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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Orthogonal Latin squares - Origin of the word “Orthogonal”

Is there any linear-algebraic link with the use of the word "orthogonal" in orthogonal latin squares? I thought about it a little bit and the closest I got to linear algebra was this definition : if $...
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126 views

Which subject deals with questions like “Distance from hyperplane to a point”

What subject covers topics like the following for N dimensions? Given equation of plane like Ax + By + Cz + D = 0, the normal vector N is given by $[A,B,C]^{T}$? What is the distance between a point ...
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422 views

Comparing two probability distributions

In my research I have to find two discrete probability distributions by solving two separate linear programs. The domain of optimization is the probability space of $m^n$ atomic events, where $n$ is ...
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112 views

Lower bound on Gram determinant

Is there a lower bound for the determinant of the Gram matrix when the elements are linearly independent? I am talking about Hilbert space setting not just vectors. Thanks.
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84 views

Inverses of the sums of all possible subsets of a set of symmetric and positive definite matrices

I have a set of $c$ matrices $A_1 ... A_c$ which are all symmetric and positive definite. I would like to calculate the inverses of all the possible sums, i.e. $(A_1+A_2)^{-1},(A_1+A_3)^{-1},(A_2+A_3)...
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The dimension of centralizer $\gamma=\{B\in M_n(\mathbb{R}):AB=BA\}$

Let $A$ be a $6\times 6$ matrix with charpoly $x(x+1)^2(x-1)^3$. We need to find the dimension of $$\gamma=\{B\in M_n(\mathbb{R}):AB=BA\}.$$ What is the relation of charpoly of $A$ with dimension of ...
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60 views

Proving an optimization problem has a rational optimum.

Consider the function $$ J_\gamma(X) = \det\left( I - \tfrac{1}{\gamma^2} (A+BXC)^\mathsf{T}(A+BXC)\right) $$ where $A$, $B$, $C$, $X$ are matrices of real numbers. Further suppose that $B^\mathsf{T}B$...
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eigen problem for direct scattering method

Consider the KdV equation $$u_{t}+6uu_{x}+u_{xxx}=0$$ with initial condition $$u(x,0)= \begin{cases} 1 &\text{if } x \in [-1,0] ,\\ 0 &\text {if } x \in (-\infty,-1)\cup(0,\infty),\end{cases}$...
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221 views

Attempting to find a specific similarity (equivalence) matrix

Apologies if this has already been asked - I searched but couldn't find. My question is regarding matrix similarity (equivalence): I have a square matrix $A$ that is of permutation form and (square) $...
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88 views

Intelligent choice of basis

Let $V$ be the vector space of all polynomials of degrees $\le 3$. Let $U$ be the subspace of polynomials of the form $a_3 x^3 + a_2 x^2$. Say you want to compute an orthogonal projection of some ...
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57 views

A proof in vectors

If it is given that: $$ \vec{R} + \dfrac{\vec{R}\cdot(\vec{B}\times(\vec{B}\times\vec{A}))} {|\vec{A} \times \vec{B} |^2}\vec{A} + \dfrac{\vec{R}\cdot(\vec{A}\times(\vec{A}\times\vec{B}))} {|\vec{A} \...
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70 views

Intuiting Product of Elimination Matrices (and NOT by Matrix Multiplication)

I want to intuit, and NOT compute with matrix multiplication, $M:=\color{green}{E_{P_3 \rightarrow P_4}}\color{#CA790F}{E_{P_2 \rightarrow P_3}}E_{P_1 \rightarrow P_2},$ where: $E_{P_1 \rightarrow ...
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1answer
64 views

Find a function $u(x,y)$ such that a line integral $I=u(B) - u(A)$ where B and A are limits of the integral

As the title, where function $u(x,y)$ can satisfy $I=u(B)-u(A)$ the line integral $I$ is already shown to be path independent and is defined as $I=\int_A^B(1+e^\frac{x}{y})dx+e^\frac{x}{y}(1-\frac{...
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Least Squares “analytic expression” for fitting a 2D quadratic function to measurements

I have n scattered elevation measurements: $ \{x_i,y_i,z_i\}_{i=1..n} $ that I want to fit a quadratic function to: $ z = ax^2 + by^2 + cxy + dx + ey + f$. The problem can be written as a vector ...
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152 views

relation between minimal polynomial and jordan normal form

I just solved some exercises on minimal polynomials and i remember that there is a relation between the minimal polynomial and the jordan normal form. But my question is the following : knowing the ...
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81 views

What is the significance of the matrix in the LAPACK logo?

This is the LAPACK linear algebra library logo: What is the significance of this matrix?
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132 views

Changing a simplex grid to an orthogonal grid.

