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 inner direct sum of projection operators

I would like to understand part of a proof that is unclear to me. Theorem: Let $V$ be a $K$-vectorspace and $q_1, \dots, q_r$ a set of projection operators on $V$ such that $\sum_{i=1}^r{q_i} = ...
10
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2answers
349 views

Characteristic polynomial of a matrix with zeros on its diagonal

Let $p(x)=x^n+a_{n-2}x^{n-2}+a_{n-3}x^{n-3}+\cdots+a_1x+a_0=(x-\lambda_1)\cdots(x-\lambda_n)$ be a polynomial with real coefficients such that every $\lambda_i$ is real. Is there always a symmetric ...
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2answers
90 views

Confusion on Eigenvalues of Similar Matrices

Please help me to identify where I went wrong: The completely reduced normal form of the real matrix $A= $\begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ ...
4
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1answer
97 views

Connection between dual space V* and negation P^c

Notice the following similarity between the vector space dual and negation in propositional logic: $$ V^* \equiv V \rightarrow F $$ $$ P^c \equiv P \rightarrow \bot $$ Is there some general notion ...
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2answers
178 views

Graphs with commuting adjacency matrices

Let A and B be adjacency matrix of two undirected simple graphs. Can we assign some combinatorial interpretations to this pair of graphs if A and B commute?
7
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1answer
357 views

How to geometrically interpret $\sum^{p}_{1}\lambda_{i}(A)=\operatorname{tr}(A)$?

$A$ is a $p\times p$ real matrix and $\lambda_{i}$ are its eigenvalues. $\operatorname{tr}(A)$ is the trace of $A$. How to geometrically interpret $\sum^{p}_{1}\lambda_{i}(A)=\operatorname{tr}(A)$? ...
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2answers
114 views

Is there a simple proof of the following result?

I have been trying to prove the following result: If $A$ is real symmetric matrix with an eigenvalue $\lambda$ of multiplicity $m$ then $\lambda$ has $m$ linearly independent e.vectors.
3
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1answer
168 views

Normal and unitary Matrices

Can you help find a $2\times 2$ matrix with eigenvalues $1,-1$ that is not a normal matrix? I really tried to find one but the matrix I found is also normal!! $A$ hermitian and $B$ unitary matrix and ...
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1answer
501 views

Iterative Methods: Jacobi Method vs. Gauss-Seidel

In most circumstances, given a linear system: $A\mathbf{x}=\mathbf{b}$ for which G-S and Jacobi converge to the solution $\mathbf{x}$, G-S converges faster. However it is possible to concoct examples ...
4
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1answer
467 views

dimension of the space of all symmetric matrices with trace $0$ and $a_{11}=0$,

I want to know the dimension of the space of all symmetric matrices with trace $0$ and $a_{11}=0$, I can show that the dimension of space of all symmetric matrices $S$ is $n(n+1)/2$, now I give a ...
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3answers
133 views

how can a matrix vector product reduce to a scalar?

I have an Excel spreadsheet with the following formula (paraphrased): ...
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3answers
1k views

Adjoint Operators and Inner Product Spaces

My linear algebra textbook gives the definition of the Adjoint Operator and then says, You should verify the following properties: Additivity: $(S + T)^* = S^* + T^*$ Conjugate ...
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0answers
43 views

maxcut and the minimal eigenvalue

For an adjacency matrix $A$ that represent a graph $G=\langle V,E\rangle$, I need to show that the maxcut is bounded by: $$ \mathrm{maxcut} \leq \frac{1}{2}|E| - \frac{|V| \lambda_{\min}(A)}{4}, $$ ...
0
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1answer
56 views

Equivalent representation of the determinant of matrix polynomial

I have to show that the determinant of \begin{pmatrix} \lambda I_p & -I_p & & 0\\ & \ddots &\ddots \\ & & \ddots & \ddots \\ 0 & & & \lambda I_p & -I_p ...
2
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1answer
279 views

Criteria for the difference of two matrices to be positive semidefinite when the eigenvectors are known [duplicate]

Let $A$ be a rank 1 positive semidefinite matrix and $B$ a Hermitian matrix. Suppose I know the eigenvectors of both $A$ and $B$ and that $A-B$ is also positive semidefinite. Apart from Weyl's ...
5
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2answers
85 views

Eigenvalues of specific block matrices

Is there any simple relation between the eigenvalues (or the characteristic polynomials) of two matrices $A$ and $B$ with that of matrix $C$ defined as $$ C=\begin{bmatrix} 0 & A \\ B & 0 ...
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2answers
71 views

Showing that the vector space $V$ cannot be generated.

Simply asking how can someone show that the vector space $V$ of all polynomials on a field, say $K$ cannot be generated with any finite set of vectors? I don't know where to tackle the problem. :( ...
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2answers
91 views

Cancelling a direct product of a subspace in isomorphic ambient spaces?

