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Questions tagged [linear-algebra]

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 concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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Similarity of Jordan Canonical Form

Show that if $A\in \mathbb{F}^{n×n}$ such that $\sigma (A)=\lbrace 1\rbrace$, then $A$ is similar to $A^k$ for all positive integer $k$. Show that the converse is not true. My attempt: Let $A$ has ...
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minimum eigenvalue of sub rows of a projection matrix

Consider a matrix $P = I - X (X^\top X)^{-1} X^\top$ where $X$ has dimension $n \times s$ and $s < n$. Clearly $\lambda_{\min}(P) = 0$. But can we say something about the minimum eigenvalue on the ...
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Why is the second coefficient of the characteristic polynomial for a 3x3 matrix the sum of cofactors?

In another post here someone mentioned that the second coefficient of the characteristic polynomial is the sum of cofactors. https://math.stackexchange.com/a/1721776/665392 Can you give me an ...
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13 views

Imaginary part of a complex matrix

Let $A\in\mathbb{R}^{n\times n}$ be a diagonal matrix with positive diagonals and $U\in\mathbb{C}^{n\times n}$ be a unitary matrix, i.e., $U^{*}U=I$, where $U^{*}$ is the complex conjugate of $U$. ...
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6 views

Methods of verifying the Smith Normal Form of a matrix

I have an algebra exam in a few weeks and, if the past papers are anything to go by, it seems likely that there will be a question on finding the Smith normal form of a 4x4 matrix with entries in $\...
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8 views

How to quantify the 'completeness' of a basis set?

As far as I know, a set of functions/vectors is complete or it is not. In many fields incomplete sets are used to approximate functions/vectors. I understand that given two sets of different ...
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11 views

Minimization problem with matrices

I am trying to solve a constrained minimization problem involving matrix calculations, which leads to a well known result in the financial literature. Let $s$ be a random variable taking values in $\...
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1answer
8 views

Quotient algebras of nilpotent Lie algebra are nilpotent

For the following proposition I found a proof in some notes that I don't understand. Below definition 1 defines the terminology I'm using, and proof attempt 1 gives my attempt at the proposition. ...
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1answer
23 views

One Quaternion two different 4x4 matrix representations and a same result just multiplying unit quaternions.

Working with different papers I found out two ways on writing Quaternions as 4X4 matrices by "two ways" I'm trying to say that the signs on the coefficients of the matrices are a little bit different ...
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1answer
18 views

Any trick to re-write in vector form $\sum_{i=1}^n -\ln(1 + \eta_i x_i) $, where $x, \eta \in \mathbb{R}^n$

Sorry, for a stupid question probably. Is there any trick to re-write this in vector form $$\sum_{i=1}^n -\ln(1 + \eta_i x_i) $$ where $x, \eta \in \mathbb{R}^n$?
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1answer
35 views

Diagonizability of $n \times n$ matrix with two distinct eigenvalues and eigenspace dimension $n-1$

Let $\mathbf{A}$ be an $n \times n$ matrix with two distinct eigenvalues $\lambda_1$ and $\lambda_2$. If the dimension of the eigenspace $E(\lambda_1)$ is $n-1$, then $\mathbf{A}$ is diagonizable. ...
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1answer
33 views

How to remember n modulo m operations better?

In Linear Algebra, we need to solve linear equation systems in fields $\mathbb{Z}_n$. As an example: In the field $\mathbb{Z}_6:2 \odot 3\equiv 2\cdot3\mod6\equiv 0$ My question is now, how can I ...
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1answer
16 views

Chebyshev center of a polyhedron: nonnegativity issue

Let us have a polyhedron, defined by the inequalities of the form: $$ \mathcal{P} = \{ x \ | \ a_i^T x \leq b_i, \ i=1,\ldots,m \} $$ Here on page 19, the way to calculate Chebyshev center is given ...
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18 views

Determinant as sum of cofactors

I have just gone through the proof that by summing the product of the elements and cofactors along any row of a matrix, you obtain the determinant. I understand the mechanism of the proof, but could ...
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28 views

If I'm looking for a basis, how do I know how many vectors it is supposed to have?

