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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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 $O(N \times \mathrm{log}_2(N))$. ...
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How to find significance of time varying regression coefficents in a regression model? ex student-t

I am in a requirement of finding the significance of time varying coefficent b(t) in the below multi linear regression model: $$y(t) = a(t)x_1(t)+b(t)x_2(t).$$ I have looked into student-t test to ...
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How can I explain the result of of multiplications by matrices

I have a vector $x$ with size $N \times 1$, it's multiplied with a $Z$ matrix $N \times N$, the resulted $N \times 1$ vector is $y = Zx$. I know that each value of $y_i$, where $i = 0, 1, 2, .., N-1$ ...
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Can I reduce the complexity of Fourier transform and equalization into one matrix multiplication

I have a signal $x \in \mathbb{R} ^{N \times 1}$, I need to perform Fourier transform, then zeros forcing equalizer (which means wise-element division by constant) and then inverse Fourier transform....
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Expressing mathematically the number of real addition operation for a vector after dividing it

I assume I have the length of such vector $y$ is $N$. In the first time I divide that vector into two columns and then sum them point-wise summation. The second time, I divide the same vector $y$ ...
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What is the complexity of multiplication a real matrix with real vector

I have a real matrix $Z$ which is following the form as following: $Z = \begin{bmatrix} x_1& 0& 0& 0& 0& 0& 0& 0\\ 0& x_2& 0& 0& 0 & 0&...
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How can I express the flipped output of multiplication in function of original inputs?

I have the vector $y = Dx$ where $D$ is a complex matrix with dimension $N \times N$, and $x$ is a complex vector of dimension $N \times 1$. If the vector $y_2 = [y'_N, y'_{N-1}, y'_{N-2},.... , y'_{...
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Multiplication of Fourier repeated matrix with a higher dimensional matrix

I have a square Fourier matrix $F_{2\times 2}$ with size of $2 \times 2$. It’s repeated twice following this way: $F_r = F \bigotimes I_{2 \times 2}$, where $\bigotimes$ is the Kronecker tensor ...
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Converting 2 variables (obtained by ICA decomposition) named A_ffdiag and W_ffdiag into mixing/unmixing matrix

I have two variables as a result of an ICA (Independent Component analysis) decomposition (decomposed using joint diagonalization algorithm which uses frobenius norm formulation) named A_ffdiag and ...
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Extended Kalman Filter: Error Covariance

I am trying to implement an EKF for orbit determination of a spacecraft(SC). The state which i am interested to estimate is the following $x = [r_{SC}\,v_{SC}\,\Delta Cd\, \Delta Cs\, b\,d]$ where $r_{...
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Convolution between a vector and another symmetric vector

Let's have the vector $y = h * x$ where $*$ is the convolution operation, $h$ is the channel with length $N$ and $x$ is a symmetry vector which means $x = [x_M, x_{M-1}, ....,x_0, 0 , x_0, x_1, .... ...
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Can we recover a vector from one element of resulted vector after multiplication?

I have a matrix $X = \begin{bmatrix} 0.5000 + 0.5000i & 0.5000 - 0.5000i\\ 0.5000 - 0.5000i & 0.5000 + 0.5000i \end{bmatrix}$ multiplied with a column containing a complex number and its ...
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Camera Calibration: Intuition for the extrensic parameters in relation to the camera center and intrinsic parameters

In our lecture we have learnt that the extrinsic parameters would transfer a point $X_\text{World}$ described in world coordinates to a point $X_\text{cam}$ in camera-coordinates In in-homogeneous ...
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Constructing a Hankel matrix for n4sid with multiple inputs\outputs

I am trying to construct a Hankel matrix to write my own code for n4sid algorithm (page 47 and 22). $ \mathcal{H}= \underbrace{\begin{pmatrix} U_{0|2i-1}\\ Y_{0|2i-1} \end{pmatrix} / \sqrt{j} }_{2(m+l)...
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Signal Processing on non-Euclidean domains

I have a very simple yet fundamental question. Suppose I have a vector of data $x \in \mathbb{R}^N$. Without additional information, I guess the majority of people think this vector as defined over ...
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2 votes
1 answer
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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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how do you know if your matrix is sparse after sparsifying transform?

