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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109 views

Is Grassmann-Plucker relation implied by 3-term Grassmann-Plucker relation?

It's a problem from book "Oriented Matroid", problem 3.16. More exactly: For $n$ elements $i_1,\dots,i_n$ and an anti-symmetric function $\det$ on${\{i_1,\dots,i_n\}}^r$. we have: ...
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50 views

Points of discontinuity

What are the number of points of discontinuity of the function: $[|x^2-4|]$ [] is greatest integer. || is mod function. Also we have to find in range $(-4,4)$. I am getting $29$ but answer says ...
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73 views

Definition of a dual Space

My question is: When the book says that a linear functional is a map from $V$ to $F$, is $F$ restricted to only the set of two elements 0 and 1? If not, why can't $v_1$ map to say 8, in the first ...
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25 views

Finding the solution of simple equation

I want to find the possible positive solutions of the following equation group: $$\dfrac{a_{11}}{a_1}=\dfrac{a_{10}}{a_0}$$ $$\dfrac{a_{00}}{a_0}=\dfrac{a_{01}}{a_1}$$ $$a_0+a_1=1$$ ...
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65 views

analytic solution to structured algebraic Riccati equation

In solving a model I have written down for a research paper, I am left with two Algebraic Riccati Equations, that is I need to solve for $X$ in the equation \begin{align*} X = A^\top (X + XB(R + ...
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84 views

Prove image of basis is basis of vector space

With V and W being vector spaces, and T: V -> W being a linear transformation: c) Suppose B: (v_1, v_2, ..., v_n) is a basis for V and T is one-to-one and onto. Prove that T(B) = {T(v_1), T(v_2), ..., ...
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27 views

Mathematical standard term for a function of (different) operator arguments

In quantum mechanics, one often considers functions of linear operators, like $$f(A,B) = A\cdot B + e^A \cdot B^2$$ where $A,B$ are linear operators. In physics this often causes confusions, as some ...
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36 views

Finding projections

I'm not completely sure about this kind of question: $$\begin{align}V&=\operatorname{span}\{(2,2,2,1),(1,0,1,1),(1,4,1,-1)\}\\ ...
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35 views

Dimension Theorem modification

The Dimension Theorem says $$ \dim(U+W) = \dim(U) + \dim(W) - \dim(U \cap W) $$ The proof of this theorem uses the bases of $U$, $W$, and $U\cap W$. Is it possible to prove this theorem with just ...
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41 views

Differentiate a vector product chain rule

I have a vector V=[v0,v1...vn]' which is a function of another Vector W=[w0,w1..wn]' Here ' is transpose. The scalar field is f=V'(W) V(W) where V and W are of the same dimension (n+1), n=0,1...n ...
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83 views

Will the projection of a singular matrix into an orthonormal space be non-singular?

I'm working through an implementation of the solution from 16.3.1 Dealing with the nullspace in the case of a singular within-class scatter matrix when performing discriminant analysis. In this ...
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230 views

How to determine yaw-pitch-roll orientation by specifying a plane via 3 points?

[Note, this question is an attempt at rephrasing the one posted here, as it has not garnered any attention, unfortunately] Hello, Let's say you have three points in 3D space: A, B and C. Together, ...
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32 views

The group $O(n)$ is contained in a sphere of radius $n$.

So I have this exercise where I have to show that the group $O(n)$ is contained in a sphere of radius $n$ and centered on the origin, but I keep getting the wrong answer. Previously in the exercise I ...
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45 views

How to find all the eigenvalues of a positive operator whose eigenvectors are positive semi-defintie?

A linear operator $T:\mathcal{H}_n\rightarrow \mathcal{H}_n$ is said to be positive if $T(\mathcal{P}_n)\subset\mathcal{P}_n$ where $P_n$ is the set of positive semi-definite matrices. For a positive ...
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73 views

Relations between minors of a matrix.

motivation: I'm looking at the Segre embedding, given by (for this example) $\mathbb{P}(U) \times \mathbb{P}(V) \rightarrow \mathbb{P}(U\otimes V)$, $([u],[v]) \mapsto [u\otimes v]$. This is an ...
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28 views

Checking degenerate and non degenerate

in P_n, w(x,y) = x(1) y(1) How to check whether it is degenerate or non-degenerate? I know how to show it is a bilinear form. If it is non-degenerate, I give out a case in which w(x,y) !=0 . Am not ...
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35 views

Solutions to $DCD^T\bar{v}=\lambda\bar{v}$ in terms of $C^{-1}$ elements

I've got the solutions to this eigenvalue equation: $$DCD^T\bar{v}=\lambda\bar{v}$$ In the problem I'm working at, the elements of $C^{-1}$ make more physical sense than those of $C$. Therefore, I'd ...
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19 views

