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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44 views

prove that $g$ is symmetric bilinear form

Let $J: V\to V$ be a linear opertor where $V$ is a finite dimensional vector space over $\mathbb R $. Assume that $J^2 = -Id$. Let $f:V\times V \to \mathbb R$ be a skew-symmetric non degenerate ...
2
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1answer
38 views

Product of projections and commutativity

Let $P_1$, $P_2$, $\dots$, $P_m\in\mathbb{R}^{n\times n}$ be orthogonal projections projecting onto subspaces $V_1$, $V_2$, $\dots$, $V_m$, respectively, and let $P_{1\cap2\cap\dots\cap m}$ denote the ...
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0answers
18 views

Explicit notation of matrix

I have to give the specific form of the matrix $$ A=(a_{ij})_{1\le{i,j}\le{n}} \\ a_{ij}=\delta_{i+1,j} $$ which would be $$ \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 \\ ...
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0answers
23 views

Topology property of inner product and norm

It is known that the norm can induce an inner product unless it satisfies Parallelogram law. I just want to know what topology property the inner product has while the norm doesn't have?
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1answer
23 views

Polarity is symmetric

Let X$\subseteq \mathbb{R}^n$ be a closed, convex set that contains the origin: $\textbf{0}_n \in$X. The polar of X is $X_{po}:=\{y\in \mathbb{R}^n : x^T y\leq 1,\forall x \in X\}$ I'd like to ...
1
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0answers
28 views

Meaning of kernel

If something is the 'kernel' of a transformation, say $K(x,x')$, does it mean I should take the integral $$\int K(x,x') f(x')dx'$$ There are many different meanings of kernel and I did not see their ...
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0answers
23 views

excel modification syntax [closed]

I'm trying to write this in a more elegant and sophisticated way in excel rather than hash out so many nested if statements. ...
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1answer
46 views

Column vector of simultaneous equaations' solution

Struggling with some basics of Linear Algebra. Please help. Let's restrict the discussion to 2D space & consider the following simultaneous equations: $2x + 3y = 8, x + 2y = 5$ I understand ...
2
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2answers
47 views

Is rank$(AQB)=$rank$(AB)$ if $Q$ is non-singular?

$\newcommand{\rank}{\operatorname{rank}}$We know that $\rank(PA)=\rank(AQ)=\rank(PAQ)=\rank(A)$ where $A\in M_{m\times n}(\mathbb F), P, Q$ are $m\times m, n\times n$ invertible matrices. mean to ...
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1answer
36 views

How to do this last step in this proof that inner product preserving implies linear?

Let $\tau : \mathbb R^m \to \mathbb R^m$ be a map such that $\tau (0) = 0$ and $\langle x,y \rangle = \langle \tau(x) , \tau(y) \rangle$ for all $x,y \in \mathbb R^m$. I want to show that $\tau$ is ...
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4answers
51 views

I'm trying to find the eigenvalues of a matrix. What is my mistake?

I have the matrix: $\left[ \begin{array}{cc} 3 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3 \end{array} \right]$ which I rewrote as $\left[ \begin{array}{cc} λ-3 & ...
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1answer
25 views

Three points are collinear iff the determinant of the matrix of their barycentric coordinates vanishes

Let $A,B,C\in \mathbb{R}^2$ be noncollinear points. Then we have that for every point $P\in\mathbb{R}^2$ there exist $\alpha_1,\alpha_2,\alpha_3\in\mathbb{R}$ such that $P=\alpha_1A+\alpha_2 ...
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0answers
66 views

matrices in the form $AB-BA$

Let $X$ be $n\times n$ matrix. What are the necessary and sufficient conditions of the existence of representation: $X=AB-BA$ where $A,B$ are some $n\times n$ matrices. This seems difficult. How to ...
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1answer
30 views

Determining the fourth vertex of a parallelogram knowing that its the point of intersection of two circles

This question was part of the exercises in one of the courses i'm taking. The answer was already provided. The first circle was assumed to have as its center, vector $v_1$, while its radius was the ...
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2answers
38 views

Eigenvectors and eigenvalues of matrices

Say that we have a square matrix $M$, and that a non-zero vector $v$ can be an eigenvector of $M$ if $Mv = kv$ for some real number $k$. This real number, $k$, can be called the eigenvalue of $v$ with ...
2
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2answers
17 views

Generate an integer matrix such that all submatrices are non-singular

I need to generate an $\infty \times N$ integer matrix with a few properties. The top $N$ rows (and $N$ columns) should be the identity matrix. Any square submatrix (meaning the result after ...
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1answer
13 views

Solving this n x n matrix equation with special structure

I have a simple linear equation of the form $A\vec{y}=\vec{x}$ for $A$ an $n\times n$ invertible matrix whose diagonal entries are all equal to $a$ and off diagonal entries are all equal to $b$. ...
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1answer
18 views

Orthogonal Complement and dimension [closed]

Let $Q=\text{span}(\{q_1,q_2,q_3\})$ be a three dimensional subspace of $\mathbb{R}^4$. How do you show that the orthogonal complement of $Q$ is one-dimensional?
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3answers
62 views

Vector space or vector field?

