# Questions tagged [matrix]

is a mathematical object to be recorded in a [usually] rectangular array of elements of ring or field, which (table) is a set of rows and columns are located at the intersection of its elements. The number of rows and columns specify the size of the matrix.

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### Analytical expression for the eigenvectors of a 3x3 real, symmetric matrix?

I am writing an algorithm that process 3D images based on the local moment of inertia. I have a 3x3 real symmetric matrix, from which I need to find the eigenvalues. I have found a variety of ...
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### Why do we deal with the eigenvectors of the autocorrelation instead of the data itself?

How intuitively to understand why eigenvectors of the autocorrelation matrix are used, but eigenvectors of the matrix constructed from temporal samples have no sense and aren't used? For example, in ...
887 views

### The Least Norm Solution of Under Determined Linear System

Suppose I have a matrix $$A= \begin{pmatrix} 1 & 0 & 1 & 0\\ 0 & 1 & 1& 0\\ \end{pmatrix}$$ where the variables are channel information like assume $X_1$, $X_2$, $X_3$ ...
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### Relationship between discrete deconvolution and Toeplitz matrices

I have 2 vectors, $a$ & $c$, both of length $M$. I know they are related by $a*b=c$. My goal is to recover $b$. Obviously $b = \mbox{deconv} (c,a)$. I am only interested in the first $M$ elements ...
6k views

### What is a covariance matrix?

Suppose you have k samples from each of the N elements of a uniform linear array (ULA) of sensors: What is the physical meaning of a covariance matrix? How do you form a covariance matrix with the ...
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### Deriving the Matrix Inversion Lemma for RLS Equations vs the Woodbury Derivation

Can any one help me in deriving the matrix inversion lemma rule for RLS algorithm? I don't know how to start with. Many books have just stated but they haven't derived it.
354 views

### Least Angle Regression (LARS) without Matrix Inversion

Sorry if this is too damned long. I did what I could to abbreviate it. The question is about Least Angle Regression (LARS). I'm new to numerical work with matrices. I believe I have a way to ...
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534 views

### 5.1 Rear To 5.1 Side mixing matrix

I am searching for a matrix definition for converting a audio signal (5.1 with rear) to a audio signal (5.1 with side). Currently I am using the definitions from: https://msdn.microsoft.com/en-us/...
172 views

### Wave Digital Filter Bridge-T Resonator implementation, gives expected cutoff frequency but incorrect gain and roll-off

Trying to implement the WDF in Fig 5(a) of this publication. The response of the ideal op-amp implementation is given by the black curve in Fig 7: Here is the plot I get when trying to implement the ...
53 views

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### Why is this matrix invertible in the Kalman gain?

In the wikipedia article about Kalman filters, the well-known expression of the matrix of Kalman gains is given:  \mathbf {K} _{k}=\mathbf {P} _{k\mid k-1}\mathbf {H} _{k}^{\text{T}}\mathbf {S} _{k}^...
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### why use svd() to invert a matrix?

In MATLAB, i compared elapsed time to invert a Hermitian matrix using inverse(), svd(), and chol(). svd() took the longest. So is there any reason to prefer svd() to the other two methods?
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### Ifft through Matrix multiplication

I am still new to MATLAB, so apologies if I sound lazy to you. I am attempting to model a transformation as a set of matrix operations. I start with a vector, up-sample it by $U$ (up-sampling rate), ...
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### Sensing matrix for compressed sensing

What are the differences between random binary sensing matrix  and random Gaussian sensing matrix? What the advantages and disadvantages of each matrix? How can I choose the suitable matrix for a ...
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### What does "kernel based" mean?

In my thesis I try to explain what kernel based methods are, especially the meaning for object detection. I know kernel based methods like Mean- and CamShift and I know how to use them. I understand ...
217 views

### Reflection Rotation matrix

I'm doing a rigid body between point clouds using SVD. Sometimes the routine produces an incorrect rotation matrix with a Determinant=-1, ie a reflection matrix. Any idea why ? Is there a valid way ...
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### Proving that the uncertainty can not increase during the update step of a Kalman filter - positive semidefiniteness

I am trying to prove mathematically that the update step in a Kalman filter can not result in a increase in uncertainty. I found the following proof which is based on the inversion lemma and the ...
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### linear convolution toeplitz matrix vs circular convolution toeplitz matrix