Let's say I have a matrix over entity Z(3,3). Now I understand that in order for a set of vectors to be a basis they have to: be linearly independent and generate the whole vector space (all that ...
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0answers
25 views

Scaling eigenvalues in eigendecomposition of Markov generator matrix

We start with a generator matrix for a continuous-time Markov chain $G$ with the usual properties of a generator matrix: \begin{align} g_{ii} &\leq 0 \\ g_{ij}&\geq 0 \quad \text{ for $i \neq ...
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split the set of Eigenvalues

I have a question on how to split a set of eigenvalues. I have a set of $n$ eigenvalues denoted by $\lambda_{m}$ for $=0...n-1$ given as $$ \lambda_{m}=1 -\dfrac{1}{K} \left( \dfrac{\sin(\frac{m \...
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10 views

Linear programming/seeing feasibility and unboudedness

Consider the dual linear programming problem and its simplex dually feasible table: $$\begin{array}{|c|c|c|c|c|c|c|c|}\hline -4& 0 & 1&5&16&0&4&0 \\ \hline -12& 0 &...
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19 views

Find an orthogonal matrix that diagonalises the matrix A

a) Let v1 = (1, 1, 0) and v2 = (1, 0, 1). If the eigenvalue of v1 is 2, what is the eigenvalue of v2? I found the answer to this by proving that these eigenvectors are not orthogonal by computing ...
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Working with nonsquare matrices

Given that square matrices are rare in real world data, are there methods of "squaring" a nonsquare matrix so that we can analyze the data with square matrix properties? What would some resources/...
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3answers
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How to find the smallest degree polynomial from a family of polynomials?

If I take a family of polynomials, like this one : $$\mathcal B = ( t+t^2, t^3, t^2 + t^4) $$ I'm searching for proving that the smallest degree polynomial generates the rest of polynomials (as in ...
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20 views

show this identity quadratic form

if $n$ be even number,and $x_{1},x_{2},\cdots,x_{n}$ be real numbers. show this identity $$\sum_{1\le i<j\le n}\min(|i-j|,n-|i-j|)x_{i}x_{j} =\sum_{j=1}^{\frac{n}{2}}(x_{j}+x_{j+1}+\cdots+...
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1answer
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Example of a bilinear form of abelian groups

Let $X$ and $Y$ be abelian groups. Then, a $Y$-valued bilinear form on $X$ is a $\mathbb{Z}$-module homomorphism $$\alpha: X \otimes_{\mathbb{Z}} X \rightarrow Y$$ How does this relate to the standard ...
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2answers
51 views

Why do Linear Transformations Take a circle to an ellipse

Short Version: How can it be intuitively shown that non-singular 2D linear transformations take circles to ellipses? (Also, its probably important to state I'd prefer an explanation that doesn't use ...
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34 views

Summation and Product sum of series

what is the answer of $$\sum_{m=1}^B\Pi_{n=1}^A \frac{m+n}{mn}=?$$ What I tried $$\sum_{m=1}^B\Pi_{n=1}^A (\frac{1}{m} +\frac{1}{n})$$ $$\sum_{m=1}^B[(\frac{1}{m} +\frac{1}{1})(\frac{1}{m} +\frac{...
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1answer
18 views

Standard form for the Characteristic Matrix/Polynomial

I'm currently taking Linear Algebra and Differential Equations, and in talking about eigenvalues of a matrix, both professors have given the same information: for some square n x n matrix A, the ...
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0answers
9 views

Tensor products of modules over non-commutative rings

I've been learning about tensor products over modules, but where the ring acting on the module is commutative. When $R$ is non-commutative, we consider a right $R$-module $M$ and a left $R$-module $N$...
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19 views

