Questions tagged [eigenvector]
The eigenvector tag has no usage guidance, but it has a tag wiki.
293 questions
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The proof of the invertibility of $\Big( \sin\frac{8kl\pi}{2n+1} \Big)_{k,l=1}^\frac{n}{2}$
Suppose that $n$ is even. Any suggestion/appraoch to prove that $S=\Big( \sin\frac{8kl\pi}{2n+1} \Big)_{k,l=1}^\frac{n}{2}$ is invertible?
- 15.4k
3
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1
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The eigenvalues of the matrix $\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^n$
What are the eigenvalues/eigenvectors of the matrix $A=\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^n$ when $n$ is odd?
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0
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137
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What are the eigenvalues/eigenvectors of the matrix $A=\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^{\frac{n-1}{2}}$ when $n$ is odd?
Suppose that $n$ is odd. The eigen values/eigenvectors of the skew-circulant matrix $A=\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^n$ are successfully computed in this post.
Q. What are ...
- 1,439
1
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1
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The dimension of the eigenvector space of non-negative irreducible matrices
Let $A$ be an irreducible non-negative matrix. Is it true that the eigenvectors of $A$ can span the $R^n$ ?
Or are all the eigenvalues of $A$ distinct?
- 20.2k
1
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2
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449
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Transforming matrix to off-diagonal form
I wonder if one can write the following matrix in the form $A = \begin{pmatrix} 0 & B \\ B^* & 0 \end{pmatrix}.$
The matrix I have is of the form
$$ C = \begin{pmatrix} 0 & a & b & ...
- 785
0
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0
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273
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Finding the eigenvectors of a submatrix
Let $A=(a_{kl})$ be a matrix in $M_n(\mathbb{R})$ when $n$ is even. Let $B=(b_{kl})$ be the symmetric $2n$ by $2n$ matrix whose entries are given by,
$b_{k,l}=a_{kl}$ if $1\leq k,l\leq n$.
$b_{n+k,l}=...
- 4,058
3
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0
answers
47
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What is the supremum of 1-dim Hausdorff measure of the nodal set of Neumann eigenfunction $u$ for planar convex domain
All descriptions of this question are limited to 2-dimension for simplicity. Recently, I read some papers on the nodal set of Laplacian eigenfunctions. Denote $\Sigma=\{u(x)=0\}$ be the nodal set of a ...
- 63.9k
3
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2
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394
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Eigenvalues and eigenvectors of non-symmetrical tridiagonal matrix
The question is the following: given a matrix
$$A=\begin{pmatrix}
1& 2 & & & & \\
1& 0& 1 & & & \\
& 1& 0& 1 & &\\
& &...
- 3,791
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1
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363
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Are there zero entries in the eigenvector corresponding to a simple eigenvalue?
For a real symmetric matrix $M$ and a simple eigenvalue $\lambda$, under which conditions the corresponding eigenvector has no zero entries? Perhaps, this is unconditional and one can provide a proof?
- 91.8k
4
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Least squares problem with left and right unknowns
For $i=1,...,n$, let $b_i$ be a scalar and $A_i$ be an $k\times l$ matrix. Is there a closed form solution for the following problem assuming $n>k+l$?
$$\min_{x\in \mathbb{R}^k ,y\in \mathbb{R}^l} \...
- 188k
2
votes
2
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485
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Eigenvalues in the semiclassical limit
Consider the Schrödinger operator $H_\hbar = -\hbar^2\Delta + V$ on $M=\mathbb{R}^n$, where $V$ is a potential that behaves well in a certain sense ($C^\infty$, bounded from below, going to infinity ...
- 2,821
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1
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Independence of rotated spherical harmonics
Hi,
Consider a spherical harmonic of degree $l$, denoted by $y_l^m$. I rotate this harmonic using $2l+1$ different rotations. The set of functions I get is not an orthogonal set, but the functions ...
7
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1
answer
464
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"Unimodality" of the positive eigenvector of a non-negative irreducible matrix?
