Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Proof of Singular Value Decomposition

I found an interesting property of SVD in the book of Introduction to Information Retrieval by Christopher D. Manning, Prabhakar Raghavan and Hinrich Schütze, page 408. The question is can I use the ...
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31 views

How to find a transformation matrix T?

(1.) Suppose there is matrix C of dimension "p by n" where p is less then n i.e. p I want to know is there any particular way exist to find transformation matrix T of dimension "n by n" such that ...
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24 views

Transformation matrices and hermitian/unitary/normal/… matrices

I need some help with the following - have I done the correct things or how can I solve the task? Let $f \in End(V)$, V a unitary space $\mathbb{C}^3$ given by: $A_{\alpha \beta} (f) = \frac{1}{7} ...
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27 views

Solving a quadratic matrix equation with non-squared matrix.

I would to solve $X=T^TT$ ,I assume that the matrices are real, we know $ X $ and we seek a solution in $ T $. Case 1: $\operatorname{rank}(X)>n $ There are no solutions in $ T $. Case 2: $ ...
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29 views

Solving a quadratic matrix equation with non-squared matrix

I was trying to solve the problem of finding the value of a non-squared matrix $n \times m$ which solves: $T^T T = X$ where $ X $ is a symmetric and positive semidefinite $ m \times m $ matrix, ...
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Trying to use Cholesky decomposition of covariance matrix to sample error ellipsoid

I'm trying to construct an error ellipsoid from a covariance matrix (which exists for a 3D point) and then sample consistent xyz points in this region. In a previous question when I asked about this ...
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24 views

is this a valid counter-example - function is not locally invertible

Let $S_n$ be the set of all symmetric matrices with real entries of size $n$x$n$. We are asked if the function $f:S_n \to S_n$, $f(A)=A^2$ is locally invertible for every $A$ (Using the Inverse ...
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32 views

Subset Sum represented as a perfect number

Can we form a set of $29$ distinct integer elements such that every subset of elements possible has a sum which is a perfect power? A perfect power is a positive integer that can be represented a p^q ...
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16 views

Solving a linear combination

I have the follow vectors: V = (a, b, c); S=(d, e, f) and Q=(g, x, i). I have to solve a problem with particular values, but I need some guidance in doing so. In particular for finding x. My ...
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17 views

Let $G_1 = \{v_1 + \lambda w_1 | \lambda \in \mathbb{R}\}, G_2 = \{\{v_2 + \mu w_2 \}$ be two skew lines, derive a formula for $d(G_1,G_2)$

Let $G_1 = \{v_1 + \lambda w_1 | \lambda \in \mathbb{R}\} \subseteq \mathbb{R}^n, G_2 = \{\{v_2 + \mu w_2 |\mu \in \mathbb{R}\}\subseteq \mathbb{R}^n$, with $v_1, v_2, w_1, w_2 \in \mathbb{R}$ be two ...
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18 views

Orthogonal polynomials induction proof

I tried writing this all out but cannot seem to get anything sensible. Basically I want to prove that assuming w(x) is the weight function of a Gram Schmidt orthogonalization process and w is an ...
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51 views

Finding criteria for a household financial budget falsification

I’m working on a financial problem about budget of households. Households in a state fill a form about their net budget in every year and our insurance company investigate their financial status and ...
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32 views

The eigenvector does not minimize the error

Consider the cost function J: $J=|P_1-\beta P_2|^2+\lambda(\pmb{q}^H\pmb{q}-E)$ where $P_1, P_2$ and $\beta$ are complex scalars, $\lambda$ is the Lagrange multiplier and E is the constraint applied ...
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30 views

Approximation of optimum for two linear programs

Suppose you got two linear programs. They are the same except that one has a shifted objective by a positive constant (1) $$\min c^Tx$$ (2) $$\min c^Tx + d$$ For (2) there exists a ...
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43 views

Let $T1$, $T2$ be two linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^n$

Let $T_1$, $T_2$ be two linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^n$. Let $\{x_1, x_2,...,x_n\}$ be a basis of $\mathbb{R^n}$. Suppose that $T_1(x_i)$ not equal to $0$ for every $i = ...
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27 views

How can I find the coefficients in a orthonormal linear combination with the most accuracy?