Well I'm on my way in learning noise, a computer algorithm that's used to create real life structures and textures, etc. The noise I'm trying to learn is Simplex Noise, I already have it in the ...
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1answer
64 views

Inverse of a matrix defined on C

Let $ A $ be a real symmetric matrix and form the matrix \begin{equation} R(z)=(zI - A)^{-1} \end{equation} for complex values of z, whenever it is defined. Prove: The elements of R(z) are quotients ...
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1answer
127 views

Does a “typical” unitary matrix have an entry of magnitude 1?

I guess that a "typical" unitary matrix (or "almost every" unitary matrix) in $d \geq 2$ dimensions does not have an entry with magnitude 1. I would like to make this statement more precise and see a ...
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64 views

Crossing Orbits

I have a question here that I am stumped with. The path of an orbit of a planet around a distant sun is $2K^2 + 2I^2 = 50$. The planet orbits the sun at roughly $900$ million kilometers. The path of ...
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132 views

Transpose of Operator

Say $T$ is the operator defined on class of smooth functions supported on some subset $U$ in $\mathbb{R}^{n}$ where $Tu = -i \sum_{j=1}^{n}{\frac{\partial_{j}\phi}{|\nabla\phi|^2}\partial_{j}u}$ for ...
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Problem reduced to analyzing solutions of a family of nonlinear systems of equations

This was posted on mathoverflow about two weeks ago and I got no response so I'm asking here in case anyone has any ideas. Original post is here. I was able to reduce a research problem relating to ...
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1answer
66 views

Solve for symmetric matrix

I have the following equation: $$ \left( \begin{array}{c} \mathbf{I}\\ \mathbf{K} \end{array} \right)\mathbf{x} = \mathbf{y} $$ where $\mathbf{K}$ is a symmetric $2\times 2$ matrix, $\mathbf{I}$ is ...
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The possible set of eigenvalues of a $4\times 4$ skew symmetric, orthogonal matrix

The possible set of eigenvalues of a $4\times 4$ Real skew symmetric, orthogonal matrix is $1.\{\pm i\}$ $2.\{\pm i,\pm 1\}$ $3.\{\pm 1\}$ $4.\{\pm i,0\}$ As it is real skew ...
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1answer
450 views

How can I calculate the correct rotation output when input needs modification?

I am making a GPS for vehicles in a game and I need them to turn accordingly to the rotation between the ending point and starting point. I have calculated the rotation, and before I can apply the ...
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42 views

Projection between subspaces of $\mathbb{R}^d$

Suppose I have two $k$ dimensional subspaces of $\mathbb{R}^d$, which I call $A$ and $B$. Let $\text{proj}_S (x)$ denote the projection of an $x \in \mathbb{R}^d$ onto subspace $S$. Now is given that ...
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1answer
35 views

Give a bound of norm in terms of norm

Supppose $A$ is a nonsingular $n\times n$ matrix and $x,y,b,c\in \mathbb{R}^n$ satisfy $$Ax=b$$ $$A(x+y)=b+c.$$ Give a bound on $\|y\|/\|x\|$ in terms of $\|c\|/\|b\|$. Can someone give me ideas?
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1answer
221 views

FFT of a matrix and its square.

I am doing something computationally intensive that requires that I compute the fast fourier transform of a matrix, let's say $A$, and also compute the FFT of its square, $A^2$. I am wondering if ...
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Second Linear Algebra Text for Budding Mathematician [duplicate]

I have already worked through Gilbert Strang's Introduction to Linear Algebra and am interested in learning more Linear Algebra. What is a good follow-up to Strang for a budding mathematician? EDIT: ...
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1answer
42 views

how to choose positive symmetric matrix?

What are the ways to find a positive symmetric matrix $P$ such that $ A^{T}P+PA=-Q$ where $Q$ is also positive symmetric matrix, $A=\left[\begin{array}{cc} 0 & I_{n}\\ -K_{v} & -K_{p} \end{...
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Spanning Hadamard product powers (Schur products)

Fix two vectors $\mathbf u$ and $\mathbf v$ in $\mathbb R^k$, and let $\circ$ denote the coordinate-wise Hadamard / Schur product, i.e. $\mathbf u\circ\mathbf v$ has coordinates $w_i=u_iv_i$. Write $\...
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Computing distances between hyperspheres and sides of a hypercube?

Suppose you are given the $n$ dimensional hypersphere: $$\left(x_1 - \frac{1}{2}\right)^2 + \left(x_2 - \frac{1}{2}\right)^2 +\ldots+ \left(x_n - \frac{1}{2}\right)^2 = \frac{n}{4}$$ And the ...
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1answer
289 views

Getting a 3d linear equation knowing the rotation of an object

I have an object, a simple rectangle I rotate it by a certain degree using Euler Angles, in this case around Z, to make it easy lets say it's 45 degrees. Right now I want the yellow: Y-Axis linear ...
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1answer
71 views

Finding a point on a plane.

