Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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Derived series of a Lie algebra

I've been studying semisimple Lie algebras and solvability and was wondering if someone could explain to me the meaning of the derived series of a Lie algebra L and this part: $$L^{(1)}=[LL]$$ I don't ...
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31 views

Linear homogeneous ODE system of first order

Good afternoon. I recently encountered the following problem to which I couldn't find a solution anywhere so far: Given $A:D\to\mathbb C^{2\times 2}$, $D\subset\mathbb C$ open, with holomorphic ...
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36 views

Diagonalizing and finding the eigenvalues of matrix of type $T$.

I have seen some of the solutions type within the math.stackexchange but didn't able to get the clear idea. Consider here n to be $\ge$ 5. $$ T = \begin{bmatrix} \alpha_1 & \beta & & & ...
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72 views

Is this a geometric series?

A geometric series is, in general, defined by: $$ \sum_{k=0}^{n-1}a\cdot r^{k}=a\cdot\dfrac{1-r^n}{1-r},\quad\quad \quad\quad \quad\quad r\neq1 $$ If I have instead the following: $$ ...
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103 views

Change in Singular Value Decomposition of a matrix on addition of a single row

Given that I know the svd decomposition of a matrix, is there any way to compute the svd decomposition of the matrix obtained by adding a single row to the original matrix? Is there any relation ...
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92 views

If symmetric matrix in a least-square deconvolution problem positive definite?

I want to apply Gauss-Seidel method in a least square deconvolution problem. The convolution of two vectors is written in: $h * x = z$. $$z(n) = \sum_{i=0}^{N-1}h(i)x(n-i)$$ It is a linear transform ...
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36 views

element subset for adjacency matrix

I am trying to create an element of a matrix that is a subset of a larger matrix. However, I am told that my subscripts do not match. I wanted other people's opinion as to what I am doing wrong and ...
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31 views

Extension of Schur-Cohn for quadratic matrix equation

Starting from a quadratic in $z\in\mathbb{C}$ with real scalar coefficients $b,c$: $z^2 + bz+c=0$ and using the Schur-Cohn recursion, I can get the following conditions on $a,b,c$ such that $|z|\leq ...
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56 views

Block Diagonalization related to Direct Sum and Single Eigenvalue?

I'm just a beginner in Linear Algebra, and I've proved myself the following: A matrix $A^{n \times n}$ is block diagonalizable if and only if the base field $F^n$ can be divided into at least two ...
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73 views

product of two vectors equal a positive definite matrix?

Given two vectors $v$ and $w$ in $\mathbb{R}^n$, what are conditions on $v$ and $w$ so that $vw^\intercal$ is a positive definite matrix?
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91 views

How to find the orthogonal complement of C(A)

So I'm doing this review sheet for an exam and I've come across a question I'm stuck on. It gives me some $3\times 4$ matrix $A$, asks me to determine whether or not $v$ is in the column space of $A$ ...
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53 views

Nilpotent endomorphism and Jordan matrix

I have a question about nilpotent endomorphism. Suppose you have an endomorphism $N:V \rightarrow V$ which is nilpotent, such that the degree of nilpotency is equal to the dimension of $V$, i.e. ...
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24 views

How do I generate some sample solutions for an underdetermined system?

I have a system of 379 linear equations and 6325 unknowns. Does anyone know of a tool that can generate some (non-negative) solutions that satisfy this system? I know there are infinitely many, but it ...
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35 views

Combining ±% with ±dB in measurement uncertainty

Firstly apologies if this is not the correct place to post this but wasn't sure which site would be good to ask regarding about measurement uncertainty calculation. I am trying to calculate the ...
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55 views

derivative of a matrix inverse

I wonder how to differentiate with respect to the diagonal matrix $X_d$, the following matrix : $$ X_d^T (\Sigma_d + X_d C X_d)^{-1} X_d $$ where $X_d$ and $\Sigma_d$ are diagonal matrices with ...
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51 views

How do you evaluate $a^b$ where b is irrational using only basic operators.

