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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113 views

Finding for every parameter $\lambda$ if matrix is diagonalizable

Given: $$A = \begin{pmatrix} 1 & i & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & i \end{pmatrix} \; , \; \lambda \in \mathbb C$$ For every value of $\lambda$ I have to know if the matrix ...
0
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4answers
92 views

Prove that if $A \in M_{2\times2}\mathbb {(R)}$ is symmetric then A is diagonalizable

Given that: $$A \in M_{2\times2} \mathbb {(R)}$$ we have to prove that $A$ is diagonalizable. As in: $$\text{There exists a turnable matrix } P \; (\text{det(P) != 0 }) \; \text{such that}:$$ ...
0
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2answers
48 views

Quick question about proofs of theorem concerning Jordan basis

I have a question about proofs of this theorem: Let $K$ be an algebraically closed field, $V$ be a finite-dimensional space over $K$ and $f : V → V$ be a linear operator. Then there exists a Jordan ...
0
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1answer
37 views

Inverse of Eigen value

What is the physical meaning of inverse square root of the eigen value? Is it possible to use it as stretch factor to decorrelate the data.
2
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2answers
139 views

Minimum distance of the linear code $\{0,1\}$

Let $H$ be a check matrix for a linear code $C$. Then the minimum distance of $C$ is $d \in \mathbb N$ such that there exists a set of $d$, but no set of $d-1$, linearly dependent columns in $H$. ...
10
votes
1answer
188 views

Isomorphism between $E_8$ lattice and lattice defined by Extended Hamming Code

I have read that the following two lattices are isomorphic, and of course it seems believable, but it would be nice to have a sketch of how to construct the bijection. Let $C$ be some extended ...
1
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0answers
50 views

Solution of a Quadratic Optimization Problem

Let $\mathbf{A_1}$ and $\mathbf{A_2}$ be two given $N\times N$ hermitian matrices. Then how do I solve the problem, \begin{align} ...
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0answers
222 views

Showing no non-trivial t-invariant subspace has a t-invariant complement.

The question is from Hoffman and Kunze Let T be a linear operator on a finite-dimensional vector space V. Suppose that: (a) the minimal polynomial for T is a power of an irreducible ...
5
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3answers
70 views

$f\colon \Bbb R^3 \to \Bbb R^3 $ be defined by $f(x_1,x_2,x_3)=…$

I am stuck on the following problem: Let $f\colon \Bbb R^3 \to \Bbb R^3 $ be defined by $f(x_1,x_2,x_3)=(x_2+x_3,x_3+x_1,x_1+x_2).$ Then the first derivative of $f$ is : 1.not invertible ...
1
vote
1answer
727 views

Linear Algebra: Least-Squares Approximation & “Normal Equation”

I am reviewing Example 1 from Chapter 6, Section 4 (Least-Squares Approximation and Orthogonal Projection Matrices) in "Elementary Linear Algebra - A Matrix Approach 2nd Edition [ISBN] ...
2
votes
1answer
67 views

I need to diagonalize this matrix but I'm not sure it can be

This is the matrix I need to diagonalize: $A=\left[\begin{matrix}3&2\\0&3\end{matrix}\right]$. So I found the eigenvalue by taking the determinant of $(A-\lambda I)$ and solving for ...
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1answer
130 views

Find all eigenvalues and corresponding eigenvectors for the matrix?

Find all eigenvalues and corresponding eigenvectors for the matrix: $$ \left(\begin{array}{cr} 0&-1 \\ 2&3 \end{array}\right) $$ Not looking for a answer, but I don't know what an "eigenvalue" ...
0
votes
3answers
154 views

Underdetermined System and Minimizing Cost

I need to minimize 4x + 4y subject to the following constraints: $4x + 8y = 40$ $x + 2y = 10$ Any ideas? Answers must be integers, as they represent physical units.
4
votes
1answer
83 views

Rank after addition of positive definite matrices

I have two positive semidefinite matrices $A$ and $B$. Is it necessarily true that $$ rank(A+B) = rank(A^2+A+B) $$ ? It is easy to see that $rank(A+B) \le rank(A^2+A+B)$, but for any example I try, ...
2
votes
2answers
139 views

Picard iterations of a matrix

I need help with this problem. I think i got the first three questions of the exercise, but i'm stuck at the fourth one. We consider the map $T:{\mathbb{R}^2}\longrightarrow{\mathbb{R}^2}$ defined ...
1
vote
1answer
153 views

how to calculate the “variance OF the covariance” matrix : E[vech(x x') vech(x x')'] for normal distributed x?

Supposing a vector x follows normal distribution. I want to calculate the expectation of the "variance Of the covariance matrix" (not variance-covariance matrix) in a vector form, meaning E[vech(x ...
0
votes
2answers
53 views

Matrix Algebra (Elementary)

I have $\hat\xi =\lambda_1\textbf{1V}^{-1} + \lambda_2\textbf{rV}^{-1}$ and sub it in to my two constraints, namely, $\xi\textbf{1}^T = 1$ and $\xi\textbf{r}^T = \mu$. My lecture notes then say set ...
2
votes
2answers
83 views

$\psi:A\rightarrow A^{-1}$ is continuous

Define a map that takes a matrix to it's inverse. Give $A\in M(n\times n)$ over reals field, define: $$\psi:A\rightarrow A^{-1}$$ Is it always continuous, where defined? How do I prove this? Thanks.
1
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1answer
212 views

How to prove that $D := ABC$ is also positive definite?