I seem to be having a problem distinguishing between a vector space (which I know to be a set of vectors over some scalar set) and a vector field. I know that in Multivariable Calculus a vector field ...
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votes
5answers
409 views

What is the intent of this problem, disguised as an eigenvalue - eigenvector problem?

Let $$ A= \begin{bmatrix} a & b & c \\ b & c & a \\ c & a & b \\ \end{bmatrix} $$ $a,b,c >0$. Find eigenvalues and a basis of ...
4
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1answer
129 views

If $AA^*=AA$, how to prove $A$ is an Hermitian? [duplicate]

If $A$ is an $n \times n$ matrix and $AA^*=AA,$ how to prove $A$ is Hermitian?
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0answers
21 views

Is a composition of two $n-1$-dimensional symmetries a composition of $n-2$-dimensional symmetries?

Let $X$ be a finite dimensional real Euclidean space and $S,T$ be symmetries with respect to $n-1$-dimensional subspaces of $X$. Is it possible to write $ST$ as a composition of symmetries with ...
2
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2answers
37 views

Generalized Diagonal

I was given the following definition: For all $\sigma\in S_n$, the product $\prod_\limits{i=1}^n a_{i,\sigma(i)}$ contains one entry from every row and every column, the entries of these ...
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0answers
21 views

Finding a jordan basis for a jordan form

Let $$A = \left(\begin{array}{cccc} 1&0&0&0\\3&-2&0&0\\14&0&-2&0\\8&-1&1&-2 \end{array}\right)$$ Easy to verify that $f_(x) = (x-1)(x+2)^3$. So the ...
3
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1answer
32 views

Eigenvalues of a binary matrix

Let $A = (a_{ij})$ be an $n \times n$ matrix with all entries equal to 0 or 1. Suppose that $a_{ii} = 1$ for $i = 1, \cdots, n$ and that $\det A = 1$. Then all the eigenvalues of $A$ are equal to 1. ...
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1answer
36 views

Linear Algebra Problem of two vectors

Say I have 2 vectors, $C_1$ and $C_2$ of differing lengths. If I know the matrix $C_k$ for which each element i,j is the product of the ith element in $C_1$ and the jth element in $C_2$, is it ...
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0answers
32 views

A question in matrix polynomial

Suppose ${P_\Delta }(\lambda ) = ({A_m} + {\Delta _m}){\lambda ^m} + ....... + ({A_1} + {\Delta _1}){\lambda ^1} + ({A_0} + {\Delta _0})$ is a matrix polynomial, and $\lambda $ is a complex ...
0
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1answer
19 views

Prove that $T^*N=NT^*$ [duplicate]

Let $T,N:V\to V$ such that $N$ is a normal operator and $TN=NT$. Prove that $T^*N=NT^*$ and $N^*T=TN^*$. I'd be glad for help because I don't even have an idea how to approach this
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2answers
44 views

Showing a basis for a polynomial

I am having trouble with this basic basis problem. Need to show that $\{z^4,z^4-z^3,z^4-z^3+z^2,z^4-z^3+z^2-z,z^3-1\}$ is basis for $P_4$. I figured out that it is linearly independent but having ...
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2answers
52 views

(a+b)+(c+d)=(a+c)+(b+d) — How to justify this using commutativity and associativity? [closed]

Please show steps in how you prove this using commutativity and associativity.
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5answers
50 views

Basis Proof question [closed]

If $\{q_1,q_2,q_3\}$ is a linearly independent subset of a vector space Q where the dimension of Q is 4 and $q_4$ is not in span of $\{q_1,q_2,q_3\}$. Prove that $\{q_1,q_2,q_3,q_4\}$ is a basis for ...
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1answer
15 views

Showing a set is nonempty for subspace

I need to show $A=\{(0, q, w): q, w \in R\}$ is a subspace of $R^3.$ I am having trouble showing that the set is nonempty because of the parameters. Do I say, Let $q = 0$ and $w = 0$ and thus $(0,0,0) ...
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0answers
56 views

Needs an explanation on why I obtain this covariance matrix

Let's say $n$ is an even integer. I'm playing with a column vector $\mathbf{v}$ which must satisfy the following three requirements: It's a length-$n$ vector of +1s and -1s. It has the same number ...
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2answers
27 views

Subset and Basis Question

If $Q ={(−2,0,1),(5,−2,1),(11,7,−5),(−1,4, −2),(2, −3,1)}$ How do you find a Subset of Q which is also a basis for $R^3.$
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1answer
36 views

