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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Determining whether a given matrix is regular

The definition of a regular matrix if some power of the matrix gives a matrix with all positive entries then it is said to be regular. Is there a way of finding this "power" without having to go on a ...
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2answers
102 views

Matrix decomposition definition

Wikipedia says "In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different ...
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1answer
58 views

Prove positive definiteness

I want to prove that the matrix $$\begin{pmatrix} 1 &\cfrac{1}{2} &\cfrac{1}{3} &\cdots &\cfrac{1}{n} \\ \cfrac{1}{2} &\cfrac{1}{3} &\cfrac{1}{4} &\cdots ...
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1answer
16 views

How to find the norm of the image of the vector $v$ = $(-3, 5, 1)$ under the map $B \circ A$?

Let $A$, $B$ be two linear maps $\mathbb R \to \mathbb R$, corresponding to rotations by $30$ degree around x-axis, and rotation by $22\frac {1}{2}$ degree around z-axis respectively. what is the ...
6
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2answers
74 views

How prove or disprove is positive define matrix?

prove or disprove following matrix is positive definite matrix ? $$\begin{bmatrix} \dfrac{\sin(a_1-a_1)}{a_1-a_1}&\cdots&\dfrac{\sin(a_1-a_n)}{a_1-a_n}\\ \vdots& &\vdots\\ ...
4
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3answers
106 views

Eigenvalues of an $n\times n$ symmetric matrix

So I had a student come to me with a question which followed as such: Given $\{c_1 , c_2 , c_3, \dots, c_n \}$ are real numbers, form the matrix A whose entries are given by $a_{ij} = c_i \cdot c_j $ ...
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3answers
59 views

Show that $\lim_{n\rightarrow\infty} A ^{n} = B$

Lin alg final coming up, getting the study sesh in Fun fact:just learned latex so yay The full problem comes in two parts. Part A is simple/ Given $A$= \begin{pmatrix} 2 & -3/2 \\ 1 & ...
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0answers
41 views

Updating the determinant of a matrix

Suppose I have the determinant of a matrix $A$, whose entries are polynomials. If I set one entry in the matrix equal to $0$, what is the fastest way to recompute the determinant? Do I have to ...
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1answer
122 views

Question About the Solutions to the Eight Queens Problem [closed]

How is $a_{15}n_8e_9k_5f_{10}d_7b_4m_6$ a solution to the Eight Queens problem? J. W. L. Glaisher, On the Problem of the Eight Queens, Philosophical Magazine, 1847 says that each one of these terms ...
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1answer
29 views

How do I prove the following equality properly?

In a linear algebra assessment, I had to show that $W$ is subspace of $\mathbb{R}^3$ for: $$W = \left \{ (x,y,z) \in \mathbb{R}^{3} \mid \frac{x}{3} = y = 2z \right \}$$ I showed that it is a ...
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1answer
265 views

Relation between rank of a symmetric positive semi-definite matrix and its number of non-zero eigen values (or singular values)

Is there any relation between the rank of a symmetric positive semi-definite matrix and its number of non-zero eigenvalues (or singular values)? For a matrix $\mathbb{P}$ Can we find the ...
2
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2answers
34 views

Motivations for Shi-Malik Algorithm

So I've been trying to make sense of the clustering algorithm on page 6 of this paper. Are the "first" k eigenvalues they refer to the smallest eigenvalues? What are the $y_i$ exactly? I don't see ...
3
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1answer
75 views

tangent/normal space to set of symmetric isospectral matrices

Let $\Lambda = \{\lambda_1, \ldots, \lambda_n\}$ be a set of $n$ distinct real numbers. $M_n(\mathbb{R})$ denotes the set of all $n \times n$ real matrices, and for $B\in M_n(\mathbb{R})$, $B^T$ ...
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2answers
35 views

Orthonormal basis, decompse then add back

This is kind of a stupid question and I am taking some risk of getting some down-votes here, but, I can't resist posting it. Suppose $(u_1, u_2)$ is an orthonormal basis for $R^2$, and let $x$ be an ...
5
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1answer
55 views

How to prove that $A^{-1} + B^{-1}$ is invertible given the conditions

If $A$ and $B$ be two invertible $n \times n$ real matrices and $A + B$ is invertible, how to prove that $A^{-1} + B^{-1}$ is also invertible?
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(Unitary) diagonalization of $A = I-xy^*$

I'm continuing to prepare for a Linear Algebra exam and found another problem that puzzles me. Let $A = I+xy^*$, where $x,y \in \mathbb{C}^m (\neq 0)$. (a) Determine a necessary and ...
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2answers
27 views