I read today that for a vector space $V$ and any subspace $S$, all complements of $S$ in $V$ are isomorphic to $V/S$, and thus to each other. I want to ask, is there a case where $V\cong W$ are ...
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2answers
251 views

Projection matrix equation

I am trying to solve this old exam question: Show that $P = A(A^TA)^{-1}A^T$ is a projection matrix if $A = \begin{bmatrix}1 \\ m\end{bmatrix}$. I don't understand what I'm doing wrong here: $P = ...
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1answer
112 views

Matrix solving problem

Hi there math experts. I have the following matrix: $$ \begin{equation} \begin{pmatrix} -1x & 0y & 0 & 0 & 0.004 & 0\\ -1x & -1y & 0 & 0 & 0.001 & 0 ...
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3answers
2k views

Eigenvalue problem: Prove that all of the eigenvalues of $A$ are $1$.

Here's a cute problem that was frequently given by the late Herbert Wilf during his talks. Problem: Let $A$ be an $n \times n$ matrix with entries from $\{0,1\}$ having all positive eigenvalues. ...
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2answers
4k views

Minimal polynomials and characteristic polynomials

I am trying to understand the similarities and differences between the minimal polynomial and characteristic polynomial of Matrices. When are the minimal polynomial and characteristic polynomial the ...
2
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1answer
434 views

bound on eigenvalues of sum of positive-semidefinites matrix

I have 2 positive-semidefinites matrix A and B. I know that A+B is also a positive-semidefinites matrix. I need to prove that the maximal eigenvalue of A+B is bounded by the sum of the maximal eigen ...
4
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1answer
518 views

Why are hyperbolic toral automorphisms (e.g. Arnold's cat map) ergodic?

Let $\varphi : \mathbb{T}^2 \to \mathbb{T}^2$ be a hyperbolic automorphism of the torus, induced by a linear map $A : \mathbb{R}^2 \to \mathbb{R}^2$ of determinant $\pm 1$ with no eigenvalues of ...
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284 views

Linear transformation of a polygon maximizing its area with respect to its perimeter.

Given a polygon $P$ on the plane, is there a rigorous method or algorithm to compute or approximate a linear transformation $T$ which maximizes the following ratio? ...
0
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1answer
219 views

Zeroes in a 3x3 Matrix Determinant

My professor found the cubic roots of a 3x3 matrix by doing the following. I don't understand how step 2 came about and why he applied the same for step 4 on row 1 instead of row 2. Step 1: ...
2
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2answers
115 views

Eigen-values of $AB$ and $BA$?

Let $A,B \in M(n,\mathbb{C})$ be two $n\times n$ matrices. I would like know how to prove that eigen-value of $AB$ is the same as the eigen-values of $BA$.
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2answers
205 views

Common eigenvalue of two matrices

When matrix $A$ and $B$ have a common eigenvalue, is it true that the matrix $A - B$ will have the eigenvalue $0$?
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1answer
778 views

How to find integer solutions for an equation? [duplicate]

Possible Duplicate: Pythagorean Triplets with “Bounds” I typed following equations in into Wolfram Alpha - $$a^2 + b^2 = c^2\\ a+b+c=1000 $$ It showed me multiple possible integer ...
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3answers
59 views

Quesion about creation of subspace with some properties

Let $V$ be a vector space with finite dimension and $K, H$ are subspaces of $V$. Prove that there is subspace $M$ of $V$ s.t $M+K=M+H$ and $M\cap K=M\cap H=\{0\}$.
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1answer
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Finding the Dimension of a Matrix Polynomial: $W$ = { $p(B)$ : $p$ is a polynomial with real coefficients}

Let $W = \{ p(B) : p \text{ is a polynomial with real coefficients}\}$, where $$B= \begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0 \end{pmatrix}$$ The dimension $d$ of the ...
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0answers
27 views

Find a basis such that two linear transforms can be diagonalized at the same time [duplicate]

Possible Duplicate: Simultaneous diagonalization Assume $V$ is a n-dimension vector space over a number field $\Bbb{K}$, $\mathscr{A}$ and $\mathscr{B}$ are two linear transforms in $V$, ...
9
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2answers
277 views

Finding the order of an element in $GL(2,\mathbb{R})$

I am working on a problem involving basic abstract algebra/group theory and am getting confused. I am following an online course by Dr. Bob found here, and am currently on assignment two. My ...
17
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5answers
3k views

How can you explain the Singular Value Decomposition to Non-specialists?