So, I have a set of 3 points that form a triangle on the surface of a wall, from a depth camera. graph representation of wall The camera is roughly where the green dot is, and the points correspond ...
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Singular value decomposition with positive values?

We know that thin SVD for a matrix $A$ minimizes the squared error between $U \Sigma V'$ and $A$ where $\Sigma$ is a diagonal matrix of some of the singular values and $U$ and $V$ are unitary matrices....
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86 views

Prove that $Ax = b$ with complex solution, actually is rational solution [duplicate]

Let $Ax = b$, a system of linear equations, $A$ is matrix of rational numbers, and this system has only one complex solution. Prove this solution is also rational. Can someone help me please? I ...
2
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2answers
89 views

Linear algebra in Hilbert space

Let $M,N$ be closed subspaces of a separable Hilbert space. How to prove rigorously the following: $\operatorname{dim} M >\operatorname{dim} N => \exists u\neq0 \in M, u\in N^\perp$ ...
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1answer
61 views

Computation of Eigenvalues

I am studying a linear algebra course and there's a problem with the calculation of the eigenvalues of a matrix. It's probably due to my own error, due to a wrong method or unsound algebraic concepts. ...
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137 views

Root-subspaces are $\mathfrak{g}$-invariant

I need some help with technicalities concernig root-subspaces of nilpotent Lie algebras of operators. Let $\mathfrak{g} \subseteq \mathfrak{gl}(V)$ be nilpotent Lie algebra, $\alpha \ \colon \ \...
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37 views

The dimension of birkoff polytope

Let $P_m$ be a subset for R^mxm be the polytope given by: $x_i,_j \ge 0$ $x_i,_1 + ... + x_i,_m \le 1$ $x_1,_j + ... + x_m,_j \le 1$ $\sum_{1 \le i,j \le m } \ x_i,_j \ge m-1$ Contruct a ...
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1answer
200 views

Characteristic polynomial of self-adjoint map

A is self-adjoint map on $R^3$. I need to determine if $$-\lambda^3 + 3\lambda^2 - 2$$ $$-\lambda^3 - 3\lambda - 2$$ may be its characteristic polynomials? The only relevant fact I know about such a ...
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Quadratic Integer Programming

Would anyone mind helping me solve this problem $$ \min\space f(x) = \frac12 x^\mathrm TQx + bx + c \qquad \text{s.t. } \sum_i x_i=\lambda $$ where $x$ is a vector whose entries are positive ...
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64 views

Basis of $(\mathbb{R}^2)^*$

Let $dx,dy$ be the standard basis of the dual space $(\mathbb{R}^2)^*$. Let $T(x,y)=(T_1(x,y),T_2(x,y))$ be a diffeomorphism. Is $(dT_1,dT_2)$ in turn a basis? Why? How do we express differentials ...
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129 views

Invariant polynomials over symmetric matrices under Euclidean transformations

It is a simple question, but I haven't still had a course on this topic and I'm finding it hard to understand some basics. Consider a $2\times2$ symmetric matrix over a field (for example $\mathbb{C}$...
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83 views

Checking if a binary vector lies in the affine span of given binary vectors

Let $x_1, \ldots,x_N \in \{0,1\}^D$ be $N$ binary vectors, assumed affinely independent (in the field of reals). Is there an efficient algorithm for determining whether a new binary vector $x_{N+1}$ ...
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1answer
111 views

A Unique Invariant subspace for a set of matrices

Im wondering if anyone can give me a good reference or answer this question which may have already be solved. For a set of generic $n\times n$ matrices $A_1,A_2,...,A_k$, such that they share only ...
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328 views

Proof of the spectral theorem for normal operators from two lemmas

I have the following lemmas that I can prove: Let $T$ be a linear operator on a Hermitian space $V$ and let $W$ be a $T$-invariant subspace of $V$ . Then $W^⊥$ is $T^*$-invariant Let $T$ be a normal ...
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152 views

Ideals (one-sided ideals) of $n×n$ upper triangular matrices

Is there any characterization of ideals (one-sided ideals) of $n\times n$ upper triangular matrices? I have just seen in monthly journal about $2 \times 2$ matrices in the below article Left and Right ...
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308 views

This matrix is an attractor?

I'm trying to find for which values of $\gamma$ the matrix A is an attractor: $$ A=\begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ -1 & 0 & \gamma \\...
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36 views

the det of identity replacing column

Let $A\in{\mathbb{R}}^{n\times n}$, $b\in\mathbb{R}^{n}$, and $x\in\mathbb{R}^{n}$ be given, where $A$ is nonsingular and $Ax=b$ holds. Let $X_{j}$ be the matrix obtained from the $n\times n$ identity ...