Dimension of sub-spaces

Given U,W Sub-spaces of linear space V. 1.If $dim(U \cap W)= dimU$ then $U \subseteq W$ I tried to prove it like that: if $dim(U \cap W)= dimU$ then $(U \cap W)= U$ and because $U \subseteq U ...
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1answer
249 views

find the gradient of trace of the matrix

Prove that $\nabla_A Tr(AA^T) = 2A$, where A is any square matrix I did simple derivative with product rule,but i don't know where i messed up, I started with $\frac{\partial}{\partial A} ...
2
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0answers
28 views

Sub-set of Rows and Columns

Given, K and T are sub-sets of Vector space V. Denote K as row sub-sets, and T column subsets of matrix A (mxn). Denote $K_1$ as row sub-sets, and $T_1$ column subsets of row equivalent matrix A ...
2
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2answers
79 views

Zeroes of polynomial

$$c_1,c_2 \text{ are polynomial's }g(x)=x^2+ax+b \text{ roots } \Leftrightarrow \begin{cases} g(c_1)=c_1^2+ac_1+b=0 \\ g(c_2)=c_2^2+ac_2+b=0 \end{cases}$$ Prove that for every polynomial with integer ...
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1answer
44 views

Vector space and Unions.

Quick question, Given K and T are subsets of vector space V. If $V = Sp(K) + Sp(T)$, Does it mean that $K \cup T$ is a basis of V? I proved it that way: If $V = Sp(K) + Sp(T)$, Then $Sp(K) + ...
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26 views

direct sum of sub sets

K and T are subsets of Linear space V. If $ K \cap T = \varnothing$ Then, the sum of $Sp(K) + Sp(T)$ is a direct sum. Well, the answer says its wrong and i'm trying to prove it. I said that:$$ K ...
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1answer
55 views

Solving two variables system equations with parameter above $\mathbb{Z}_7$

Let: $$(1+5a)x +y = 1$$ $$a^2x + y = 2$$ Eliminating the $y$ variable we have: $$(-a^2 +5a +1)x = 6$$ Now, I should have find $y$ such that $(-a^2 +5a +1)y = 1$, but obvoiusly I can't do that ...
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1answer
24 views

converting equation from vector to matrixes

I have an equation to calculate the values of a $3\times 1$ vector from another vector: $$ \left( {\begin{array}{c}y\\c_b\\c_r\\\end{array} } \right) = \left( {\begin{array}{ccc} a_1 & a_2 ...
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1answer
183 views

how to solve this problem of liner equation in two variable

in NCERT Exampler, pg: 34, question: 10, A railway half ticket costs half the full fare, but the reservation charges are the same on a half ticket as on a full ticket. One reserved first class ...
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1answer
98 views

Inner Product Space vs. Vector Space

I had no trouble understanding what a vector space is: a constraint on the type of vectors you can create, such that certain operations could be performed with them. For example, a vector space of ...
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1answer
25 views

Linear Algebra (Matrix) [duplicate]

An n×n matrix M of integers has the property that Mab=Mxy iff a+b=x+y. How many distinct elements are there in M?
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1answer
264 views

Incremental Calculation of the Sample Covariance

The formula to calculate the sample covariance given $n$ vector samples $x_{i}$ for $i = 1, \ldots, n$ is as follows: \begin{align*} S &= \frac{1}{n-1}\sum\limits_{i=1}^{n}(x_{i} - m)(x_{i} - ...
12
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1answer
338 views

A sharper bound for $\|\cos(kA)\|_{\infty}$ for symmetric stochastic matrices

Given $A \in \mathbb{R}^{n \times n}$ that is symmetric, stochastic and diagonalizable, and $k \in \mathbb{N}$, I am interested in bounding $\|\cos(kA)\|_{\infty}$ from above. $\| \|_{\infty}$ is ...
2
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1answer
117 views

Prove that matrices have equal rank.

If $P$ and $Q$ are $n \times n$ matrices of real numbers such that $P^2=P$ and $Q^2=Q$ and $I-P-Q$ is invertible where $I$ is an $n \times n$ identity matrix, Show that $P$ and $Q$ have the ...
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0answers
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Banachewicz identity

Please, help solve this question: I need to factorize/simplify this matrix, a.k.a. Banachewicz identity: Given the partitioned matrix \begin{equation} P=\left( \begin{array} {c,c} A \quad B \\ C ...
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3answers
117 views

Calculating a root of a complex number with euler formula

Let $z= 1+i$. The polar form of $z^{1/5}$ can be easily calculated: $$z = 2^{1 \over 10}\left\{\cos({\pi/4 + 2\pi k \over 5}) + i\cdot \sin({\pi/4 + 2\pi k \over 5})\right\}$$ the "principal root" ...
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1answer
39 views

Linear Algebra Question.

