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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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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0answers
18 views

Matrix polynomial

Suppose: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is a ...
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1answer
13 views

Is it possible to determine if a matrix is not diagonalizable via row operations?

Suppose a matrix can be row reduced to the identity matrix, is this enough to say that it is not diagonalizable? If so, what theorem(s) or logic figures this out?
3
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2answers
15 views

Equivalent definitions of an orthogonal matrix.

I wish to show that the following definitions of an $n \times n$ real matrix $Q$ are equivalent: $QQ^T=I$, $Qx\cdot Qx=x\cdot x$ for all $x\in \mathbb{R}^n$. I found it easy to show that ...
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1answer
54 views
+50

Solving a matrix equation using numerical optimization

To my knowledge, if $A \in \mathbf{S}^n_{++}$, then given any $b \in \mathbb{R}^n$, the system of linear equations $Ax = b$ has a unique solution $x^* \in \mathbb{R}^n$. Moreover, the solution $x^* ...
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2answers
44 views

Which $n$-forms are pullbacks of top forms on $\Bbb R^n$

Let $V$ be a finite-dimensional vector space. I write $F_n(V)$ for the $n$th exterior power of the dual vector space. Which elements of $F_n(V)$ can be pulled back from a top form along a linear ...
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1answer
19 views

Determinant inequality and positive definite matrix

Let $B$ and $C$ be $n\times n$ hermitian matrices, with $B$ positive definite and $C$ positive semi-definite. (1) Show that $B+C$ is positive definite (2) Show that $\det(B)\leq \det(B+C)$. What ...
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13 views

E as an expectation of a quadratic form [on hold]

if E(expectation of quadratic form) is an operator, show that E(AB+C) = AEB + EC. where b and c are variables.
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10 views

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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12 views

Let T be the bounded operator and T* be the adjoint operator of T.Show the following. [on hold]

Let T be the bounded operator and T* be the adjoint operator of T. Show the following. 1.||TT||=||T|| 2.||TT||=||TT*||=||T||^2 3.(T+S)=T+S* 4.(αT)=α ̄T (α∈C) 5.(TS)=S T* 6.(T* )* =T
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2answers
40 views

Showing that the set of $2 \times 2$ real orthogonal matrices has a particular parameterization

Theorem Every orthogonal matrix in $\mathbb{R}^{2, 2}$ is in the form \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} or \begin{bmatrix} \cos\theta ...
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1answer
22 views

Simple question - represent vector with respect to a basis

Basic question here, I've always been weak at this stuff. Suppose that we have a situation $U=WX$ where $U,W,X$ are matrices that are known to us. You can suppose that $U$ is invertible. I want to ...
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0answers
15 views

System of Equations for 3-digit number [on hold]

This is a rare word problem where I've had trouble: Find system of equations and use elimination. The sum of the digits of a three-digit number is 9, and the tens digit of the number is twice the ...
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2answers
947 views

Gauss-Seidel method convergence algorithm

From Wikipedia: The convergence properties of the Gauss–Seidel method are dependent on the matrix A. Namely, the procedure is known to converge if either: ...
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1answer
33 views
+50

Asymptotic series of a matrix-valued function.

Consider the following matrix $$f(\lambda)=\left( \frac{\lambda-1}{\lambda + 1} \right)^{\nu \sigma_3} \ \ \ \lambda \in \mathbb{C} \setminus [-1,1]$$ where $\sigma_3=\begin{pmatrix} 1 & 0 \\ 0 ...
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1answer
52 views

A question about matrix algebras

Let $A,B \in M_n$, $n \geq 2$. If $A$ and $B$ do not share a common eigenvector, why is $\mathcal{A}(A,B) = M_n$? Notation and definitions: $M_n$: the set of $n \times n$ matrices over ...
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1answer
85 views

When and why can functions “take on” the role of vectors in defining vector spaces?