How would you evaluate $a^b$ where b is irrational and you can only use +,-, multiplication, division, and rational powers. For example $2^\pi.$ We know $2^2$ = $2\times2$ etc... but when the power ...
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63 views

Orthogonal Matrices and similarity

Two questions: Is it necessary true that, Every two Invertible matrices with the same dimension are similar to each other. Every two row equivalence matrices with the same dimension are similar to ...
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35 views

Characterizing direct sums

Let $U,V$ be vector spaces. Let $T: U \to V$ be a linear map. The codimension of $T$ is defined to be $\mathrm{dim}(V) - \mathrm{dim}(\mathrm{im}(T))$. My questions are: (1) given the subspace ...
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15 views

Finding normal components of a vector

If \begin{align} \notag A_{1}=c_{1}y_{1}^{2}y_{2}^{2}u(u+2t)(u+y_{2}^{2})\frac{\partial}{\partial x_{2}} \end{align} and \begin{align} \notag A=-c_{11}\frac{\partial}{\partial ...
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34 views

Morphisms of algebras

Let's put : $ j = e^{i \dfrac{2 \pi}{3}} $, $ J = \begin{pmatrix} 1 & 0 & 0 \\ 0 & j & 0 \\ 0 & 0 & j^2 \end{pmatrix} $ and : $ \mathrm{Circ}_3 ( \mathbb{C} ) $ is the set of ...
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68 views

rank-one approximation for minimization of Frobenius norm out of kronecker products of unknown matrices

Let $A\in M_{pq}$ be a given matrix. We want to find matrices $X\in M_p$ and $Y\in M_q$ whose Kronecker product approximates $A$ as well as possible in the Frobenius norm; that is, we want to find ...
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35 views

Changing of basis from conjugate diagonal to rotation matrix

Given matrices, $$D = \begin{pmatrix} z & 0 \\ 0 & \bar{z} \end{pmatrix}, \hspace{12pt} B = \begin{pmatrix} 1 & -i \\ 1 & i \end{pmatrix}, \hspace{12pt} U = \begin{pmatrix} Re(z)& ...
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208 views

composition of rotation matrices

We wish to construct a general rotation $\mathbf{R}$ of a coordinate system by composing three elementary rotations $\mathbf{R}_1, \mathbf{R}_2, \mathbf{R}_3$, so that a vector $\mathbf{v}$ is rotated ...
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67 views

Numerical algorithm to solve quadratic eigenvalue problem.

Given the equation $$-4 \left(a^2+a (n-1) (2 t^2-1)\right) \left(\sum _{i=0}^n \alpha_i t^{2 i}\right)^2 \\ +\frac12 \left(\sum _{i=0}^n \alpha_i t^{2 i}\right) \left(t \left(8 a (t^2-1)+1\right) \sum ...
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649 views

What does a dot in a circle mean?

I'm looking at some formulas involving matrices (in the context of machine learning, but I'm not sure it's relevant) and I came across $\odot$. What could this mean? The context is $M \odot N$, where ...
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36 views

Is there any significance to this matrix/operator?

I am working on a problem involving the the polarized Hessian covariant in Cartesian coordinates on $\mathbb{R}^2$ $[a,b] = \frac{1}{2} \frac{\partial ^2 a}{\partial x ^2} \frac{\partial ^2 ...
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142 views

How to estimate/determine surface normals and tangent planes at points of a depth image (point cloud)?

I have depth image, that I've generated using 3D CAD data. This depth image can also be taken from a depth imaging sensor such as Kinect or a stereo camera. So basically it is depth map of points ...
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32 views

Determine a transformation matrix

Determine a transformation matrix that projects on the line $$4x-2y=6$$ My attempt: Now the line can be rewritten: $$y=2x-3$$ So my book suggests displacing the line through the origin with this ...
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56 views

Finding the adjacency matrix for any given quiver and some collection of words.

For a directed graph (quiver) $Q$ with $n$ vertices and without multiple arrows, we have the adjacency matrix $A$, in which $A(i,j)=1$, if there is an arrow from $i$ to $j$, and $0$ elsewhere. This ...
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75 views

Frobenius action on $\overline{\mathbb Q_p}$

Let $p$ be a prime number and let $F_p$ be the Frobenius automorphism of $\overline{\mathbb F_p}$. Given an explicit element $x $ of $\overline{\mathbb Q_p}$, how do I compute $F_p(x)$? Does it even ...
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112 views

Meaning of $A^A$

Let $A$ be a matrix. I know that there is definition for $A^k$ power of $A$, and $e^A$ expontential of $A$. Is there any meaning of $A^A$?
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42 views

Entries of tensor product

If I have a pure tensor $v_1\otimes\dots\otimes v_n$ in an $n$-fold tensor product $V^{\otimes n}=V\otimes\dots\otimes V$ what is the canonical name for the vector $v_j$? Do I call it the $j$-th entry ...
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40 views

spectrum of a special class of tridiagonal matrices

Consider a real and symmetric tridiagonal matrix with zero diagonals and where subdiagonals and superdiagonals are equal to 1 except the (1,2)-th component being equal to $a$, i.e., $\begin{bmatrix}0 ...
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32 views

Can Anyone Check my Proof that all linear maps $T:U_1\oplus U_2\longrightarrow V_1\oplus V_2$ are of the form below?