So I have a set of functions in spherical coordinates $f_k(r,\theta,\phi)$ and $g(r,\theta,\phi)$. Both functions sets are real and defined in the unit ball and I want to write the function ...
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34 views

Vector spaces question difficulty

I know how to do ALLL the question parts up to (iv) I just don't know how to show the last part (v) . Please help me.
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Name search for special Linear Mixed Integer Programm

I am looking for a name for the following question in literature! (and if you know it, then it would be great) I couldn't find it and due to wide audience here, presumably you know more. Thank you ...
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11 views

Verification of step in a proof of the decomposition of primary f.g torsion modules over PIDs?

I was reading about the decomposition of finitely generated primary torsion modules over PIDs, and though of an alternative way to do the "inductive" step. Since it is substantially simpler than the ...
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113 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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32 views

Proof Verification of a statement regarding dimension of the sum of subspaces

This is the statement I am trying to prove. Suppose $V$ is finite dimensional. Prove that if $U_1,\ldots,U_m$ are subspaces of $V$ such that $V = U_1 \oplus \cdots \oplus U_M$ then ...
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89 views

Maximize the maximum Eigenvalue under a diagonally constrained matrix

Suppose we have $N\times N$ Hermitian matrix $\mathbf{A}$ I want to find the real $N\times N$ diagonal matrix $\mathbf{D}$ that maximizes the sum of the maximum Eigenvalues : $\mathbf{D}=\arg\max ...
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21 views

Let $ W_1 $ and $ W_2 $ be subspaces of $ V^* $. Prove that $ W_1 = W_2 $ iff $ Ann(W_1) = Ann(W_2) $

Let $ S $ be any subset of $ V^* $ for some finite dimensional space $ V $. Define $ Ann(S) = \lbrace v \in V ~|~ f(v) = 0 $ for all $ f \in S \rbrace $. Let $ W_1 $ and $ W_2 $ be subspaces of $ V^* ...
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28 views

Principle of superposition for linear systems

I have always learned about the superposition principle for systems of the form $$\dot x = A x + B u,$$ $$ y = Cx.$$ where $A$, $B$, and $C$ are time invariant matrices of appropriate dimensions. ...
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10 views

Does one condition unambigously set a linear form?

Let $x$ be a non-zero vector in linear space $V$ and $f: \ V \rightarrow K$, where $K$ is a field of scalars. Does condition $f(x)=1$ unambiguously set a linear form $f$?
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29 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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23 views

Matrix similarity & Characteristic polynomial (linear algebra)

I need to practice this kind of material, so as a self practice, I though about this: A1 and A2 are both nxn matrices. A1 is Invertible matrix. A1A2 & A2A1 are necessarily similar? and how about ...
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36 views

Prove solution does not exist for inequalities system

I have an inequalities sytem like the following: Example > x+y+z <= A > x+y <= B > x+z > C > y+z > D > x >= E Let A,B,C,D,E be any ...
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43 views

Convergence of Matrix Power Series

If $A$ is a square matrix with complex entries, then $\| A\|$ is defined as the sum of the absolute values of the entries of $A$. I have shown that this matrix norm is homogeneous, subadditive, and ...
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25 views

What different block-diagonal forms can a matrix have?

A square matrix over $\mathbb C$ is always similar to at least one block-diagonal matrix: its Jordan normal form. Is that the only one? If not, is there any kind of complete classification of all the ...
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38 views

Looking for ranks.