In what I call "advanced" linear algebra, we examine the properties of vectors in a vector space like an inner product space by checking that they satisfy e.g. the Cauchy-Schwarz inequality, the ...
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3answers
39 views

Inverse of partitioned matrices [on hold]

A matrix of the form $$A=\begin{bmatrix} A_{11} & A_{12}\\ 0 & A_{22} \end{bmatrix}$$ is said to be block upper triangular. Assume that $A_{11}$ is $p \times p$, $A_{22}$ is $q \times q$ and ...
3
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1answer
329 views

Vector Project onto Subspace

So the question is: Let S be the subspace of $\mathbb{R}^3$ spanned by the vectors $ u_2 = \begin{pmatrix} \frac{2}{3}\\\frac{2}{3}\\\frac{1}{3}\end{pmatrix} u_3 = \begin{pmatrix} ...
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1answer
45 views

Verify the following assertion:

Suppose that $U=\{(x,x,y,y)\in F^4:x,y\in F\}$ and $W=\{(x,x,x,y)\in F^4:x,y\in F\}$.Then $$U+W=\{(x,x,y,z)\in F^4:x,y,z\in F\}.$$ Not sure how to add these subsets. Please provide explanation.
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1answer
28 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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8answers
139 views

Show that $B$ is invertible if $B=A^2-2A+2I$ and $A^3=2I$

If $A$ is $40\times 40$ matrix such that $A^3=2I$ show that $B$ is invertible where $B=A^2-2A+2I$. I tried to evaluate $B(A-I)$ , $B(A+I)$ , $B(A-2I)$ ... but I couldn't find anything.
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0answers
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How can I find $\det(A)/\det(B)$, when individual determinants blow up

I am interested in the quantity: $\frac{\det(A)}{\det(B)}$ of positive definite matrices $A$ and $B$. The problem I am running into now is that for large $A$,$B$, (around $200 \times 200$), the ...
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0answers
20 views

How can I tell if two lines will cross using vectors [on hold]

things I need 1. a visual recpinatation 2. A explanation on how to solve it and 3. a problem to do
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0answers
17 views

Maximization of sum of convex 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) is concave ...
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0answers
27 views

Don't understand about how to solve the first equation with Gauss Elimination

Please take a look at the picture. My quesion is: How to solve the first equation with Gauss Elimination that is displayed in the picture. I don't understand HOW.
2
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0answers
24 views

Eigenvalues of the product of two “incidence” matrices

I am trying to solve the following problem. Let the following incidence matrix of an undirected graph with four nodes $$ B = \begin{bmatrix}1 & 0 & -1 & 0 & 0 \\ -1 & 1 & 0 ...
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24 views

Linear independent vectors

I am stuck in an argument. Is the following true: Let $\{v_1,\ldots,v_k\}$ and $\{u_1,\ldots,u_k\}$ be sets of linear independent vectors. Set $u:=\sum_{i=1}^k \alpha_i u_i$, for some $\alpha_i$. Now ...
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1answer
22 views

On the projection onto the image set of an $m\times n$ matrix

I came accross as statement that: "If $K$ is the image set of an $m\times n$ matrix $A$ with full column rank, then $$P_Kx=A(A^TA)^{-1}A^Tx."$$ How do I verify this? I know that the inequality ...
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5answers
718 views

Show that if $T_1$, $T_2$ are normal operators that commute then $T_1+T_2$ and $T_1T_2$ are normal.

Let $V$ be a finite dimensional inner-product space, and suppose that $T_1$, $T_2$ are normal operators on $V$ that commute. How to show that $T_1+T_2$ and $T_1T_2$ are then normal? It is clear if ...
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2answers
352 views

If $V$ and $W$ are subspaces of the same dimension such that $V$ meets $W^\perp$, then $W$ meets $V^\perp$

I'm having a little difficulty understanding the proof for orthogonal complements. I kind of understand orthogonal complements, but I cannot seem to find a logic to this. I'm trying to follow along ...
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0answers
11 views

Determining dimension of a sum of subspaces in terms of a parameter

Problem: Consider the linear subspaces \begin{align*} U = \text{span} \left\{ (1,0,1,0), (1,a,0,a)\right\} \quad \text{and} \quad W = \text{span}\left\{(-1, a, a^2, 0), (0,1,0,-1)\right\} \end{align*} ...
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2answers
29 views

Solving system of nonlinear equations

Say I have a system of 4 equations, 4 unknown (A,B,C,D), how would you solve it analytically, assuming a, b, C1, C2, C3, C4, C5, C6, F, G, H, I are just some constants? If using Gaussian Elimination, ...
3
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1answer
23 views

A linear functional on the space of transformations is basis independent

I've been working on this problem for a bit and am not sure how to proceed: let $V$ be an $n$ dimensional $\mathbb{ R } $-vector space, and denote by $\mathcal{L}(V)$ the space of linear operators $V ...
3
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8answers
85 views

For two vectors $a$ and $b$, why does $\cos(θ)$ equal the dot product of $a$ and $b$ divided by the product of the vectors' magnitudes?