Let $A,$ $B$ and $C$ be symmetric, positive definite matrices and suppose that $D := ABC$ is symmetric. How might I prove that $D$ is also positive definite?
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votes
1answer
33 views

diagonalizing a matrix $A$: can $P$ be bigger than $A$?

can you have a P bigger than the original A matrix? in other words after I found the eigenvalues I then found all the eigenvectors so when I constructed the P vector turns out to be bigger than my ...
2
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0answers
202 views

Can a linear combination of even Legendre polynomials have common real root(s) with a linear combination of odd Legendre polynomials?

I am using the following definition of Legendre Polynomials: $P_0(x)=1$, $P_1(x)=x$ and $$P_{k+1}(x)=\left(\frac{2k+1}{k+1}\right)xP_k(x)−\left(\frac{k}{k+1}\right)P_{k−1}(x)$$ Let ...
2
votes
2answers
193 views

Continued fraction for $\sqrt{14}$

I'm referencing this page: An Introduction to the Continued Fraction, where they explain the algebraic method of solving the square root of $14$. $$\sqrt{14} = 3 + \frac1x$$ So, $x_0 = 3$, Solving ...
5
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3answers
158 views

Are Clifford groups very *non-commutative*?

Clifford groups seem to be very non-commutative by the relation \begin{equation} \gamma_{i}\gamma_{j}=-\gamma_{j}\gamma_{i}. \end{equation} But is it really so? Can we put this degree of ...
4
votes
2answers
136 views

Prove there exists $y\neq 0$ but $x\cdot y=0$

I would like to know if I'm missing something with my solution - as an earlier version was wrong and I think I've managed to patch it up - to this problem from Rudin's Principles of Mathematical ...
0
votes
1answer
53 views

Find Line with specific Angle to another Line

Given any line in 3 dimensional space $$A: \vec{X} = \vec{O} + \lambda \vec{D}$$ and any angle $\phi$, I want to find another line $B$ which fullfills the following criteria: it ...
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2answers
35 views

Recursive formulae involving a linear operator

Given a basis $e_{1}$, $e_{2}$ in the plane, define the linear operator $F$ as $F(e_{1})=3e_{1}+e_{2}$ and $F(e_{2})=e_{2}$. Furthermore, define the sequence $u_{1},u_{2},\dots$ of vectors in the ...
3
votes
1answer
169 views

Coupling of Gaussian Random variables.

Suppose I have two mean zero multivariate Gaussian random variables $X$ and $Y$ on $\mathbb R^d$, with covariance matrices $A$ and $B$. (Assumed to have full rank). I have many choices of joint ...
4
votes
1answer
338 views

Find $\det X$ if $8GX=XX^T$

I need to find $\det X$ where $$8GX=XX^T,\quad G=\left(\begin{matrix}5 & 4\\3 & 2\\\end{matrix}\right).$$ My answer is that the determinant of $X$ is $-128$ and that is correct but there is ...
2
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0answers
86 views

Full Rank Matrix with a specific construction

Assume that we have a $p \times p$ matrix $Z$ over $\mathbb{F}_{2^p}$ $$Z=\begin{bmatrix} w_1 & w_1^2& w_1^4& ... & {w_1}^{2^{\frac{p}{2}-1}} & \alpha_1w_1 & ...
0
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2answers
62 views

What is the meaning of the “essentially the same” in this question?

Although I have seen a couple of explanations about the phrase essentially the same it hasn't made any sense and I could not understand what I should do to prove the following claim: Let $S=\left\{ ...
1
vote
1answer
40 views

Robust feasibility with halfspace?

Consider a convex function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, such that for all $x \in \mathbb{R}^n$ we have $$ a_1^\top x + b_1 \leq f(x) \leq a_2^\top x + b_2 $$ for some given $a_1, a_2 ...
2
votes
2answers
192 views

Number of linear maps is less or equal than the dimension of the vector space

Problem Let $V$ be a vector space of dimension $n$. Let $T_1,T_2,\dots,T_m$ be linear maps $V\rightarrow V$ such that $\dim R(T^{2}_{i})=\dim R(T_i)=1$ for all $i=1,2,\dots,m$, and $T_i\circ T_j$ is ...
0
votes
1answer
23 views

Perturbation parameters in Eigenvalue question [easy]

I'm solving an eigenvalue/eigenvector question of the matrix: \begin{bmatrix} 2 & 1 \\ 0 & 2 + \varepsilon \end{bmatrix} where $\varepsilon$ is the perturbation parameter. Would I just solve ...
1
vote
1answer
168 views

How tell if a polyhedron contains a lattice point

So given a polyhedron $Ax \le b$ Is there an Algorithm or formula to determine whether said polyhedron contains a lattice point (integer point) I was thinking a couple things: brute force ...
1
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1answer
58 views