At loss on how to tackle this problem about linear tranformations and vector spaces

I am given this problem: https://imgur.com/a/tWg7F0A I am unsure on how to even begin solving this problem, and I am also clueless as to what the asterisk (*) is representing. Could anyone give me ...
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1answer
31 views

Determinants - adding one row to another

If we think of the determinant of a matrix as the magnitude of the space enclosed by its columns (NOT by its rows), then what's the geometric interpretation of this property: "Adding one row of a ...
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Schur complement in LUP decomposition of a block tridiagonal matrix

Section 2.2 of the article On twisted factorizations of block tridiagonal matrices explains how to do a LUP decomposition of block tridiagonal matrices by showing the process on a 4 blocks by 4 blocks ...
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0answers
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Finding lower triangular $L$ such that $A_n\cdots A_1 = I - XLX^T$, where $X=[x_1\cdots x_n]$, $A_i = I - 2x_i x_i^T$

Problem Given $A_i = I-2x_i x_i^T$ such that $\Vert x_i \Vert_2 = 1$, find the lower triangular matrix $L$ such that $A_n\cdots A_1 = I - XLX^T$, where $X=[x_1\cdots x_n]$ Try For $A_2A_1 = (I-...
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3answers
44 views

Why we never use the product between vectors like between elements of direct groups product?

We can say, that any field $\mathbb{K}$ -- $1$-dim vector space on itself: $\mathbb{K}_{\mathbb{K}}$. So any vector of one another finite-dimensional vector space $V_{\mathbb{K}}$, after choosing the ...
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0answers
5 views

Finding vertices of the intersection of a polytope and a hyperplane along with bounds on coordinates.

I am given a set of inequalities $v_1\ge v_2\ge \cdots\ge v_n\ge 0$, $\sum_{i=1}^n p_iv_i\le q$, and only one bound for the coordinates: $v_1\le a_1$, where $a_1\ge 0$. My question is: What are ...
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1answer
21 views

Rotate a Point on an ellipse by an angle and calculate the distance between them

I am given a starting point $S=(s_1,s_2)^T$, the Center Point $C=(c_1,c_2)^T$ of the ellipse, with major radius $a$ and minor radius $b$, also the major axis is rotated w.r.t. the X axis by an angle ...
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1answer
20 views

Show that $\min_X \|X\|_2 = \frac{1}{\sigma_1}$

Given $ A\in \mathbb{C}^{m\times n}$ with $\sigma_1$ as biggest singular value, and $\det(I-AX) = 0$ where $ X\in \mathbb{C}^{n\times m} $, can you show that $$\min_X \|X\|_2 = \frac 1{\sigma_1}\:?$$ ...
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1answer
25 views

Derivative of matrix as a function of a vector w.r.t a vector

I want to compute the derivative of the matrix $ diag(x)M $ with respect to $ x $, where $ x \in \mathcal{C}^{n \times 1} $ and $ M \in \mathcal{C}^{n \times m} $. This is how I have approached it, ...
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Relationship between the projection matrices from the column spaces of submatrices

Problem Say we have the matrix $X_{n\times p} = \left[X_1 \vert X_2\right]$, where $X_1 : n\times p_1$, $X_2 : n \times p_2$ with $p = p_1 + p_2$. Assume $p<n$, and $X$ has full rank, i.e. $\...
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25 views

Linear algebra: dimension of a subspace

A payoff vector $x \in \mathbb{R}^{K}$ at time one is said to be marketed if there exists a portfolio $\Theta \in \mathbb{R}^{N}$ such that $\Sigma'\Theta=x$ and there exists some $w \in \mathbb{R}$ ...
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2answers
44 views

Geometric Multiplicity of Eigenvalues

Let $A\in\mathbb{F}^{n×n}$ and define $L: \mathbb{F}^{n×n}\rightarrow \mathbb{F}^{n×n}$ by $L(X)=AX$. If $\lambda$ is an eigenvalue of $A$ with geometric multiplicity 1, show that the geometric ...
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1answer
27 views

How do the values of $a$ affect definiteness of this matrix $\textbf{A}$?