While watching a video about dot products (https://www.youtube.com/watch?v=WDdR5s0C4cY), the following formula is presented for finding the angle between two vectors: For vectors $a$, and $b$, ...
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5answers
14k views

Correlation matrix from Covariance matrix

This is for a project which I've been trying to find some information for Covariance matrix and correlation matrix. I understand that for a $n \times n$ matrix $A, AA^T$ will give me the covariance ...
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4answers
56 views

Does an $n\times n$ matrix $A$ only have an inverse if $rank(A)=n$? If so, why?

I'm currently learning about the rank and inverses of matrices, and every time I attempt to find the inverse of a matrix with a rank smaller than it's number of rows, I find I am unable. One example ...
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0answers
14 views

How to compute the Eigenvectors for a Markov matrix?

I have the following matrix for which I want to get the Eigenvectors. I know how to compute the Eigenvalues, but when I compute the vectors in the null space of the matrix, I get the wrong answer. ...
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1answer
37 views

Finding the Jordan form and basis failing

Let $$A = \left(\begin{array}{cccc} 3&4&-1\\0&-2&0\\1&-4&1 \end{array}\right)$$ Find the Jordan form $J$ and $P$ such that $P^{-1}AP = J$. So here's what I did: $f_A(x) = ...
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4answers
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Points on 3d line

Say we have $2$ points on a 3d line, point $A(x,y,z)$ and point $B(x,y,z)$. If we want to get the coordinates of a third point, beyond point $B$ but a certain distance from point $A$, how would we do ...
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3answers
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Eigenvalues of normal matrix

I want to show that $\lambda$ is an eigenvalue of a normal matrix $A$ if and only if $\overline{\lambda}$ is an eigenvalue of $A^{*}$. I am trying to show it for a while and I guess there are some ...
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1answer
64 views

Matrix Algebras: Generator

Problem Given the algebra $\mathcal{M}_\mathbb{C}(2)$. Denote the normals: $$\mathcal{N}:=\{N\in\mathcal{M}_\mathbb{C}(2):N^*N=NN^*\}$$ And their calculus: ...
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2answers
37 views

minimum eigenvalue for difference of two matrices

Let $A$ a symmetric positive definite matrix, and $B$ a matrix constructed from $A$ by setting all its off-diagonal elements to zero. Then is there a way to see for which values of positive scalars ...
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2answers
65 views

Matrix invertibility in terms of elementary operations

There is a theorem that an $n \times n$ matrix $A$ is invertible if and only if $A$ is row equivalent to $I_n$, and in this case, any sequence of elementary row operations that reduces $A$ to $I_n$ ...
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4answers
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The transpose of a permutation matrix is its inverse.

This is a question from the free Harvard online abstract algebra lectures. I'm posting my solutions here to get some feedback on them. For a fuller explanation, see this post. This problem is from ...
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1answer
21 views

Iterative solutions of linear systems

I do not understand that why $M$ must be invertible for $x^{(k+1)}$ to be uniquely specified in equation below: $$ Mx^{(k+1)} = Nx^{(k)} + b \quad (k=0,1,\ldots).$$ Why $M$ must be invertible? And ...
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0answers
57 views

Help me to prove the determinant formula

Actually it is about the question of n-linear function, but it is so relevant to the determinant formula. Here is the notation of the theorem. If $n>1$ and $A$ is an $n \times n$ matrix over $K$, ...
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1answer
27 views

Prove there exists a self-adjoint transformation $C$ s.t. $CA=B$ if $A$ and $AB$ are self adjoint

If $A$ and $B$ are linear transformations such that $A$ and $AB$ are self adjoint and such that $Ker(A)\subset Ker(B)$, then prove there exists a self-adjoint transformation $C$ s.t. $CA=B$
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1answer
31 views

Prove the theorem of ideal (about g.c.d)

If $p_1,\ldots,p_n$ are polynomials over a field $F$, not all of which are $0$, there is a unique monic polynomial $d$ in $F[x]$ such that (a) $d$ is in the ideal generated by $p_1, \ldots, ...
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2answers
19 views

a symmetric bilinear form has a basis such that it's matrix with respect to it is diagonal

I'm reviewing a proof regarding $f$, a symmetric bilinear form having a basis such that it's matrix with respect to this basis is diagonal. Here's a summarization: For $n=1$ there's nothing to ...