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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Algebraic subspaces

How do I prove that $U=\{(x,y,z)|x\text{ is an integer}\}$ is not a subspace of $\mathbb{R}^3$? I understand that I have to show $U$ is closed or not closed under vector addition and scalar ...
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20 views

The minimal poly of $T_w$ divides the minimal poly for $T$

I'm stuck on proving the following theorem. Let $W$ be an invariant subspace for $T$. The minimal polynomial for $T_W$ divides the minimal polynomial for $T$. We have \begin{equation} A ...
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1answer
21 views

Is taking the real part required in vector orthogonality and projection?

In a real inner product space, two vectors are orthogonal if $\langle \mathbf{u}, \mathbf{v} \rangle = 0$. Similarly, $$\operatorname{proj}_{\mathbf{u}}(\mathbf{v}) = ...
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39 views

Separating vectors from linear combination

Suppose I have a linear combination of vectors as follows $ \mathbf{s} = \alpha_1\mathbf{x}_1 + \dots + \alpha_m\mathbf{x}_m + \beta_1\mathbf{y}_1 + \dots + \beta_n\mathbf{y}_n $ where $\alpha_i, ...
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59 views

Properties of matrices $M=UDU^*$ with $UU^*=Id$

I recently came across some matrices of the form $M=UDU^*$ (the superscript $*$ denotes the conjugate transpose), where $U \in \mathbb{C}^{r\times n}$ with $r<n$, $D \in \mathbb{C}^{n \times n}$ a ...
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1answer
19 views

Is it possible to find a companion matrix of a polynomial which is also hermitian?

The eigenvalues of a square matrix $A$ coincide with the roots of its characteristic polynomial $p[A]$. Conversely, if I have a polynomial $$ a_0 + a_1 x + \cdots + a_{n-1}x^{n-1} + x^n ~, $$ I can ...
2
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0answers
28 views

Newton's method on a surface

I am trying to use Newton's method to find the stationary solutions of the integro-differential equation of the form $$\frac{\partial u(r,t)}{\partial t} = -u(r,t) + \int_{\mathbb{R}^{2}}w(r - ...
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29 views

Create solid torus with geometric algebra

I have created an algorithm for tracking a vessel's centerline in 3 dimensions, and traveling through that centerline. My supervisor asked me whether I could add a small theoretical section on ...
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44 views

Frobenius automorphism and non-split Cartan subgroup of $\mathrm{GL}_2(\mathbb{F}_p)$

For completeness I give some definitions. Let $p$ a prime number and consider $V$ a 2-dimensional $\mathbb{F}_p$-vector space. Consider $k$ a sub-algebra of $\mathrm{End}(V)$ that is a field of $p^2$ ...
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32 views

Does injective imply each $x$ matches to a unique $y$?

Injective means one-to-one matching, as in each $y$ is matched by only one $x$. However, does this mean that each $x$ matches only to one $y$?
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53 views

Computing $\operatorname{Tr} \bigl( \bigl( (A+I )^{-1} \bigr)^2\bigr)$

Suppose that $A \in \mathbb{R}^{n \times n}$ is a symmetric positive semi-definite matrix such that $\operatorname{Tr}(A)\le n$. I want a lower bound on the following quantity $$\operatorname{Tr} ...
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25 views

Computing eigenvalue of the adjacency matrix of a path

Let $A\in \{0,1\}^{n \times n}$ be the adjacency matrix of a path of length $n$, i.e. having ones on the two off-diagonals, and zeros elsewhere. How does one compute the eigenvalues of this? I know ...
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18 views

How Kriging, Bochner theorem and Positive definite (PD) function are related?

This question referes to the link: https://en.wikipedia.org/wiki/Kriging I can understand the relation between Bochner's theorem and PD function. But could not properly understand and connect all ...
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1answer
42 views

How to Write a Vector Multiplication as a Trace of Matrix?

Let $\mathbf{w}_j\in\mathbb{C}^{M\times 1}$ and $\mathbf{h}_k\in\mathbb{C}^{M\times 1}$ be two complex vectors. How to prove this? ...
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97 views

compute the bisecting normal hyperplane between two $n$-dimensional points.

I have two points $\mathbf{x_1}$ and $\mathbf{x_2}$, where $\mathbf{x_i}=\{x^i_1, x^i_2, \ldots, x^i_n\}$. I need to find a normal hyperplane $P$ that goes through the midpoint of $\mathbf{x_1}$ and ...
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14 views

Determinant of a Certain 3 by 3 Block Matrix with Scaled Identity Blocks

What is the determinant or/and eigenvalues of the following 3 by 3 block matrix: $$\left[\begin{array}{ccc} \frac{3}{4}I & \frac{1}{4}I & \frac{1}{4}I \\ \frac{1}{4}I & \frac{3}{4}I & ...
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How can I intuitively interpret this vector operation?

