Tagged Questions

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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2answers
21 views

Find the matrix $A$ with this condition…

If $\theta \in\mathbb{R}\setminus\{k\pi, k\in\mathbb{Z}\}$ and $A\in M_{2\times 2}(\mathbb{C})$ such that $$A^{-1} \begin{pmatrix} \cos \theta & -\sin\theta \\ \sin \theta & ...
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1answer
21 views

Equivalent of solutions of IVP

Consider the IVP $y''-2y'+26y=0$, $y(0)=1$, $y'(0)=2$. From the characteristic equation $m^2-2m+26=0$, i found the roots as $m_1=1-5i$ and $m_2=1+5i$. Then when i use the basis solutions ...
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0answers
31 views

One question on Matrix Equation

Assume $\hat{M}_1, \hat{M}_2, \hat{T}_{11}, \hat{T}_{12}, \hat{T}_{21}, \hat{T}_{22}$ are $2\times 2$ matrix. And $a, b, A, B, C, D$ are all numbers, satisfying the following relation: \begin{align} ...
2
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0answers
20 views

Is there a nice representation for KKT conditions for matrix constraints?

I have a convex programming problem: $\min \left\lVert J - R \right\rVert _2$ $J,R$ are matrices. $J$ is given for the problem. One of the constraints is: $R = KQ$ Here, $R,K,Q$ are matrices. $K$ ...
1
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1answer
31 views

If A is a Hermitian matrix then SAS* is Hermitian

If $A$ is an $n\times n$ Hermitian matrix, and $S$ is an nxn matrix, then $SAS^*$ is also Hermitian. Why is this true? I have seen this claim made in several places but can't find a proof.
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2answers
33 views

Are three vector not in one plane mutually orthogonal, or linearly independent? [on hold]

Let $u, v, w$ be three points in $R^{3}$ not lying in any plane containing the origin. Are these three points linearly independent or mutually orthogonal?
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0answers
14 views

Can the Lanczos algorithm converge very fast by choosing initial guess smartly?

Suppose I have the two lowest eigenvectors $v_1$, $v_2$ of a matrix $M$. If slightly change $M$ to $M'$. Can I use $v_1$ or $v_2$ as an initial guess for $M'$? If so, which one should be used, $v_1$ ...
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0answers
30 views

Prove $NuclearNorm(W*U*S)\geq NuclearNorm(W*S)$

Suppose $W$, $S$ is two diagonal matrices of size $n*n$. $U$ is an orthogonal matrix. For $W$, the diagonal elements satisfies: $0\leq w_{1,1}\leq w_{2,2}\leq ...\leq w_{n,n}$, and for $S$, the ...
1
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1answer
22 views

Find a basis of a subset given an equation

$W = \{(x_{1}, x_{2}, x_{3})\in $R$^3: \frac{x_{1}}{3} = \frac{x_{2}}{4} = \frac{x_{3}}{2}\}$ Find a basis for $W$ I need help. I don't know how to do this.
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0answers
27 views

Prove $NuclearNorm(W*U*S)\geq NuclearNorm(W*S)$

Suppose $W$, $S$ is two diagonal matrices of size $n*n$. $U$ is an orthogonal matrix. For $W$, the diagonal elements satisfies: $0\leq w_{1,1}\leq w_{2,2}\leq ...\leq w_{n,n}$, and for $S$, the ...
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2answers
14 views

Determining whether sets of vectors form a basis

Is $\{(1,1,0,0),(0,0,1,1)\}$ a basis for the subspace of $\mathbb{R}^4$ consisting of all vectors of the form $(a,a+b,b,b)$ with $a,b\in \mathbb{R}$? Here is how I proceeded: First note that ...
2
votes
3answers
62 views

Linear Algebra: What do vector spaces represent?

I understand what a vector can represent, but I still don't understand what a vector space represents. I understand that you can add two vectors and that becomes a vector space. What else can you do ...
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1answer
19 views

Orthogonal transformation between vectors of the same norm

Suppose $V$ is a vector space over a field not of characteristic $2$, and is equipped with an inner product. I want to show that, given vectors $v$ and $w$, there is some orthogonal ...
1
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0answers
28 views

Compute a particular solution of AX = b

$A = \begin{bmatrix} 1 & 3 & 5 & 0 & 2 \\ 2&5&8&8&9 \\ 2&4&6&0&-1 \\ \end{bmatrix} $ Compute a particular ...
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1answer
25 views

Why is the laplacian matrix for a graph positive semidefinite?

Why is the laplacian matrix for a graph positive semidefinite? Can anyone provide an intuitive explanation and a proof?
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1answer
25 views

Find the equation of the ellipse with given foci and $a$

I have to find the equation of the elipse with foci: $$(-1,-1),(1,1)$$ and $a = 3$ I could do it using the definition of elipse, which I know how to work with. But I need to do it using translation ...
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0answers
10 views

Is there such a thing as a “continuum singular value decomposition”?