Let $X$ belong to $\operatorname{Mat}_n(R)$, where X is inrevertible and let its column vectors be $X_1,X_2, \dots X_n$. Let $Y$ be a matrix that has the column vectors $X_2, X_3, \dots , X_n$. Let ...
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$K=span\{x_i,x_2,\dots ,x_k\}$, $X$ is orthonormal, then $\{y_{k+1},y_{k+2},\dots y_n\}$ is a basis for $K^{\perp}$

This is a problem from final exam in my university. I think, the problem was quite hard. I can't answer the question. Let $V$ be inner product space over $\mathbb{C}$, $\dim V\geq 2$ and $X$ be a ...
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36 views

Solving a set of equations

I have a set of n equations of the form of $Z_i((a-x_i)^2 + (b-y_i)^2 + c^2)$, i varies from 1 to n and $Z_i$, $x_i$, $y_i$ are knowns and $a,b,c$ are unknown. They are all equal to each other ...
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47 views

Transformation of quadratic formula with fractions

I'm trying to implement a nonlinear least squares based trilateration algorithm into a code project. The algorithm I'm using I found in a paper by Yu Zhou (An Efficient Least-Squares Trilateration ...
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26 views

determinant and trace of a huge positive definite matrix

I have a problem to compute the determinant and the trace of inverse matrix: $det(\Gamma^{-1}+I_n⊗\Phi^T\Phi)$ and $tr[(\Gamma^{-1}+I_n⊗\Phi^T\Phi)^{-1}]$ where $\Gamma$ is a huge positive definite ...
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36 views

Intuition/Picture - Matrix Multiplication - Product of [Row or Column Vector] and Matrix [Lay P95]

This question is not a duplicate of the original, in which user Shuchang proved the question. Presently I'm asking about further intuition or a picture, and no proofs please. $1.$ Intuitively, in ...
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33 views

Finding the constant of a function in terms of the gradient of a tangent.

Let $f : \Bbb R \to \Bbb R, f (x) = e^x+ k$, where $k$ is a real number. The tangent to the graph of $f$ at the point where $x = a$ passes through the point $(0, 0)$. Find the value of $k$ in terms of ...
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16 views

Show that there exists f-stable subspace for certain conditions

Let K be field. Assume that the characteristic of $K$ is not equal to $2$. Let $f:V \rightarrow V$ be a linear operator such that $f(f(v))=v$ for any $v \in V$. Show that there exist f-stable ...
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49 views

Frobenius normal form of nonnegative matrix

As I have understood the Frobenius normal form of a reducible matrix A is given as follows: $P^T A P$. Here my question is, if we have a non negative matrix can we say something about the diagonal ...
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97 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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28 views

Elementwise operations vs. matrix functions

Is there any notable connection in general case between elementwise matrix operations (such as matrix addition, scalar multiplication, Hadamard-product), and matrix functions (such as power of ...
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36 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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32 views

Finding a generating function to solve a linear equation

I need to find a generating function to solve a linear equation. This is the linear equation: $$X_1+X_2+X_3+X_4+X_5 = 3n+1$$ All numbers are natural numbers which can be divided by 3 with no remainder ...
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22 views

Matrix rank and linear independence

$\mathbf{a}_i,\mathbf{b}_j$ are $n$ dimensional vectors. Consider the matrix $\mathbf{M}$ defined by: $$\mathbf{M}_{ij}=\mathbf{a}_i\cdot\mathbf{b}_j$$ Prove/disprove that ...
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36 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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20 views

Largest average principal submatrix of a symmetric matrix.

I am wondering if there exists literature on the following problem: Let $X$ be an $n \times n$ symmetric matrix. How do you identify the $k \times k$ principal submatrix of $X$ with the largest ...
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11 views

Bounding column totals of right triangular matrix

Given a nxn right triangular matrix containing nonnegative entries (these can be integers as well for simplicity), and some predetermined row sum values, I would like to find a way to place bounds on ...
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68 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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18 views

Maximal angle between kernel of rows of a matrix

Consider a matrix with 2 columns $$ \begin{pmatrix} a_1 & b_1 \\ a_2 & b_2 \\ a_3 & b_3 \\ \vdots & \vdots \end{pmatrix} . $$ To each row $(a_i \;\;\; b_i)$, one draws the kernel ...
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30 views

Tensor Product over a field

This question appeared in my exam and I could not solve it. Let $L$ be a field, $K$ subfield of $L$. Assume that dim$_K(L)$=$m$, and $V$ be a $L-$vector space amd dim$_L(V)=n$. If as usual ...