I have an issue in understanding the difference between building the Toeplitz matrix when the convolution is linear and when it's circular. As I know that Toeplitz matrix $H$ can be built as following ...
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### What is about the circular convolution in OFDM

In an OFDM system, serial-to-parallel conversion for data is done, then the DFT is performed and then adding the cyclic prefix (CP). My question is related to that step of adding a CP. As I know, ...
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### Noise estimation SNR matrix

I have a signal matrix which is a $256\times 192$, and I want to calculate the SNR considering that my $259\times 192$ matrix is an average of a $256\times 192\times 330$ matrix, where $330$ is the ...
718 views

### Matrix form of STFT

IS there a Matrix form of the STFT that could be applied to a signal directly, as in the case of DFT? We know the matrix structure of the DFT Matrix. Can we derive that somehow for a STFT Transform? ...
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### Deconvolution using Toeplitz matrices [duplicate]

Lets say I have 2 vectors (1D signals that are sigmoids): $s$ and $m$, both related through the relation: $m = s * r$, my goal here is to recover the vector $r$ (should look like a gaussian). I tried ...
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### In 1D DCT, why is the input a vector?

The question is specific to this document: Image Compression. It is chapter 6 from book "A First Course in Applied Mathematics" by Jorge Rebaza. It is, a to the point explaination of DCT that is ...
324 views

### Dominant eigenvectors of an unknown matrix

Do you have any idea about how we can find the principal eigenvectors of an unknown matrix ${H}$? The elements of $H$ are unknown in general. If you are familiar with channel estimation procedure in ...
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### Digital Image Processing Framework

i'm looking for a free/opensource solution for digital image processing with reasonable performances processing quite large images (e.g. actual smartphones have resolutions of 8-10Mpx) and being ready ...
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### What is an analysis dictionary or operator in compressive sensing?

I am doing research on compressive sensing. I am new in this field. I read several papers regarding analysis dictionaries. Here are some papers that I have read so far: https://www.hindawi.com/...
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### Normalized LMS with a posteriori Error and Woodburry's Matrix Inversion

I was going through this paper and the author mentioned that we can prove the following using the Matrix Inversion Lemma (AKA Woodburry's Matrix Inversion Identity): Using matrix inversion lemma we ...
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### Deterministic method to compute “Process noise covariance matrix, Q” for a Kalman filter when parameter variations of the model is known apriori

I am implementing a Kalman filter (for a linear ODE system for now). My model represents a physical device that has 6 "parameters", i.e. those values of the device do not evolve over time (within a ...
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### Help Understanding Radial Gaussian Filter

I am currently reading through Mueller's "Fundamentals of Music Processing" and I am trying to understand audio segmentation through the use of self-similarity matrices. Currently, my matrix looks ...
226 views

### Can Cholesky outer product version result in negative square roots?

Say A is symmetric positive definite matrix , which means one necessary condition is diagonal entries of A are positive. If I do cholesky factorisation using outer product form, Can there be any ...
1 vote
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### Optimization of square matrix multiplied with another matrix to have the final result a unitary matrix

I have a square matrix $D$ whose size is $m \times m$ multiplied with another $m \times m$ square matrix $C$, I need to optimize the matrix $C$ to have a unitary matrix $DC$. I mean optimize the ...
1 vote
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### Covariance matrix explanation

I am trying to understand and visualize the concept of a covariance matrix. Suppose I have a matrix: $A = \begin{pmatrix} 2 & 3 & 4\\ 5 & 5 & 6 \end{pmatrix}$ how do I calculate its ...
1 vote
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### What is the complexity of big-$O$ $O(N \times \mathrm{log}_2(N))$ vs real operations

I usually see books/references writing the complexity of such operations as $O(N \times \mathrm{log}_2(N))$; For example, the complexity of FFT/IFFT operation is $O(N \times \mathrm{log}_2(N))$. ...
1 vote
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### The inverse of an orthogonal matrix is its transpose

In the following statement I don't understand the case for $\ i = j$: Let $\mathbf A$ be an $\ m \times \ n$ orthogonal matrix where $\ a_i$ is the $\ i^{th}$ column vector. The $\ ij^{th}$ element ...
1 vote