In reading through some very old source code that I inherited and came across a three-dimensional Euclidean vector operation that I can't seem to gain an intuition for. Transcribing the program code ...
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Eigenvalue of Block matrix: Adjacency of complete bipartite Graph

Let $A\in \{0,1\}^{mn \times mn}$ be the adjacency matrix of a complete bipartite graph with $m$ and $n$ vertices each, i.e. let $A$ be the matrix consisting of two blocks $A_1\in \{0,1\}^{m \times ...
4
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1answer
683 views

Inverse of Symmetric Matrix Plus Diagonal Matrix if Square Matrix's Inverse Is Known

Let $A$ be an $n \times n$ symmetric matrix of rank $n$ with known inverse $A^{-1}$. Let $D$ be a diagonal matrix with the same dimensions and rank. What is the fastest way to compute $(A+D)^{-1}$? ...
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32 views

Simultaneously vanishing quadratic forms?

Given a set of Hermitian matrices $\{A_i\}$, is there a simple way to check if there exists a vector $c$ such that for all $i$: $$c^* A_i c = 0?$$ Namely, when can the quadratic forms defined by the ...
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34 views

Linear combination of basis function in logarithm space. Is it possible?

I have a function $f(x)$. As theory said that it can represent by linear combination of basis functions such as $$f(x)=\sum_{i=1}^{N}\alpha_ig_i(x)$$ where $\alpha$ is coefficient and $g(.)$ is basis ...
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37 views

Maximization of sum of functions

Let $w,a\in R^n$, and $B\in R^{n\times n}_{++}$ (the set of $n\times n$ positive definite matrices). We know that the following function (which is a specific form of the Rayleigh quotient) has a ...
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60 views

Minimum matching convolution

Let $\text{SPD}^n$ and $\text{PD}^n$ be the symmetric semi-positive and positive definite matrices in $\mathbb{R}^{n\times n}$, respectively. I want to find an $X\in \textrm{SPD}^n$ that minimizes ...
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68 views

Jordan canonical form of an upper triangular matrix

Find the Jordan canonical form of the matrix. Justify your answer. $A=\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 4 \end{bmatrix} $ My Try: The eigenvalues ...
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282 views

Dual norm of the matrix $L^1$ norm is infinity norm (and vice versa)

Recall that for a given norm $\|\cdot\|$ on $\mathbb{R}^n$, the dual norm is defined as a function $\|\cdot\|_*: \mathbb{R}^n \rightarrow \mathbb{R}$ with: $\|y\|_* = \max \limits_x \{x^Ty: \|x\|\le1 ...
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32 views

Dimension of $\left(\lambda |\psi\rangle \langle\psi| +(1-\lambda)\frac{\mathrm{I}}{2}\right)^{\otimes N}$

I have the $N$-fold tensor product of a convex combination of a pure state, i.e. $|\psi\rangle\langle\psi|$ with $|\psi\rangle$ a unit vector in a complex Hilbert space of dimension two, and the ...
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1answer
35 views

How to solve the convex combination problem of matrix?

Let $A \succeq B$ denote matrix $A-B$ is positive semidefinite, and here is the definition of redundant(all the matrix dimensions are $N\times N$ ): Given a set of matrix $\{B_i\}_{i=1}^{l}$, if ...
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31 views

Can this be expressed in terms of linear constraints?

I'm attempting to find a matrix $X$ that minimizes some function $f(X)$ subject to the constraint that $$ X=W A Z $$ where $A$ is a given non-negative matrix with rows that sum to 1, and $W$ and $Z$ ...
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55 views

System of equations - What's wrong with my solution?

The system of equations below can be solved by substitution or elimination. I understand the official solution to this problem, which I will provide below. I'd like to understand why my initial ...
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Simultaneous Equation - no solution and many solutions [closed]

$$mx + 3y = 2,\\ 12x = my = 2m - 8$$ Find values for $m$ which there are a) no solutions b) infinitely many solutions
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71 views

Best way to solve specific block-tridiagonal linear system (10000x10000 and larger)

To provide more context, this system came from energy balance equation on a mesh with (n,m) nodes in each direction. It's a linear system that looks like this (size of system in blocks n = 4, size of ...
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1answer
281 views

Symmetric positive definite with respect to an inner product

Let $A$ be a SPD(symmetric positive-definite) real $n\times n$ matrix. let $B=LL^T$ be also SPD. Let $(,)_B$ be an inner product given by $(x,y)_B=x \cdot By=y^T Bx$. Then ...
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1answer
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How to find the point in convex set $C$ that is closest to $y\notin C$?

How to find the point in convex set $C$ that is closest to $y\notin C$? $C=\{ x\in \mathbb{R^2}:(x_1-1)^2+(x_2-1)^2\le1 \}$ and let $y\notin C $ but $y\notin \mathbb{R^2} $.
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2answers
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Point-Slope Equation of a line. Why is one answer incorrect and other is correct?