Example about the main result on the “New Decomposition For Square Matrices” given by Sir Julio Benitez?

can you give a specific example about the main result on the "New Decomposition For Square Matrices" given by Sir Julio Benitez? I really need it. please?
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116 views

Linear Algebra - Understanding how to use the Gram Schmidt Orthogonalization Procedure

Use the Gram Schmidt Orthogonalization Procedure to transform the basis $\{(1,2,1), (1,0,1), (3,1,0)\}$ into an orthogonal basis for $R^4$. I haven't been able to understand how to use the Gram ...
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86 views

How do I find the “radius” of a cuboid along a given axis?

Let's say I have a cuboid, which has the half extents $e_c = (e_{cx}, e_{cy}, e_{cz})$ (each component in that vector is half the length of four edges on a cuboid. This means that, for example, along ...
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195 views

Let A be an m×n matrix and let B be an n×m matrix, so that both AB and BA are defines. Prove that:

If rank(B) = m, then null(BA) = Null (A) I know that this is true I just don't know how to use the m x n matrix and n x m matrix. null (BA) = B(ax) = 0 and null (A) = ax = 0 so then B(0) = 0 ... but ...
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68 views

Is the normal equation valid over $\mathbb{C}^n$?

For reference, the normal equation over $\mathbb{R}^n$ is $$\mathbf{X}^T\mathbf{X}\mathbf{b}=\mathbf{X}^T\mathbf{y}$$ and its solution $\mathbf{b}$ represents the optimum solution of the linear ...
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41 views

Linear Algebra - Finding the basis of a column space

Find a basis for the column space of the matrix $\begin{bmatrix}1&-2&0&0&3\\2&-5&-3&-2&6\\0&5&15&10&0\\2&6&18&8&6\end{bmatrix}$. What is ...
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46 views

What type of formula am I looking for?

Let say you have a list of items with 3 columns, two are statistical the third is just a name. The statistical categories you have are Points, and Salary. You have 10 different options. Each Row ...
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68 views

Any material on complexification?

These days, I met a problem on linear algebra: Suppose $A,B$ are real matrices. If there's a complex unitary matrix $U$ such that $U^*AU=B$, where $U^*=\overline U^\top$, namely, the conjugate ...
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312 views

Linear transformations are Lipschitz and continuous

I'm a little confused about the proofs that linear transformations $f:\mathbb{R}^n \to \mathbb{R}^n$ are a) continuous and b) Lipschitz. I know that Lipschitz implies continuity. However, the only ...
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40 views

Find the projection matrix $P$

This problem projects $b=(b_{1},...,b_{m})$ onto the line through $a=(1,...1)$.The horizontal line $\hat{b}=3$ is closest to $b=(1,2,6)$ Find the projection matrix $P$. So $p$ $=(3,3,3) = P(1,2,6)$, ...
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19 views

Alternative coordinates for the complex plane $\mathrm{Re}[e^{-is}z]=a$, $\mathrm{Re}[e^{-it}z]=b $

I am defining coordintes on $\mathbb{C}$ using a "generalized" real and imaginary part. Here $a,b \in \mathbb{R}$. \begin{eqnarray*} \mathrm{Re}[e^{-is}z]&=&a \\ ...
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58 views

Cross Product - Moments :: Dynamics 2

This problem is related to Cross Product - Moments :: Dynamics Please look at that link for the background on the problem I am faced with right now, I have linked a pdf of the book that I am using ...
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669 views

Find the projection of $b$ onto the column space of $A$

$A= \left[ {\begin{array}{ccccc} 1 & 1 \\ 1 & -1 \\ -2 & 4 \end{array} } \right] $ and $b = \left[ {\begin{array}{cccc} 1 \\ 2 \\ 7 \end{array} } \right] $ I ...
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39 views

Commutativity of two endomorphims

Let $f$ and $g$ be two endomorphisms of a real vector space $E$. I want to show that if $\ker(g)$ is stabilized by $f$ then $f\circ g=g\circ f$. Thank you for any hint.
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188 views

Annihilator in dual vector space

If $y(x) = x_1 +x_2 + x_3 = 0$ whenever $x = (x_1,x_2,x_3)$ is a vector in $\mathbb C^3$, then $y$ is a linear functional on $\mathbb C^3$; find a basis of the subspace consisting of all those vectors ...
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63 views

3D Animation of object flying straight towards a surface

Lets say we have the following the orthogonal(?) 4x4 matrix, which represents a world space transformation in a right-handed coordinate system. ...
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23 views

The arbitrary function that calculates some number$ f(A)$

Theorem 3.8 Let f be an arbitrary function that calculates some number $f(A)$ for any square matrix $A$ of size $N$. Assume that f is multilinear as a function of the rows and that $f(A)$ equal to ...
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33 views

Difficulty with terminology/standard definition for Jordan-normal form.