Suppose $U_1, U_2$, $V_1$ and $V_2$ are vector spaces such that $$\textrm{dim}(U_1)=\textrm{dim}(V_1)=n-k\quad \textrm{and}\quad \textrm{dim}(U_2)=\textrm{dim}(V_2)=k$$ and ...
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76 views

Kalman Filter Predict Update of LDL Decomposition of a Covariance Matrix

From the state predict equation: http://en.wikipedia.org/wiki/Kalman_filter# $$P_{n+1}=AP_nA^T + Q$$ Suppose the $LDL^{T}$ ( http://en.wikipedia.org/wiki/Cholesky_decomposition#LDL_decomposition_2 ) ...
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Algorithm for finding only the $k$-th singular vectors

I know that we have truncated SVD that can compute the first say $k$ largest singular values (and corresponding singular vectors). However, I'd like to know if there is an algorithm that can find only ...
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426 views

What is the operation inverse to vectorization (vec operator)?

There is a well knows vectorization operation in matrix analysis $\mbox{vec}$: https://en.wikipedia.org/wiki/Vectorization_%28mathematics%29 I've vectorized my matrix equations, did some ...
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23 views

Is there any relation between positive definite operator and an operator that satisfies maximum principle?

Suppose $L$ is a self adjoint differential operator which satisfies maximum principle. Maximum principle: Assume that $u(x)$ satisfies $u(0)\geq 0$ and $u(1)\geq 0$. Now $L$ is said to satisfy ...
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44 views

Scalable QR decomposition algorithm

Suppose one has a processor for QR decomposition of complex matrix of size 4 x 4. So if it is necessary to decompose M x M complex matrix, A, one can represent it as R x R block matrix [Cij] (block ...
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37 views

Minimize the number of nonzero elements of a matrix through elementary row operations?

Is there a general method to minimize the number of nonzero elements of a real rectangular matrix through elementary row operations? I am looking for something analogous to Gaussian elimination, that ...
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46 views

Need help in deriving mathematical formula

I need a mathematical formula that would give me the specified result for given input x (x is always an integer) ...
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What does the phrase “uncoupled across coordinate directions” mean in this text?

The following paragraph is from a paper about comparison of maneuvering target tracking models.In the paragraph it talks about constant acceleration models. The above models are simple but crude. ...
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Hilbert space without the projection theorem

One succinct statement of the projection theorem in Hilbert space is $A+A^\bot=\scr H$, where $A\in\scr C$, the set of closed subspaces of $\scr H$. (We will also denote the set of all subspaces by ...
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Show that there is a T-invariant subspace W ⊂ R^4 with 0 < dim ( W ) < 4.

Here is the full problem: Let $T : R^4 → R^4$ denote a linear transformation with $f_T( x ) = (( x + 1)^2 + 5)^2$ . Show that there is a T-invariant subspace $W ⊂ R^4$ with $0 < dim ( W ) < ...
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160 views

Matrix for orthogonal projection with respect to ordered and canonical bases

Orthogonal projection onto the line $y = 2x$ gives a linear transformation $T: R2 → R2$ such that $$T(1,2) = (1,2)$$ and $$T(−2,1) = (0,0)$$ Then the matrix of T with respect to the ordered basis ...
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107 views

inverse of Vandermonde's Matrix without using determinants

I want to show, that the Vandermonde's Matrix ...
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37 views

$A$ matrix is diagonalisable if $\exists S : S^{-1}AS $ is a diagonal matrix, how can I find S?

Per definition a matrix $A$ is diagonalisable if there exists a matrix S such that $S^{-1}AS$ is a diagonal matrix. My question is how do I find the matrix $S$? Is it always the combination of the ...
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100 views

About Jordan-Chevalley decomposition

I have this problem: Let $K$ be a field. Let $J\in M_n(K)$ a Jordan matrix. Prove that there exists a diagonal matrix $D$ and a nilpotent matrix $N$ such that $J=D+N$ and $DN=ND$. I saw that this ...
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Which is the starting and ending basis? - Matrix of a linear transformation [Lay P294 Q 5.4.28]

Denote some arbitrary linear transformation as $L.$ When a question asks "to find a matrix of $L$ with respect to S and T", does this denote $[L]_{T \leftarrow S}$ or $[L]_{S \leftarrow T}$ ? How can ...
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44 views

Do signs matter in SVD?

I have written an algorithm to compute the SVD of a 2x2 matrix. I was checking against a Mathematica query, and I noticed that the signs in the $U$ and $V$ matrices do not match those from my ...