$S_n$ notation in permutations

What does notation $S_n$ stands for? For example if I have the following set $\{1,2,3,4\}$ so we say that $S_4$=24? Moreover in many examples I saw the use of following numbers like $\{1,2,...,n\}$ ...
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0answers
33 views

unable to implement linear programming for min cut max flow problems [on hold]

iam trying to solve codechef problem using linear programming(simplex). https://www.codechef.com/problems/CHEFBOOK i understood the concept of linear programming , but i was unable to implement. I ...
2
votes
1answer
30 views

Exercise 1.4.1 from Shankar's Quantum Mechanics book

Exercise 1.4.1: In an n-dimensioanl vector space $V^n$, prove that the set of all vectors $∣Vi\rangle$ , orthogonal to any vector $∣W\rangle \neq 0$, form a subspace $V^{n-1}$ of dimension $n-1$. I ...
0
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1answer
24 views

Matrix,Linear algebra,polynomial,finite field,notation

In the book by Arora and Barak,Computational Complexity,on page 168,1st paragraph, there is a notation which I do not understand. They write For every $n \times n$ matrix $A$,and $i\in [n]$,we define ...
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0answers
27 views

Inverse of Rank-one matrix [duplicate]

I need to find out the inverse of rank-one matrix ie. $A^{-1}=[\underline{x}\underline{x}^T]^{-1}$ where $\underline{x}\in \mathbb{R}^{n}$. I guess it is not possible to find out the same by using ...
1
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1answer
41 views

Finding the Jordan Form and basis

$$A= \begin{pmatrix} 2&1&2\\ -1&0&2 \\ 0&0&1 \end{pmatrix}$$ I found that $$f_A(x)=m_A(x) = (x-1)^3.$$ So the Jordan form must be: $$J= \begin{pmatrix} 1&0&0\\ ...
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3answers
83 views

Prove or disprove: $A=A^\top \land B = B^\top \Rightarrow AB = (AB)^\top$

where $A,B\in\mathbb{R}^{n\times n}$. My current solution is that this will only work iff $A$ and $B$ commute. Since: $(AB)^\top = B^\top A^\top = B A$ $\ $ ($=AB$. iff $A$ and $B$ commute.) I ...
4
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1answer
60 views

Coordinate-Free Definition of Trace.

$\DeclareMathOperator{\tr}{trace}$ I am reading the wikipedia article on the trace operator. The section titled Coordinate-Free Definition defines the trace as follows. Let $V$ be a finite ...
2
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2answers
28 views

A Proof of a False Result: If $U$ is $T$-invariant, then so is $U^\perp$.

$\newcommand{\ab}[1]{\langle #1\rangle}\DeclareMathOperator{\tr}{trace}\newcommand{\mc}{\mathcal}$ I have a "proof" of the following wrong fact: Let $T$ be a linear operator on a finite ...
1
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1answer
17 views

Double integral of a product in calculus of variations

Let's say I have an integral of the form $$ V(u) = \iint\limits_{[0,T]^2}f(x,y)u(x)u(y)\mathrm dx\mathrm dy $$ which I would like to optimize over smooth functions $u$. For the variation I get $$ ...
0
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1answer
43 views

Prove that if $A$ is a square matrix such that all of the rows of $A$ have the same sum $k$ then $k$ is an eigenvalue of $A$.

Prove that if $A$ is a square matrix such that all of the rows of $A$ have the same sum $k$ (i.e. the sum of the entries in each row is $k$) then $k$ is an eigenvalue of $A$. my effort I think that ...
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3answers
26 views

Euclidean space as a different dimension.

Clearly $ \mathbb{R}^{n}(\mathbb{R})$ is vector space of dimension $n$ under addition and scalar multiplication as component wise addition and scalar multiplication . But i am trying to define $ ...
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28 views

Article writing: How to represent a matrix by its elements?

Intuitively I guess that parantheses with subscript and superscripts is a way of representing a matrix or an array by its elements, e.g., $$ A = (a_{i,j})_{i,j=1}^n $$ (This is taken from here) ...
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1answer
26 views

matrix with eigenvalues on the unit circle

Suppose $A$ is an $n\times n$ matrix with complex entries such that there exists strictly positive constants $c_1<1<c_2$ so that $$c_1<\frac{\|A^Nx\|}{\|x\|}<c_2$$ for any integer $N\geq ...
4
votes
1answer
65 views

What are the rules for taking derivatives in linear algebra?

I was reading through a paper on beamforming and came across an equation whose derivative I don't fully understand. A cost function is given as: $$ J(\mathbf{w}) = \mathbf{w}^HR\mathbf{w} ...
2
votes
2answers
45 views

Realizing the oscillator algebra as a matrix Lie algebra

In Hilgert's & Neeb's Structure and Geometry of Lie Groups, they introduce a Lie algebra, which they call the "oscillator algebra," as an extension of the Heisenberg algebra. They give a basis ...