition is U_i,0.

Note: I did try assuming that U_i,1 = U_i,0 and the solution does seem right, but I just would like to see if there is a better approach.

https://preview.redd.it/gizjjpmqncc81.png?width=613&format=png&auto=webp&s=ad33dee5a12cf9b241bfb2ce5c96068c4adbf9d3

For lack of a better title.

I am looking to solve the equation:

V = A @ V @ A.T + Q

for a square matrix V, where A and Q are square matrices.

Any help appreciated.

The title sums it up. I am having trouble creating my sparse matrix. Different boundary conditions (dirichlet/Neumnn) might be involved so I don't know how to generalize my matrix in a way that would take care of both boundary conditions. Can someone point me to any form of written material on how would someone set up the matrix? I also am not sure on how am I supposed to number the elements

Like the text above says I'm looking for some insight into how to simplify a matrix

The example in question is the following:

-X^(T)𝜃^(T)y - X𝜃y^(T) = -2*(X^(T)y)𝜃

this is taken from : (y-X𝜃)^(T)(y-X𝜃) = y^(T)y - X^(T)𝜃^(T)y - X𝜃y^(T) + X^(T)𝜃^(T)X𝜃, for context.

How does this simplification work? Assuming the orders of the matrices in question enable the multiplication and subtraction to work.

That being said, I could have royally messed up the expansion of brackets from the contextual equation.

Hi- so I know how to solve simultaneous equations in matrix form but I had my first period of my SAC today (Australian assessment) and noticed that the two last problems I need to first write simultaneous equations in algebraic form which I have really struggled with. I have my last hour to work on it tomorrow. I try to write them out but it never makes sense because I cant use the process of elimination or substitution on them. Because it is an assessment I don't have the questions but I can do a random example:

(I know it was about basketball tickets) "Non-members and members pay a different amount for a ticket at a basketball game but it also depends on the round of basketball Round 1 Non members pay $37 Members pay $23 Round 2 Non members pay $46 Members pay $35, Round 1 totalled $5487 and round 2 totalled $3567 how many tickets were sold (im not sure exactly of the question)"

What I am struggling with is actually putting this into an equation form not matrix form. I had an idea that I could go based off of my matrix form and just kind of deconstruct (work backwards)

Thank yall so much

I think it's just the result of The Analyst's new Matrix. Trinity comments on how beautiful it was right before they jump off the roof in Resurrections. Smith also comments on it early on in the movie. It's part of the new prison and part of The Analyst's strategy.

Kinda like a "Look at how perfect everything is. You'd have to be crazyyyyyyy to deny it." But it's obviously just another manipulation and also keeps the memories of the old Matrix a fiction to Neo and Trinity.

Like The Analyst says, The Architect hated the human mind. Hence the droll and mundane look of his Matrix. The Analyst "gives the people what they want". A comforting and pretty world that caters to their fictitious view of it. It's so effective that anyone questioning it ends up questioning their own sanity over anything else.

I am a little confused about Matrix game's Campaign Series: Middle East 1948-1985 and John Tiller Game's Modern Campaign Series: MIDDLE EAST '67. What is the difference in game engines? I think the (Matrix agme version) Campaign Series Engine is also made by John Tiller, so are they in fact the same mechanism? And if not which one is better (to buy).

Thank you very much.

I would like to achieve this output. And with the current code, it looks like this

Any suggestions about how can I increase the gap between the columns and wrap the second line contents closer to the first one?

I am attaching the code here

$$
\,\, \mathbf{T}_{\mathbf{S},\mathbf{W}} = \left[\begin{matrix}
	\mathbf{D}& \overrightarrow{\mathbf{V}}\\
	\overrightarrow{0^{\mathbf{T}}}& 1\\
\end{matrix}\right]\,\,=\,\,\left[\begin{matrix}
	\cos\mathrm{b} \cos\mathrm{g}&		\sin\mathrm{a}\sin\mathrm{b}\cos\mathrm{g}&		\cos\mathrm{a} \sin\mathrm{b} \cos\mathrm{g}&\Delta\mathrm{x}\\
	&		-\cos\mathrm{a} \sin\mathrm{g}&		+\sin\mathrm{a} \sin\mathrm{g}&			\\
	\cos\mathrm{b} \sin\mathrm{g}&		\sin\mathrm{a}\sin\mathrm{b}\sin\mathrm{g}&		\cos\mathrm{a} \sin\mathrm{b} \sin\mathrm{g}&\Delta\mathrm{y}\\
	&		+\cos\mathrm{a} \cos\mathrm{g}&		-\sin\mathrm{a} \cos\mathrm{g}&			\\
	-\sin\mathrm{b}&		\sin\mathrm{a} \cos\mathrm{b}&		\cos\mathrm{a} \cos\mathrm{b}&				\Delta\mathrm{z}\\
	0&		0&		0&				1\\
\end{matrix}\right]
$$>

Hello all!

I need some help soving this question. I do not understand how they get at an equilibrium of x*=-2 and x*=3

https://imgur.com/G1HRzls

Equation is Ay=b, size of this matrix is 50 and b = [5,5,5,5,5...... 5]^T

Matrix A has 10's on diagonal, 8's just above diagonal and the rest is filled by 1's.

Just like belov, but of size 50.

| 10 8 1 1 |

| 1 10 8 1 |

| 1 1 10 8 |

| 1 1 1 10 |

I missed one class on computer algebra because of important family matter and I'm pretty much stuck with this excercise. I know that if there were not 1's but 0's I could easily save the two diagonals in two lists of size 50 and solve using back substitution but I can't come up with any other ideas other than bruteforcing it with LU decomposition. We are not actually required to solve this using matrix structure, but it looks like it could really matter.