An $n \times n$ matrix M of integers has the property that $M_{ab} = M_{xy}$ iff $a+b=x+y$. How many distinct elements are there in $M$?
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votes
1answer
164 views

How prove this $P(A+B)\leq P(A) +P(B)$

Assume $A$ and $B$ are real symmetric matrices of order $n$. I denote the number of the positive eigenvalues of matrices $A,B,A+B$ by $P(A),P(B),P(A+B)$ respectively. Show that: $$P(A+B)\leq ...
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1answer
208 views

Prove vectorspace of bounded functions with supremum-norm is complete and no hilbert space

I have the following: Consider the real vectorspace with bounded functions $$V = \{f:[0,1]\rightarrow\mathbb{R} | \exists C > 0 : f([0,1])\subset[-C,C]\}$$ and the supremum-norm $$||f||_\infty ...
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1answer
30 views

Matrices in Linear Algebra

Let: $ u: R^2 --> R^3$ be defined by: $$ u(x,y)=(x+2y, 2x-y, 2x+ 3y)$$ Give the matrix $M[u]$ in the canonical base of its definition space. This question might seem sort of stupid, but it was ...
7
votes
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360 views

What's the Jordan canonical form of this matrix?

given is the $6 \times 6$-matrix $A$: $A = \begin{pmatrix} 0 & 1 & 0 & -1 & 0 & 0 \\ 0 &0&1&1&-1&0\\ -1&0&0&0&-1&-1 \\ 1 & ...
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1answer
391 views

Householder vs Gram–Schmidt Orthogonalization. Which should I use?

When should I use them, and how the relation of speed and precision changes? Which are the advantages, and disadvantages of them?
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0answers
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Interior Products

Over on the Wiki page for interior products: http://en.wikipedia.org/wiki/Interior_product There is a line that says $\iota_X \alpha = \alpha(X) = \langle \alpha,X \rangle$ where $\alpha$ is a ...
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3answers
73 views

Proof help, span and invertible matrix

Let $A$ be an $n \times n$ invertible matrix and $v_1,v_2,...,v_m$ an element of $\mathbb{R}^{n}$. Prove if $\{v_1,v_2,...,v_m\}$ spans $\mathbb{R}^{n}$ then $\{Av_1, Av_2,...,Av_m\}$ also ...
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1answer
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Find a linearly dependent subset

I need to find a linearly dependent subset of R3 in which no two vectors are scalar multiples of each other, however I thought for a set to be linearly dependent at least two vectors had to be scalar ...
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1answer
41 views

Can a subset of $\mathbb{R}^3$ span $\mathbb{R}^2$?

Does the $\operatorname{span} \{(1, 0, 0), (0, 1, 0)\}= \mathbb{R}^2?$ I was told the span of this set has dimension $2$ but what is the exact span?
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Two subsets with the same span?

I need to find two subsets of $\mathbb{R}^3$ whose spans equal each other but their intersection is the empty set. I was thinking $v_1=\{(1,0,0)\}$ and $v_2=\{(0,1,0)\}$ but I'm not sure..would this ...
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Francis Algorithm (Implicit QR Algorithm)

In Numerical Analysis, we are touching upon QR and Francis Algorithm. I understand that for Francis's Algorithm, we reduce the matrix to its upper Hessenberg form using Householder transform. What I ...
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2answers
53 views

show that $\det(A)=0$ in this case

(a) Let $x$ and $y$ be $n\times 1$ matrices, $n \ge 1$, and let $A=xy^T$. Show that $\det(A)=0$. (b) Explain why the statment in part (a) is false if $n=1$.
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1answer
261 views

Solving Cubic Equations Using Origami

I have to write a research paper on a mathematical topic for my class; I chose the above topic. I understand that a parabola can be formed using a focus and directrix, both created by origami folds, ...
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1answer
32 views

Need to check the meaning of a Transition matrix

Is the transition matrix just the change of basis matrix from a non-standard basis to the standard basis?
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77 views

Determine Cross Product with Left Hand vs Right Hand

If I perceive http://en.wikipedia.org/wiki/Cross_product correctly, then to determine the cross product With a right hand, let: the 1st vector in the cross product = your index finger = in red ...
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1answer
75 views

Scalar Matrix and Diag Matrix

A set of all diagonal matrices (nxn) over R is a field relatively to additive and multiply operations on matrices. A set of all scalar matrices (nxn) over C is a field relatively to additive and ...
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1answer
34 views

Matrix inequality: conjugating positive matrix by $R<-I$

Consider a symmetric positive definite matrix $P$ and arbitrary matrix $R<-I$. Does the following inequality hold? $$ P < RPR^T $$ If yes, provide some references. If no, guide me under what ...