Let $$\textbf{A} = \begin{bmatrix} -2 & 0 & 1 \\ 0 & -2 & a \\ 1 & a & -2\end{bmatrix}$$ Given that one of its eigenvalues is equal to $-2$, how does its definiteness vary ...
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How to show for positive semi-definite, there exist $x \in \mathbb{R}^n$ such that $Bx+c = 0$ and and $\|x\| \leq \Delta$?

Let $B \in \mathbb{R}^{n \times n}$ be symmetric and positive semi-definite such that $B = U\Lambda U^T$, where $U = [u_1,\cdots,u_n]^T$ is an orthogonal matrix with $u_i \in \mathbb{R}^n$, and $\...
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1answer
28 views

Symmetric matrix diagonalizable

How can we get an invertible matrix $W \in$ $\mathbb{R}$$^{3\times3}$ to make $W\hspace{-1mm}AW^{T}$ a diagonal matrix? Here, $$ A = \begin{bmatrix} 2 & 1 & 3 \\ 1 & 0 ...
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22 views

Finding a differential of a matrix function

I am trying to find a differential of the following function: $f(x)=a^t*X*B$, where B is an $n{\times}n$ matrix with the Frobenius norm (I got no idea, how this information could assist me), $ a\in \...
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0answers
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Reference on the $GL_n$ representation theory of $A = Sym ( \oplus_{i = 1}^m V^*)^{\Sigma_m}$.

Let $V$ be the defining representation of $GL_2(\mathbb{R})$. Let $\Sigma_m$ be the permutation group, acting on $ \oplus_{i = 1}^m V^*$ by permuting coordinates, and let $GL_2(\mathbb{R})$ act by the ...
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2answers
20 views

Characterize the sum of subspaces

Let $U,V$ be the following subspaces of $\mathbb R^3$ $U= \{(a,b,c) \in \mathbb R^3,a+b+c=0 \}$ $V=\{(a,b,c) \in \mathbb R^3,a=c \} $ $W=\{(0,0,c)\in \mathbb R^3\}$ Carachterize : a)$U+V$ I ...
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1answer
31 views

Non similar to orthogonal Matrix

Give an example of a 3-by-3 matrix with determinant 1 which is not similar to an orthogonal matrix. I know that a matrix is orthogonal if $AA^T=I$ and a matrix $A$ is similar to a matrix $B$ if there ...
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25 views

Prove that a normal linear map whose only eigenvectors are 0, 1 is an orthogonal projection

The question is as stated above. I proved the complex case with the spectral theorem but I'm not sure how to deal with the real case. It suffices to show $T^{2} = T$ but I am not sure how to do this. ...
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2answers
47 views

Summation of ${\sum_{k=1}^n k\binom{n}{k} = n2^{n-1}}$ using equations

I was trying to solve the summation: $${\sum_{k=1}^n k\binom{n}{k} = n2^{n-1}}$$ I started something like: $${\sum_{k=0}^n \binom{n}{k} = 2^n}$$ $${\Rightarrow \sum_{k=1}^n \binom{n}{k} = 2^n - 2^...
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1answer
52 views

When can you uniquely determine a square matrix if you know the sum of its rows and columns?

There is a square matrix consisted of only $0, 1, i$. You know the sum of all the numbers in each row and each column. When can you uniquely determine the matrix? Edit: For clarification, there is a ...
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0answers
16 views

How to show that $\text{tr}\left(A^T B\right) \le \|A\|_{op} \|B\|_1$?

Let $A$ and $B$ be two matrices. How can I prove that $$ \text{tr}\left(A^T B\right) \le \|A\|_{op} \|B\|_1$$ where $\|\cdot\|_{op}$ is the largest singular value of $A$, and $\|B\|_1$ is the sum of ...