Consider an eigenvalue / eigenvector problem for a matrix $A$ that is known to be non-negative and irreducible (so the Perron-Frobenius theorem applies):
$$\sum_j A_{ij} x_j = \lambda x_i$$
Here $\...
- 884
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2
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473
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Support of eigenvectors
Consider the $N$ by $N$ matrix
$$M_N= \begin{pmatrix} 1+3\lambda & -1-2\lambda & - \lambda & 0 & 0 &0 &0\\
-1-2\lambda & 2+3\lambda & -1 & -\lambda & 0 & 0 &...
- 192
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2
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372
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How close to uniform are Perron-Frobenius eigenvectors?
Let $A=(a_{i,j})$ be a square matrix with non-negative entries. (Assume $A$ is symmetric, if it helps.) Let $v$ be a Perron-Frobenius eigenvector. What do we need to assume about $A$ in order to have ...
- 1,354
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Local energy estimate in a semiclassical regime
Let us consider $h_n=(2n+1)^{-1/2}\to 0$ as $n\to \infty$ be a small parameter, which we just write as $h$ for convenience, and $u_h : \mathbb{R} \to \mathbb{R}$ be functions satisfying $Pu_h=0$ (I ...
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1
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141
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On the eigen vectors of a diagonalizable matrix
Let us consider the space $M_n(\mathbb{C})$. By a unitary matrix $U=(u_{ij})$ we mean that $U^{-1}=(\overline{u_{ji}})$.
Q. Let $U$ be a unitary matrix. I am looking for the pairs of matrices $(D,A)$ ...
1
vote
0
answers
274
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Find the eigenvectors from the QR algorithm in the unsymmetric case
It is possible to find many references describing the QR Algorithm with more or less refinements to approximate the eigenvalues of a square matrix $A\in\mathbb{R}^{n\times n}$.
I implemented a version ...
- 11
2
votes
1
answer
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Diagonalizing a symmetric block matrix
Let us consider the matrix
$$ A = \begin{pmatrix} a & c+ib \\ c-ib& a \end{pmatrix},$$
then this matrix has eigenvalues $a\pm \sqrt{c^2+b^2}.$
Now, let us consider a block matrix
$$ A = \begin{...
- 52.4k
2
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0
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301
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Eigenvector of Hadamard matrix functions
Let $X\in\mathbb{R}^{n\times n}$ with SVD $X=UDV^T$. Are there known results regarding the eigenvectors of $Y=X^{\odot g}$? I am mainly interested in simple functions such as $g(z)=z^2$, i.e. $Y_{ij}=...
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1
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Statistical estimation of singular values and vectors
My question is about the well known and well studied singular value decomposition (SVD). What I am working on right now requires performing an SVD repeatedly on a slowly varying matrix. Since I don't ...
- 63.9k
1
vote
1
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468
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Trace minimization for generalized eigenvalue problem
In [1], it is shown in theorem 1.2 that for symmetric $n \times n$ matrices $A$, $B$, we have
$$
\min_{Y \in Y^*} \text{tr}(Y^TAY) =
\text{tr}(X^TAX) =
\sum_{i=1}^p \lambda_i,
$$
with
$$
\text{
$X^...
- 21
5
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0
answers
208
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Perturbation of Neumann Laplacian
Consider the $N \times N$ matrix
$$A_{\alpha}=\begin{pmatrix} \lambda_1 & -1 & -\alpha & 0 & \cdots & 0\\
-1 & \lambda_2 & -1 & -\alpha & \cdots & 0\\
-\alpha &...
- 73
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474
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When must an eigenvector have only non-negative entries?
What would be a reasonable sufficient condition on a real symmetric matrix that would force its eigenvector with largest eigenvalue (or one of its eigenvectors with maximal eigenvalue) to have only ...