To successfully compress the data using Compressive Sensing method, I need to have sparse vector, theoretically a vector is sparse if the entries of the vector has many zero or nearly zero. My ...
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3 votes
1 answer
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Is it possible to detect the sparse vector based on a non-invertible matrix

Given a non-invertible matrix $X \in \mathbb{R}$, let's say that matrix is, e.g. : $X = \begin{bmatrix} 0.7500& -0.2500 &-0.2500 & -0.2500 \\ -0.2500& 0.7500& -0.2500 & -0....
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Estimating Correlation Matrices

I am trying to obtain the correlation matrices of two random signals. Both of them, $ X $ and $ Y $, are white Gaussian Noise, with unitary variance. However, they are correlated, with correlation ...
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Matrix multiplication computational complexity based on radix 2

I am wondering, can we use Radix 2 based computational-complexity calculation for any matrix multiplication whose size is $N$ x $N$ ?? where $N$ = $2^K$ and $K > 1$ is an integer ?? Or it can only ...
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2 answers
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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 ...
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3 votes
1 answer
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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 ...
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2 votes
0 answers
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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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3 votes
2 answers
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Why do we need to estimate eigenvalues?

I am not working in signal processing field, but recently I happen to read a paper which estimates source numbers using Gerschgorin radii, and I feel kind of confused about why we need to estimate ...
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1 answer
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Matrices of complex numbers multiplication

I'm trying to implement the multiplication of two matrices something like this picture in c langage. I want to read the numbers from a text file of x and store it later in an array the code that i ...
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1 vote
1 answer
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What is the relation between eigenvalues and state-space response in control systems?

I understand the mathematics behind it but I want to know what happens physically in a real-life system. How do the eigenvalues come into the picture from a non-mathematical (physical) point of view? ...
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Is FFT sub matrix degeneracy a problem in OFDMA for some type of noise?

Say that the discrete Fourier transform (DFT) is used in OFDMA. There are a number of degenerate (singular, non invertible) sub matrices of some DFT matrices. Does this result in any problems? One ...
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1 answer
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Property of the trace and expectation

I'm reading the paper Model-Driven Deep Learning for Joint MIMO Channel Estimation and Signal Detection by Hengtao He, Chao-Kai Wen, Shi Jin, and Geoffrey Ye Li on Orthogonal Approximate Message ...
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1 vote
1 answer
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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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How can I find expansion coefficients of the a vector in a given basis?

How can I find the coefficient of the vector $\mathbf y$? And how can the inner product be done on these vectors? Let $\mathbf y = \begin{bmatrix}1\\2\\0\\1\end{bmatrix}$ What are the expansion ...
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1 vote
0 answers
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How to find parameters of Kalman filter using matrix information?

I'm trying to understand concepts on Kalman Filters. Consider the overdetermined system $Ax=y$; $$\begin{bmatrix}1 \\ 1 \\ 1 \\1 \end{bmatrix} x = \begin{bmatrix} 3 \\ 5 \\ 4 \\ 8 \end{bmatrix}$$ Let $...
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1 answer
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Relation between the matrix trace and the amplitude of each element

Assume a diagonal matrix $\mathbf X$ whose size $N\times N$ and its diagonal elements are $0.5 + 0.5i$, and the vector $\mathbf p$ of size $N\times 1$ whose elements have similar amplitude. I have ...
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2 votes
1 answer
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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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Stability of $x(n) = A x(n-1)+b$

I am looking at the following system: $x(n) = A x(n-1) + b$ where x and b are vectors and A is a matrix. How can I derive the stability and causality conditions for such a system using Z transform? If ...
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1 vote
0 answers
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What is the order of the output of HoughCircles in OpenCV, when having several circles on a frame?