I am reviewing basic algebra. I am using quiz from the link this, and I solved the equation on paper and I get answer which is showing incorrect, I do not understand why is it wrong? It says my ...
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1answer
27 views

What condition do I have set to have $x_j$ for $j=1,…,n$ to be non-negative?

I have $\sum_{j=1}^n q_{jj}(x_j-y_j)^2\le1$ What condition do I have set to have $x_j$ for $j=1,...,n$ to be non-negative? The book I am reading says $\sqrt{q_{jj}}\ge1/y_j$ but why? edit: ...
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Why the SVD is named so…

The SVD stands for Singular Value Decomposition. After decomposing a data matrix $\mathbf X$ using SVD, it results in three matrices, two matrices with the singular vectors $\mathbf U$ and $\mathbf ...
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39 views

Polynomial Interpolation and Data Integrity

This question is about polynomial interpolation and security. Please consider a scenario where we have a polynomial $f$, one of whose roots is $a$. We evaluate it at some ...
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1answer
125 views

Matrix Inequality

Suppose $A$ is positive-definite, symmetric and real. Show that for $u,v\in\mathbb{R}^{n}$ that $$2|Au\cdot v|\leq\rho^{\frac{1}{2}}(u\cdot Au+|v|^{2}),$$ where $\rho$ is the maximal eigenvalue of ...
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3answers
35 views

Changing the subject of a formula

How do i make the subject of the formula $$A=\frac{m}{n}+\frac{n}{k}$$ into k? Also, does anyone know a website that solves this type of question for you? Mathway works fine, though I don't have ...
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1answer
26 views

inner product of positive semi definite symmetric matrices [on hold]

I have a positive semi definite symmetric matrix $X$, $(n\times n)$. let $X=vv^T$ s.t $\|v\|=1$. I came to a point where I am stuck to show which is: $v^TYv=\langle X,Y\rangle$ (How to show this ...
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For what $p$ is the condition number of a given matrix $A$, using the $p$ norm on matrices, minimal? [duplicate]

For what p is the condition number of a given matrix $A$, using the p norm on matrices, minimal? The condition number on $A$ is given as: $$K(A,p) = \|A\|_p \times \|A^{-1}\|_p$$ I tried ...
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Why is this a useful way to prove the characterisation of bases?

So I have come across the following theorem(see below)in my linear algebra notes and am slightly confused. I feel I understand both the statement and the proof, my confusion arises from the fact that ...
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How to calculate power series with a large norm matrix.

Take this matrix representing the Markov map for seeing a Heads followed by a Tails in a sequence of fair coin flips: $$M = \begin{pmatrix} \frac{1}{2} & \frac{1}{2} & 0 \\ \frac{1}{2} & ...
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1answer
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Root Locus Diagrams - “Breakaway Point”

Say that we have a root locus diagram with n poles and m zeroes. And we determine that the root locus on the real axis lies between two of these poles and breaks away from the real axis and tends to ...
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3answers
62 views
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What is Homogeneous Coordinates? Why is it necessary in 2D transformation?

What is Homogeneous Coordinates? Why is it necessary in 2D transformation of objects in computer graphics? The concept of homogeneous coordinates in effect converts the 2D system a 3D one. So, why ...
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1answer
764 views

Find the line in $\mathbb{R}^3$ that passes through the point $(1,2,-3)$ and is parallel to the vector $u=(4,-5,1)$.

Find a vector equation and parametric equation of the line in $\mathbb{R}^3$ that passes through the point $(1,2,-3)$ and is parallel to the vector $u=(4,-5,1)$. Find two points on the line that are ...
2
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1answer
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Affinity of lorentz transformations

Lorentz transformations are often defined to be linear. But suppose instead we only consider transformations that preserve the spacetime interval. Is it possible to prove that those transformations ...
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1answer
29 views

Let $W$ be a subspace of $\mathbb{R}^n$ spanned by $n$ non-zero orthogonal vectors. Show that $W=\mathbb{R}^n$.

Let $W$ be a subspace of $\mathbb{R}^n$ spanned by $n$ non-zero orthogonal vectors. Show that $W=\mathbb{R}^n$. My approach: Suppose span$\{u_1,\dots,u_n\}=W$, where$\{u_1,\dots,u_n\}$ is a set ...
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1answer
525 views

Equivalent systems of Linear equation

I've just begun to re-learn linear algebra because is so important, the book that I chose is naturally the Hoffman's for a lot of reason. Well, In the first chapter I'm stuck with the following, ...
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Hyperplane in a complex vector space

This is my first question on MSE, I'm sorry if there already exists similar questions, I couldn't manage to find it. My friend, who studies Physics, asked me about the meaning of "functional" so I ...