# Tag Info

Accepted

### 2D Convolution as a Doubly Block Circulant Matrix Operating on a Vector

The point is that circular convolution of two 1-D discrete signals can be expressed as the product of a circulant matrix and the vector representation of the other signal. The circulant matrix is a ...

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

There's a newer (2017) closed-form formulation for the eigendecomposition of 2x2 and 3x3 Hermitian matrices here: Charles-Alban Deledalle, Loic Denis, Sonia Tabti, Florence Tupin. Closed-form ...
Accepted

### The inverse of an orthogonal matrix is its transpose

An orthogonal matrix has orthogal columns, i.e. the scalar product of two different columns is zero (the case $i\neq j$). For the case $i=j$ you have $a_i^Ta_i=\|a_i\|^2>0$. So, all you can say ...
Accepted

### Circular Convolution Matrix of ${H}^{H} {H}$

If $H$ is a matrix form of Circular Convolution then it is a Circulant Matrix. Being a Circulant Matrix means it can be diagonalized by the Fourier Matrix ${F}$: $$H = {F}^{H} D F$$ Where the ...
Accepted

### Why do we need to estimate eigenvalues?

It seems to me that we can directly calculate ... which can be done in a few second in matlab. Who says that Matlab is calculating it directly, or that it isn't using Gershgorin's circle theorem in ...

### Why do we need to estimate eigenvalues?

We do use eigenvalues, because they behave like invariants in linear systems (invariant inputs are well connected to outputs) and generally finding invariants of unknown or model systems provides a ...

### Optimization of square matrix multiplied with another matrix to have the final result a unitary matrix

Could it be that you are indeed looking for the closest orthogonal matrix $Y$? Then, there is a solution which involves computing the square root of $D^TD$ . If $E=(D^TD)^{1/2}$ were invertible, the ...
Accepted

### Sensing matrix for compressed sensing

A sensing matrix maps input vector to measurement vector through linear wighted summation of input. What makes a specific matrix good, is application dependent. Now, both distributions more or less ...
Accepted

### On the simplification using trigonometric functions

If you use the Euler's formula, you can simplify like this: $$[d]_{k,n} = \frac{\sqrt{2}}{N}\left( \cos{\left[ \frac{(k-1)(2n-1)\pi}{2N} \right]} e^{j\frac{2 \pi nk}{N}} \right)$$ I think we can't ...

### why use svd() to invert a matrix?

The two methods differ, above all, by their applicability to matrix classes. col (cholesky) decomposes Hermitian, positive-definite rectangular matrices into the ...
Accepted

### What is the complexity of multiplication a real matrix with real vector

If you multiply an $M \times N$ matrix with an $N \times 1$ vector you get a vector of size $M \times 1$ For the generic case you will need $M \cdot N$ multiplications and $M \cdot (N-1)$ additions. ...

### What is the complexity of big-$O$ $O(N \times \mathrm{log}_2(N))$ vs real operations

Big O abstracts away knowledge about multiplies and adds and complex math, and focuses on how (whatever operations) scale when you increase N. For the case of FFTs, the core operation that motivate ...
Accepted

### Understanding y=Hx+n equation in detail?

The result $E[nn^\dagger] = I_r$ comes from writing out explicitly the diagonal, and the off-diagonal terms in the $r\times r$ matrix $nn^\dagger$, paying special attention to that $\ \dagger$ ...
Accepted

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

If you want to solve for single value of $\lambda$ in the model: $$\arg \min_{x} \frac{1}{2} {\left\| A x - b \right\|}_{2}^{2} + \lambda {\left\| x \right\|}_{1}$$ Then you can use Coordinate ...
Accepted

### How is the sound converted to matrix in Matlab?

The function audioread doesn't generate any values, it just reads audio samples stored in a file. If you want to generate the sound of a guitar, you need to look ...

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

Let me take a stab at it. You agree that $\mathbf{R}_k$ is positive definite. Since it is the variance. Now, $\mathbf{P}_{k|k-1}$ is also positive definite as it is a covariance matrix, as ...

### Maximising each element in a matlab array

Try using bsxfun if your version has it. A = bsxfun(@max, B, C.') As per their documentation, ...

### Maximising each element in a matlab array

If the matrices are not too big, repmat could work: ...

### Deriving the Matrix Inversion Lemma for RLS Equations vs the Woodbury Derivation

I'm not sure if the OP was looking for a proof or derivation. In my mind a derivation is bit different than what Royi provided. I have looked for but never seen a derivation of the various versions of ...
Accepted

### How can I get the uncertainties for peaks on an image?

One way is to simply model each peak with a Gaussian, with mean $\mu_i$ and variance $\sigma_i$. In fact what you mean by uncertainty corresponds to the variance. You can iteratively fit Gaussians ...

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

It means you will have a non-unique solution or redundancy. In your formulation, $X_4$ is completely free and $X_3$ is an offset parameter. You can deflate your matrix and obtain the full rank part to ...
Short answer: just use $\sigma = 10^{-8}$. Covariance matrices have eigenvalues $\geq 0$ (theoretically), so $Ri + 10^{-8} \, I$ will have eigenvalues $\geq 10^{-8}$, safely non-singular. A longer ...