So I am trying to use stereo vision to calculate the depth of several red balls. With the function HoughCircles, I get an output at one frame from one of the cameras like for example: ...
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1 answer
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Translating SFFT expression to MATLAB code

The Symplectic Finite Fourier Transform (SFFT) of a 2D periodized sequence $x[k,l]$ with periods $(M, N)$ is defined as $$X[n,m] = \sum_{k=0}^{M-1} \sum_{l=0}^{N-1} x[k,l] e^{-j2\pi \left(\frac{mk}{M} ...
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MATLAB: How to copy column vectors from a matrix into a cell array? [closed]

I have some sensor data stored in a 1000 x 5 matrix. I'd like to copy each column into a 5 x 1 cell array such that each cell contains a 1000 x 1 vector. I tried the num2cell(mymatrixname,1) function, ...
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1 vote
1 answer
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Which one can be accompanied by linear filters?

I have a matrix : \begin{bmatrix} 1 & 2 & 3\\ 1 & 4 & 5\\ 2 & 6 & 7 \end{bmatrix} After doing operation 1 , I get \begin{bmatrix} 0 & 0 & 0\\ 1 & 2 & 3\\ 1 &...
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1 vote
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correlation matrix vs. correlation function?

Can someone help me understand the difference between a 1-dim autocorrelation function and a 2-dim autocorrelation matrix of a random process aka time series? My Leon-Garcia textbook defines CX(τ) and ...
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1 answer
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Solve equations with conjugates multiplications

If I have a two variables $x_1$, $x_2$ and two equations. In the first equation the first variable $x_1$ is multiplied with the conjugate of same number which is multiplied with $x_2$, and in the ...
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1 answer
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Taking inverse Fourier transform in column-wise and solve it in row-wise

$\DeclareMathOperator{\FFT}{FFT}\DeclareMathOperator{\IFFT}{IFFT}$Assuming I have a matrix $X$ of size $64\times16$. Taking the $\IFFT$ for it in column-wise, I means that $Y = \IFFT(X)$; Is it ...
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4 answers
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On the simplification using trigonometric functions

Assume I have a matrix $D$ whose its entries are as below : Where $A$ and $B$ can be written using using the trigonometric functions for (1) as: My question, Is it possible to simplify (1) more? ...
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unitary matrix complexity multiplication

Having a unitary matrix $X$ whose size is $n \times n$ and a vector $z$ whose length is $n$, and let's have: $$y = X^H {\rm diag}(z)X$$ where $X^H$ is the conjugate transpose of $X$. My question,...
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1 vote
1 answer
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Real time signal processing use cases for eigenvalues of symmetric matrices

I realize that this might be somewhat of an unusual and specific question. I know that the eigenvalues of symmetric matrices are used in a number of ways in scientific computing, such as for finding ...
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2 answers
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How can I find expansion coefficients of the y vector in a given basis?

Consider the following vectors in $\mathbb R^4$: $$\mathbf{v}^{(0)}=\begin{bmatrix}\frac{1}{2}\\\frac{1}{2}\\\frac{1}{2}\\\frac{1}{2} \end{bmatrix} , \mathbf{v}^{(1)}=\begin{bmatrix}\frac{1}{2}\\\...
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Expectation of a constant diagonal matrix

Is the expected value of a diagonal matrix with constant entries equal to the mean value of the entries? My question stems from the following observation in a paper. Given a real diagonal matrix $\...
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On the use of Permutation matrix to perform iFFT

I have a question about using the permutation matrix for performing $iFFT$ for such matrix and then reshape it row-wise and column-wise way. Let's say that we have a random matrix $x$ whose size is (...
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Effect of Adding the cyclic prefix on the toeplitz matrix in OFDM

Assuming we have $N$ symbols to transmit encoded in block $k$, Performing $N$−iFFT at the transmitter, we now have The resulted signal $x(k)$ has length of $N$. inserting a cyclic prefix $CP$ ...
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1 vote
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Calculate sampling lattice matrix in 2D

The pattern in which the sample points are distributed in 2 dims, is called a sampling lattice, and can be defined by a generator matrix.. In 2 dimensions, the generator matrix consists of 2 vectors. ...
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3 answers
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Why do convolution kernels such as Gaussian, Laplacian, LoG almost always seem to be expressed in integers?

I'm a total newb in search of some deeper understanding, but I'm not able to read the maths behind these on Wikipedia. If I understand correctly, you get the new value for each pixel by multiplying ...
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