- 20.2k
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0
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Eigenvalues and eigenvectors of non-symmetric elliptic operators
We know that the operator $A=\Delta$ with domain $D(A)=\{u\in W^{2, 2}(\Omega): u=0 \ \ \text{on } \partial\Omega\}$ (say $\Omega$ is a bounded nice domain) has eigenvalues $\lambda_1>\lambda_2\ge \...
- 11
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0
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327
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Eigenvector to zero eigenvalue of general Laplacian
I was wondering what we can say about the eigenvectors of a matrix $A$ fullfilling $Ax =0$ where $A$ is symmetric with a diagonal equal to one and every row sums up to 0. Obviously this is a ...
0
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1
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878
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Centrality measures in a network with negative correlations
I have a bidirectional network where the weights of edges are based on partial correlation matrix. I have both positive and negative values as weights. Now, I want to compute centrality measures as ...
- 1,444
1
vote
1
answer
176
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Derivative of eigenpair with respect to matrix
Suppose that $A$ is real and symmetric matrix (or tensor) of dimension $3 \times 3$, with its spectral decomposition
$$A = \sum_{i=1}^3 \lambda_i\ n_i\otimes n_i$$
where $\lambda_i$, $n_i$ and $\...
- 188k
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2
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Eigenvectors of graph Laplacian for spectral clustering
I have the following questions regarding the graph Laplacian for spectral clustering:
What is the intuition behind projecting the Laplacian (D-A, where D is the degree matrix and A is the affinity ...
- 19.4k
13
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1
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697
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$\ell^1$-norm of eigenvectors of Erdős-Renyi Graphs
Setting. Let $G(n,p)$ denote the usual Erdős-Renyi (random) graphs. For each such graph there is an associated Laplacian matrix $L = D - A$ where $D$ collects the degrees on the diagonal and $A$ is ...
CommunityBot
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1
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Eigenvectors of a tensor in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$
I want to find the critical point of tensor $f=a_0b_0c_0 + a_1b_1c_1$ in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$, and I followed this construction:
First, I take the following partial ...
- 309
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0
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Fast decay of eigenvector elements
Let A be a set of similar (symmetric) matrices, sharing the same eigenvalues. I understand that their eigenvectors would be different. Let us focus on one eigenvector (e.g. corresponding to the lowest ...
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2
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Lower eigenvectors of nonnegative matrices with zero trace
Let $A$ be an $N\times N$ nonnegative matrix with all diagonal entries equal to zero and such that there is $n_0$ such that all entries of $A^{n_0}$ are strictly positive. Let $\lambda_1,\ldots, \...
- 12.6k
2
votes
1
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Distribution of eigenvectors of random matrices and link with the components of the matrix
Let $M$ be a real symmetric matrix of size $N$ with its components $M_{ij}$ following a normal distribution centered around 0.
Let $x\in\mathbb{R}^N$ be an eigenvector of $M$ with eigenvalue $\lambda\...
- 188k
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votes
1
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Can I assign the term "is eigenvector" and "is eigenmatrix" of matrix $P$ in my specific (infinite-size) case?
Remark: I asked this in MSE, the question got views and votes but seemingly no one had an answer so far.
Background: I'm rereading a couple of my exploratory (surely not research-level) math-essays ...
CommunityBot
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Conjugate gradient and the eigenvectors corresponding to the large eigenvalues [closed]
I am working on an optimization problem (for example, conjugate gradient) to solve $Ax=b$, where $A$ is a symmetric positive definite matrix. I can understand that the CG (conjugate gradient) has ...
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Derivative of eigenvectors of an Hermitian matrix
In the question "Derivative of eigenvectors of a matrix with respect to its components", Liviu Nicolaescu has provided an answer valid for a real matrix. As outlined in the following, the ...
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1
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Find a way to apply the MLE on Fisher or Covariance matrix to make cross-correlations
I have 2 Fisher matrixes which represent information for the same variables (I mean columns/rows represent the same parameters in the 2 matrixes).
Now I would like to make the cross-correlations ...
- 61.9k
2
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Infinite positive matrices with probability eigenvector
Let $A$ be an infinite non-negative matrix with integer entries ($a_{ij} \geq 0, \forall i,j \in \mathbb N$).