I have a question about expressing 2D functions as sums of separable functions. As a concrete example, consider the Gaussian circle function, ...
0
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1answer
16 views

Question about a subset not being a subspace in R^n

The question is: "Find an example of $S_{1}$ and $ S_{2}$ which are non-subspace subsets of $\mathbb{R}^3$ such that $S_{1}\cup S_{2}$ is a subspace of $\mathbb{R}^3$" I'm having trouble ...
2
votes
2answers
57 views

Proving ($\left|\left|Ax\right|\right| = \left|\left|x\right|\right|$, for all $x\in\mathbb{C}^n$) $\implies A$ is unitary

As the title states, I'm trying to prove that $\left|\left|Ax\right|\right| = \left|\left|x\right|\right|$ for all $x\in\mathbb{C}^n\implies$ $A$ is unitary, where $A$ is a square matrix. This is ...
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1answer
28 views

Vector subspace of polynomials

If I have a set of polynomials of degree at most $2$, such that $p(x) \geq 0$ for any real $x$. It isn't a vector subspace because I can multiply by a negative number such that $p(x) < 0$?
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0answers
9 views

Does the operation of addition on the subspaces of V have an additive identity? Which subspaces have additive inverses?

I was reading Linear Algebra Done Right. I came across the following question (Ch-1, Q12), for which I have solution , but I am having little confusion regarding it: Q12. (a) Does the operation of ...
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0answers
11 views

help with proof involving matrix derivations

So, Ive been trying to learn the research in a particular article, which can be read http://www.sciencedirect.com/science/article/pii/0024379580902219# Specifically lemma 2. So far, I have understood ...
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3answers
48 views

Can $A$ be singular

$A^2 + A + I= 0$ Can $A$ be singular? Justify your answer. I do not know where to start.
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0answers
31 views

Determining the set of a linear transformation of elements in a polyhedron

I have a set defined by linear inequalities of the form: $X = \{x : Ax \le b\}$. For any $x \in X$, I write $y = Gx$ where $G$ is a matrix (the dimension of $y$ is less than the dimension of $x$). ...
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2answers
43 views

Linear Algebra - Orthogonal problem (from exam can I appeal this?)

I have this problem : Let $A=\{v_1,v_2....,v_k\}$ in $R^n$ while $2 \leq k$. Prove if $A^\perp=(A-\{v_1\})^\perp$, then A is not linear independant. Please take a look at my solution since this is ...
1
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1answer
23 views

Relationship among $b_1$, $b_2$ and $b_3$ to have a solution

$$ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{bmatrix} $$ If $b= \begin{bmatrix} b_1 \\ b_2\\ ...
0
votes
1answer
18 views

Show that $X_{\textrm{null space}} + X_{\textrm{particular}}$ is a solution of $AX = b$.

If $X_{\textrm{null space}}$ is a vector in $N(A)$ and $X_{\textrm{particular}}$ is a particular solution of $AX = b$, then show that $X_{\textrm{null space}} + X_{\textrm{particular}}$ is also a ...
1
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0answers
20 views

Commuting exponentials of non-commuting matrices

For two non-commuting matrices $A,B \in M(2,\mathbb{K})$, $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, can be shown that: $$ e^C=e^{A+B}=e^Ae^B=e^Be^A \iff \begin{cases} ...
2
votes
2answers
26 views

Linear Algebra: Symmetric matrices, diagonalization (help with proof)

I need a bit of help with an IFF proof, here it is: {Let X be a symmetric n × n-matrix. Show: $$X=Y^2$$ for some symmetric matrix Y iff X has only non-negative eigenvalues. } My thinking: This ...
0
votes
2answers
38 views

Equivalent quadratic form with 4 varibles

Consider two quadratic forms: $Q(x,y,z,w)=x^{2}+y^{2}+z^{2}+bw^{2}$ and $P(x,y,z,w)=x^{2}+y^{2}+czw$. For what type of values of $b$ & $c$ (real or complex or negative or positive or zero) $P$ ...
2
votes
1answer
17 views

Help with determining if a function is onto (surjective)

The question is to determine if the following function $T(x,y,z) = (y\sin x,z\cos y,xy)$ is onto. So far I have only learned of creating a coefficient matrix and checking if the determinant is $0$ to ...
0
votes
1answer
32 views

Find a $3\times3$ matrix whose reduced row echelon form has two leading ones and whose row space intersects its column space by the line $x_1=x_2=x_3$

I am confused on how a matrix can exist I tried doing something like this $$ \begin{bmatrix}1& 0& 1\\0& 1& 1\\0& 0& 0\end{bmatrix} $$ but this only intersects with $x_1=x_2$ ...
0
votes
1answer
24 views

Proof concerning basic solutions

Prove that every basic solution of $Ax=b$ (where $A$ is a matrix of rank $r$) is set by $r$ linearly independent columns of matrix $A$ (so it is $[A^{k_1}\dots A^{k_r}]\bar{x}=b$ where $A^{k_1},\dots ...
2
votes
1answer
38 views

Is $g(A)$ diagonalizable?