I am giving a presentation in two days about a search engine I have been making the past summer, and my research involved the use of singular value decompositions, or in other words, $A=U\Sigma V^T$. ...
2
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2answers
147 views

Complex Eigenvectors

This is a relatively simple question and I've Google'd this topic but I can't seem to grasp the method. One site I visited helped be a bit, but it simply used substitution to solve the problem rather ...
4
votes
2answers
1k views

Show that the direct sum of a kernel of a projection and its image create the originating vector space.

I got the following question as my homework. Given $V$ is a vector space with $P \in \operatorname{End} V$. $P \circ P = P$ ("P is idempotent"). Show that $V = \operatorname{Ker} P \oplus ...
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1answer
69 views

Using the inequality of Cauchy-Schwarz in Euclidean space $\mathbb{R}^3$

Using the inequality of Cauchy-Schwarz in Euclidean space $\mathbb{R}^3$(with the usual product), show that if $a,b,c \in \mathbb{R}^+$ different from zero, then:$$9 \leq ...
2
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3answers
801 views

What's the rule for solving nested sums?

I have the following nested double sum : $\sum_{t=1}^{T}\sum_{u=t}^{T} Z(u) \cdot (1+i)^{-u}$ with $0<i<1$ and $Z(u)$ being a non-specified function. By working with an example of $T=3$ I ...
2
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5answers
5k views

Find the rotation axis and angle of a matrix

$$A=\frac{1}{9} \begin{pmatrix} -7 & 4 & 4\\ 4 & -1 & 8\\ 4 & 8 & -1 \end{pmatrix}$$ How do I prove that A is a rotation ? How do I find the rotation axis and the rotation ...
2
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1answer
100 views

Orthonormal basis question

Let $U⊆ℝ^4$ a subspace given by the equations: $$x_2+2x_3=0 \mbox{ and }x_2+x_4=0$$ Find an orthonormal basis of $U^⊥$. I think these two vectors are in $U^⊥$: $(0,1,2,0)^T,(0,1,0,1)^T$ How ...
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1answer
277 views

which are diagonalizable over $\mathbb{C}$

a) Unitary matrices are normal matrix hence diagonalizable as a consequence of spectral theorem b)same as a) c)No idea.but I think it may not be diagonalizable unless it has one eigen value with ...
0
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1answer
175 views

Help with understanding the general formula for the determinant? [duplicate]

Possible Duplicate: What’s an intuitive way to think about the determinant? Could anyone give an intuitive explanation of the determinant? I know mostly what the determinant means and I can ...
12
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3answers
1k views

What's the Clifford algebra?

I'm reading a book on Clifford algebra for physicists. I don't quite understand it conceptually even if I can do most algebraic manipulations. Can some-one teach me what the Clifford algebra really ...
3
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1answer
117 views

$T:\mathbb R^2 \rightarrow \mathbb R^2$ is a linear transformation such that $Ty=\alpha x$ and $Tx=0.$

I was thinking about the following problem: Let $x,y$ be linearly independent vectors in $\mathbb R^2$. Suppose,$T:\mathbb R^2 \rightarrow \mathbb R^2$ is a linear transformation such that $Ty=\alpha ...
3
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2answers
86 views

Dimension of null space of linear map?

Suppose V is the real vector space of real-valued functions with a derivative. What is the dimension of the null space of the linear map $$Tf = x\frac{df}{dx} - 4f\;\;?$$ The basis I found for the ...
2
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2answers
70 views

n×n matrices A with complex enteries

Let U be set of all n×n matrices A with complex enteries s.t. A is unitary. then U as a topological subspace of $\mathbb{C^{n^{2}}} $ is compact but not connected. connected but not compact. ...
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1answer
68 views

A matrix $A = (a_{ij})_{n\times n}$ such that $a_{ij} = 0$ for $i>j$ and $a_{ij} = 1$ for $i=1\dots n$. Multiple choice question about the inverse

Consider a matrix$ A = (a_{ij})_{n\times n}$ with integer entries such that $a_{ij} = 0$ for $i>j$ and $a_{ii} = 1$ for $i=1\dots n$. Which of the following properties must be true? $A^{-1}$ ...
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2answers
154 views

Linear Subspace/vector subspace

I need to prove that this is a linear subspace: $$x-y+2z=0$$ $$x+y+4z=0$$ How am I going to do this?
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2answers
97 views

A question concerning unipotent matrices and a basis choice

Let $G$ be a subgroup of $\mathbb{GL}_n$, the group of all invertible $n\times n$ matrices over an algebraically closed field $k$, which consists of unipotent matrices. I don't understand a step in ...
2
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3answers
116 views

What kind of software can be used to solves systems of equations?

For example, I have to solve the following equations: $$\left\{\begin{align*} &x^2 + y^2 + z^2 = 1\\ &Ax + By + Cz = 0 \end{align*}\right.$$ for $y$ and $z$, where $x$, $A$, $B$ and $C$ are ...