Suppose that $A$ is irreducible, aperiodic, and recurrent. So that it satisfies the ...
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votes
1
answer
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Condition number for matrix of eigenvectors of a diagonalizable matrix
Let $A$ be a diagonalizable matrix, i.e., $A=SDS^{−1}$. For any matrix $A$, condition number is defined as $\kappa(A)=\|A\| \|A\|^{-1}$.
For $A$ being a diagonalizable matrix, define $G_A=\{{S: S^{-1} ...
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If there are eigenvectors with largest components $i$ resp. $j$, then is there an eigenvector with two largest components $i$ and $j$?
Let $G=(V,E)$ be a connected (finite simple) graph with vertex set $V=\{1,...,n\}$ and let $\theta_2\in\Bbb R$ be the second-largest eigenvalue of its adjacency matrix. I wonder about the following ...
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Requirements for finite backward derivatives of degenerate eigenvectors
A matrix, $\mathbf{A}(\theta)\in\mathbb{R}^{n\times n}$, has elements that depend on a parameter $\theta$. The $j$-th eigenvalues and eigenvectors of the matrix are denoted as $\lambda_j$ and $\mathbf{...
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References on discrete Sturm-Liouville eigenvectors convergence
Let $ L : u_n \mapsto a_n u_{n + 1} + b_n u_n + a_{n - 1} u_{n -1} = \nabla ( a_n \Delta u_n ) + (b_n + a_n + a_{n - 1}) u_n $ be a discrete Sturm-Liouville operator, with $ \nabla u_n := u_{n + 1} - ...
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Real eigenvectors of complex matrices
Let $A$ be a nonsingular complex $(3 \times 3)$-matrix (that is, an element of $\mathrm{GL}_3(\mathbb{C})$).
Then what are some of the best-known criteria which guarantee $A$ to have real eigenvectors ...
- 108k
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1
answer
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Calculating second derivatives of eigenvectors of a matrix with some degenerate eigenvalues
Given real symmetric matrix $\mathbf{M}$ with eigenvalues $\lambda_i$ and eigenvectors $\mathbf{v}_i$, the derivative of an eigenvector is $$\dot{\mathbf{v}}_i = \sum_{j \ne i} \frac{\mathbf{v}_j \...
- 13k
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How to find eigenvalues of following block matrices?
Is there a procedure to find the eigenvalues of A?
$$A=\begin{bmatrix}X & I &&&&&&&&& 0\\I & 0 & P &&&&&&&&\\& P^t ...
- 123
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0
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166
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Minimize a vector from a matrix operation
I want to minimize a certain vector that results from a matrix operation with some constraints and i don't exactly know how to tackle this problem.
Lets say we have
$$
(L+A)*s = v
$$
L is the ...
- 101
11
votes
2
answers
10k
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Derivative of eigenvectors of a matrix with respect to its components
Suppose that $B$ is a real, positive-definitive symmetric ($3\times3$) matrix (more accurately, $B$ is a tensor) with distinct eigenvalues, and that we can write it as
$$
B= \sum_{i=1}^3 \lambda_{i}(...
- 1
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0
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79
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Eigendecomposition of $A=I+BDB^H$
Suppose that we have $$A = I_m + BDB^H$$ where matrix $A$ is $m \times m$, matrix $B$ is $m \times k$, $BB^H \neq I_m$ and $D$ is a $k \times k$ diagonal matrix. Can we obtain the eigendecomposition ...
0
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0
answers
51
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What transformation is required to find a unique solution of this problem instead of multiple solutions?
$$
\max\limits_{\mathbf{f},\ \|\mathbf f\|=1 } \log_2\left(\prod^K_{i=1} \ \frac{ \mathbf{f}^H {\mathbf E} (\mathbf{W}_i, \Theta, \tau_i) \mathbf{f}} { \mathbf{f}^H \mathbf{G}_i ( \mathbf{W}_i, \...
- 1