Let $A$ be an $n \times n$ diagonalizable matrix; let $g(x)$ be a polynomial. Is $g(A)$ diagonalizable? If not, what are the minimum hypothesis one needs to make so that it works (if any?) (As ...
0
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0answers
20 views

Find non degenerate linear programming problems

I have to find non degenerate linear programming problem in a canonical form such that: a) it has no solutions b) it has solutions, but but doesn't have an optimal solution A ...
0
votes
1answer
37 views

Find the Basis and dimension of orthogonal complement of W

$$U = \pmatrix{ a_1 & a_2\\ a_3 & a_4 } $$ $$ V = \pmatrix{ b_1 & b_2\\ b_3 & b_4 } $$ $$ \langle U,V\rangle = a_1b_1+a_2b_2+a_3b_3+a_4b_4 $$ $W= \{t(2, 0, 0, -1): t \in \Bbb R ...
1
vote
1answer
23 views

3D rotation around arbitrary axis

I have a 3D rotation matrix, R which is a combination of rotations around x-axis , y-axis and z-axis. I know how to calculate n(the arbitrary axis around which a point rotated about theta angle and ...
0
votes
1answer
22 views

Subspaces -vector spaces

Let V be a nonempty subset of R^n. Show that V is a subspace of R^n if and only if for all u,v ∈ V and c∈R,u+cv∈ V. Any1 can help with this ques?I don't really know how to show this.appreciate ur ...
1
vote
3answers
27 views

Elementary matrix proof

I am supposing that $E$ is the elementary matrix obtained from $I$ (the identity matrix), by adding $\mu$ times the $m$-th row to the $l$-th row for some $\mu \in \mathbb{R}$ and $1\leq l,m\leq n$ and ...
0
votes
2answers
32 views

Are all vector spaces also a subspace?

I am currently learning about vector spaces and have a slight confusion. So I know that a vector space is a set of objects that are defined by addition and multiplication by scalar, and also a list ...
3
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1answer
32 views

Is it true that $\|\text{diag($\pi$)} P\|_2 \leq 1$ for $P$ stochastic and $\pi P = \pi$.

$\| \cdot \|_2 $ is the matrix norm induced by $L_2$. $P$ is any given real square $n \times n$ non-negative matrix with rows summing to one, i.e. $P1 = 1$, where $1$ is the vector of ones. There is ...
0
votes
1answer
25 views

How to find out the linear transformation?

Is it linear transformation? Let the transformation be defined as $T:\mathbb{R}^3 \to \mathbb{R}$ $$T([x, y, z])=x^2-2y+3z$$ Well actually I have no idea how it works with an equation like ...
0
votes
1answer
12 views

Incorrect elementary row operation in an augmented coefficient matrix

When solving the matrix $$\left(\begin{array}{ccc|c} 1 & 1 & 1 & 4\\ 1 & 3 & 1 & 4\\ -1 & 2 & 3 & -2\end{array}\right)$$ I somehow made an error with the ...
1
vote
1answer
11 views

When is $\| X \| _\star = \| F X \| _2$ submultiplicative?

All matrices are real. By $\| \cdot \|_2$ denote the matrix norm induced by $L_2$. Assume $F$ is an invertible matrix. Consider the norm $\| X \| _\star = \| F X \| _2$. What is the condition on ...
0
votes
1answer
25 views

Linearisation of non-linear models

I'm asked to linearise the following model: $$y=\alpha xe^{\beta x}$$ I know I have to find an equation along the lines of $Y= A+BX$, but when I apply natural logarithms and use identities I still ...
0
votes
1answer
24 views

How to solve systems of linear equations of multiple variables (more than 3 to 100s)?

This was a question asked during an interview for programming job. And the bottom line was to write an alogrithm to solve such equations. As much as it numbed my neurons - it really provoked me. I had ...
2
votes
2answers
59 views

Prove that $0 < x < y$ implies $\|x\| < \|y\|$ for any norm.

All vectors are real. Prove that $0 < x < y$ (element-wise) implies $\|x\| < \|y\|$ for any norm. This is probably very basic, but I don't seem to get the hang of it. Edit: it turns out this ...
0
votes
1answer
19 views

Prove that, if $x(k) \in \mathbb{R}^n $, then $\sum x(k)^T (N^{-1} \sum x(k) x(k)^T)^{-1} x(k) = Nn$

Prove that, if $x(k) \in \mathbb{R}^n $, then $\sum_{k=1}^N x(k)^T (\sum_{k=1}^N N^{-1} x(k) x(k)^T)^{-1} x(k) = Nn$, where the sums are over k ranging from 1 to N. We have N>n. Assume also that ...
6
votes
5answers
267 views

How to solve this to find the Null Space

What I did : I put this into reduced row echleon form: $$\begin{bmatrix} 1 & -2 & 2 & 4 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 ...
0
votes
0answers
21 views

Camera Calibration

In a camera model, in order to find the camera calibration, how do we find the the parameters from the vector a in the equation $Ca=0$? I know that the camera matrix to convert a world point to image ...