(Perhaps this is something rather for MOverflow, don't know) I've understood the concept of the Jordan-normal-form such that it is similar to the idea of diagonalization of a matrix, but can be ...
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111 views

Kronecker product and characteristic polynomial

Let A and B are two square matrices, is there any relation between minimal polynomial of each of them and their Kronecker product?
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34 views

In the Schwarz inequality, what happens when $a$ and $b$ lie on the opposite sides of the origin?

The Schwarz inequality is an equality if and only if $b$ is a multiple of $a$. The angle is $\theta = 0$ or $\theta = 180$, so $\cos \theta = 1$ or $-1$. In this case $b$ is identical with its ...
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58 views

Directional derivatives, linear maps, and uniform convergence

The Exercise Let $f(x,y)=x$ if $|y|>x^2$ and $f(x,y)=0$ otherwise. Show that all the directional derivatives of $f$ exist at the origin but there does not exist a linear map $D$ such that ...
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35 views

Convert function form to variate polynomials form

Let $f:{\Bbb Z}\longrightarrow{\Bbb Z}$ be a function, example $f(0)=3,\ f(1)=2,\ f(2)=0,\ f(3)=1$. Now that was "in decimal". We can think of it "in binary" format: $f(00)=11,\ f(01)=10,\ f(10)=00,\ ...
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47 views

How to solve the following equality

Is it possible to solve the following equation analytically for $\beta$: $$y'(A+\beta B)^{-1}y = \alpha,$$ where $A$ and $B$ are both positive-semidefinite and symmetric matrices (essentially, some ...
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47 views

Quotient rule for the Jacobian

Is there an analog to the quotient rule that can be applied to the calculation of the Jacobian? Example: Can the jacobian of a quotient of two functions be decomposed into some series of linear ...
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42 views

Eigenvalues of $B$ and $-B$

For a real symmetric matrix $B$, if we assume $QBQ^{T}=-B$ for some orthogonal matrix $Q$, then we know that the eigenvalues of $B$ and $-B$ are the same. My question is whether matrices of this ...
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59 views

How does SVD work?

Trying to find information, and, no-one seems to know the answers. I have a time-series, represented by $T = [0, 1, 1, 0, \ldots, n]$ the time series is then transformed into the Spectral results: ...
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50 views

Tensor vector bundle construction

$\newcommand{\p}{\partial}$Let $M$ be a smooth manifold, and define $$T_{r,s} := \bigsqcup_{p \in M} (T_p M)_{r,s} = \bigsqcup_{p \in M} \big( \underbrace{T_p M \otimes \dots \otimes T_p M}_r \otimes ...
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72 views

How to show that $v_1\otimes\cdots\otimes v_k = 0$ if and only if at least one $v_i = 0$?

I'm trying to show that given vector spaces $V_1,\dots,V_k$ (not necessarily finite dimensional) over the same field $F$ then if $v_i\in V_i$ we have $v_1\otimes\cdots\otimes v_k = 0$ if and only if ...
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40 views

relation of eigenvalue of cross product and weighted cross product

If I have a matrix $X$ that is, say $n\times p$ and a diagonal matrix $W$ $n\times n$ with all positive values on the diagonal, is there a relationship between the eigenvalues of $X^TX$ and $X^TWX$? ...
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72 views

Properties of matrix functions

Can I say that a certain matrix function is absolutely continuous or monotonically increasing in $\lambda$(assuming that the matrix is a function of $\lambda$)? In other words, are these ...
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44 views

Finding matrix with given special eigenvalues

Given an arbitrary matrix $A=\left[a_{i,j}\right]\in\mathbb{C}^{2\times2} $ Now define the set $$ \mathcal K = \left\lbrace z\in\mathbb C : ...
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116 views

Diagonalization of matrices and linear transformations

I'm reading Lawrence Perko's Differential Equations and Dynamical Systems, and he writes the following : Theorem. If the eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_n$ of an $n \times n$ ...
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295 views

Why is the special linear group generated by elementary matrices that add a multiple of row$ j$ to row $i$

The general linear group is generated by elementary matrices that add a multiple of row $j$ to row $i$ and elementary matrices that multiply row $i$ by a scalar. This is because you can write an ...