Inclusion of orthogonal complements: if $U_1\subset\ U_2$, then $U_2^\bot\subset\ U_1^\bot$

Show if $U_1\subset\ U_2$, then $U_2^\bot\subset\ U_1^\bot$ I'm thinking using contradiction. $\exists v\in U_2^\bot$ s.t. $v\notin U_1^\bot$ $\Rightarrow$ $\exists u\in U_1$ s.t. $u\notin U_2$ ...
1
vote
1answer
366 views

Proof of adjoint(ab) = adjoint(b)adjoint(a)

So I'm trying to prove whether $\operatorname*{adjoint}(AB) = \operatorname*{adjoint}(B)\operatorname*{adjoint}(A)$. Here, for any matrix $C$, the matrix $\operatorname*{adjoint}(C)$ is defined as ...
1
vote
2answers
70 views

Identifying matrix vector multiplication

I have the following question in a book: According to the book, the answer is (D). But I don't understand how. Isn't this just scalar multiplication? The solution in the book says that I have to ...
3
votes
1answer
44 views

Underdetermined System Cost

$\begin{bmatrix} 1 & 3\\ 2 & 6 \end{bmatrix} \cdot \begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} 10\\ 20 \end{bmatrix}$ Is an underdetermined system. How do I find the shortest ...
1
vote
1answer
81 views

Which line is better fit with least squares matrices

Given two best fit curves determined by the least squares method, how can I determine which line is a better fit. Specifics $X\boldsymbol\beta=\bf{y}$ is the representation of the "perfect fit" ...
0
votes
3answers
796 views

Finding Best-fit Curve from Points

I am given the set of data points: $(1,2), (0,1), (-1,0), (-2,3)$ I am trying to find the best fit curve in the space: $f(x) = ax^2 + bx + c \;\;\;a,b,c \in R$ How do I go about doing this? I was ...
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1answer
342 views

Find an orthogonal matrix

I wonder if i can find for 2 unit vectors $v,w$ only one orthogonal matrix $Q$ such that $Qv=w$ is there any proof for that?
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2answers
74 views

Least Squares Approximation

Given: $\begin{bmatrix} 1 & -2\\ -2 & 4 \end{bmatrix} \cdot \begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} 1\\ 5 \end{bmatrix}$ How do I find the least squares approximation and ...
6
votes
2answers
106 views

let$ G=\{M_1,M_2,…,M_k\}$ be a finite group if $\sum _{i=0}^k \operatorname{tr} (M_i)=0$ then how prove $\sum _{i=0}^k M_i=0$

Let $G=\{M_1,M_2,...,M_k\}$ be a finite set such that $ M_i\in M_n(\mathbb R)$ and $(G,\;\cdot\:)$ is group with operations of matrix multiplication If $\sum _{i=1}^k \operatorname{tr} (M_i)=0$ ...
1
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1answer
288 views

3 tank mixing problem

There are 3 tanks filled to capacity with fresh water, all with a 100 liter capacity. At t=0, brine with .5 kg/l salt concentration flows into tank 1 at a 3 l/min rate. The other flows are: tank 1 -> ...
4
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0answers
32 views

what to do if it's not direct sum?

Suppose $X=Y+Z$ is Banach, $Y$ and $Z$ are closed subspaces. I want to show there exists $\alpha>0$ such that $\forall x \in X, \exists$ $y \in Y$ and $z \in Z$ such that $x=y+z$ and $\|y\|+\|z\| ...
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3answers
122 views

Eigenvalue and algebraic multiplicity of some matrix

Determine the eigenvalue and algebraic multiplicity of $$P=\pmatrix{0 & 1 & 0 \\ 0 & 0 & 1 \\1 & -3 & 3 \\ }$$ My calculations led me to the eigenvalues $0,\pm 1,3$ each of ...
0
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1answer
193 views

Addition of homogeneous vectors with different w component

If I have two homogeneous vectors say $v_1 = (x_1, y_1, z_1, w_1)$ and $v_2 = (x_2, y_2, z_2, w_2)$, then is their addition defined if $w_1 \ne w_2$? If $w_1 = w_2 = w$, then I can simply do $v_1 + ...
1
vote
1answer
47 views

Is this set $\{ p(x): x\in \operatorname{bco} A\}$ bounded in $\mathbb{R}$?

$\newcommand{\bco}{\operatorname{bco}}$Here are some terminologies. Definition. Let $X$ be a real vector space and let $A\subseteq X$. The balanced-convex hull of $A$, denoted $\bco A$, is the ...
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0answers
35 views

Is the polar decomposition useful in the real case as well?

I'm reading Roman's Advanced Linear Algebra p.252, where he talks about the Polar Decomposition. He states the theorem only for the case of $V$ a complex inner product space. Wikipedia also states the ...
0
votes
1answer
158 views

Solution of a system of linear equations with n variables

I have a system of linear equations with n variables \begin{cases} a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n = \frac{1}{2}x_1\\[4pt] a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n = \frac{1}{2}x_2\\[4pt] ...