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how to sum m-by-n matrix A, to m by n by p matrix B, using Matlab without a For loop. Where the result C should be m by n by p matrix, the direct addition results in
Error using +
Matrix dimensions must agree.
?
If you are looking to add A to each of the p slices of B then you should use bsxfun:
bsxfun(#plus,A,B)
First modify A to have the same size of B by replicating A, p times:
A = repmat(A ,[1 1 p]);
Now A is m by n by p the summation then can be done as
C = A + B
Where C is a m by n by p matrix
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Working in Matlab with time-homogenous Markov Chains and looking to figure out how I can perform matrix multiplication in Matlab for matrix A, similar to R's matrix multiplication, i.e., A %*% A. It would be even better if I could perform A^n instead for a given n instead of having to use A %*% A %*% A, when n = 3, for example.
Any help is greatly appreciated!
First of all you can raise a Matrix to a power in MATLAB:
A ^ n = A * A * A * ... * A
Actually MATLAB uses pretty sophisticated algorithms behind the scene to accelerate this.
For example, if the Matrix is diagonalizable, MATLAB will use that to accelerate the calumniation.
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I know how to solve the problem
Ax=B
with matlab, I just use mldivide to obtain x: x=A\B
But what if I have multiple basis A_i and multiple data B_i but the nature of the problem suggests me that the solution x must be the same for every i?
You could try stacking the A matrices and B vectors to obtain a larger least squares system. That is, form
A = (A_1)
...
(A_n)
and
B = (B_1)
...
(B_n)
and then solve
A*x = B
in the least squares sense
The solution x to such a system will be the value that minimises
Sum{ || A_i*x - B_i ||^2 }
If I understand correctly, this is an image "unmixing" problem, which requests the solution of a (highly) overdetermined system of W x H equations (image area) in K unknowns (number of end members).
One wants to solve
X1.U1ij + X2.U2ij + X3.U3ij = Vij
(assuming K=3) where i, j cover the whole image.
The standard solutions will be least-squares minimization, with two caveats:
if there are outliers the results canbe badly biased and robust methods should be preferred;
if this is really a mixture problem, then the coefficients are constrained to be positive and the question should be recast as a linear programming problem.
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Suppose I have a function of 2 variables F(i,j) which depends on the row index and column index of a matrix, and I want to fill the matrix with the values M_ij = F(i,j)
Of course it's possible to do a loop through i and j, or even only i or j if the function F can be vectorized, but I'd like to know the neat way to do that.
It's impossible to answer without seeing your F but let's assume that F is vectorized such as
F = #(x,y)x+y;
then you could use ndgrid:
[I,J] = ndgrid(1:m,1:n);
M = F(I,J)
In the above case, and this may well apply in your case as well, you might be able to vectorize the function directly using something like bsxfun:
M = bsxfun(#plus, 0:m-1, 1:n);
Regardless of whether your function F is vectorized or not, you have to evaluate it for every value of i and j. If F is not vectorized, you will have to do a loop over the indices manually. If F is based on MATLAB builtins like sin, log, etc., it is most likely vectorized. In this case, you can pass in i and j that are the same size as M and get the result in one step:
[j, i] = meshgrid(1:size(M, 2), 1:size(M, 3))
M = F(i, j)
Note that meshgrid takes and returns the parameters as X, Y, which is the opposite of matrix indexing order row, col.
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For certain measurements i need to obtain only the numeric value of the first principal component from the matrix. Can someone please tell me how do i go about it?
the most straight forward way is just to get the top eigenvector/value of your data's covariance matrix using eigs
say the data matrix x is N by D, or # of data by dimension of data
you can simply do
C = cov(X);
[V, D] = eigs(C, 1);
in fact, you can get the top k principal components by running
[V, D] = eigs(C, k);
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I'm having a matrix of 10 rows and 5 columns, thatr I have called A
I would like to make AX=0 and determine X which contains 5 unknown parameters.
What I did is
null(A)
But it seems that it's considering what I did as a linear algebra.
I would like to introduce another matrix which contains a matrix of errors . which would give me a priori:
AX + E = 0
because the results are not accurate, indeed, but still, I would like to find the closest parameters of the unknown vector (X) and get an error matrix.
Can you help me with that?
You can do eigen value decomposition of A^T.
[Q, D] = eigen(A^T)
And take the eigen vectors that corresponds to the smallest eigen value.
That is take y vectors from the right hand side of the Q matrix.
y is the column number of X.
And then E = -AX
====edit=====
OK. actually you want to minimize E. But since E is a matrix I assume you want to minimize the sum of squares of all the elements in E.
That is:
||E||_F^2 = ||-AX||_F^2 = trace(X^T(A^TA)X)
You can do eigen decomposition of A^TA
[Q, D] = eigen(A^TA)
QA^TA = DQ => D = QA^TAQ^T
To make it smaller, X should be the columns of Qs that corresponds to the smallest eigen value in the diagonal of D.
If there is a eigen value equals to 0, then AX can be 0 and E=0