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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1answer
99 views

How can I find a matrix $\bf B$, with positive eigenvalues, such that its square $\bf B^2$ is another matrix $\bf A$?

I've been given a 2x2 matrix $\mathbf A$, its eigenvalues $\lambda_1$ and $\lambda_2$, its eigenvectors $\mathbf v_1$ and $\mathbf v_2$, and a diagonal matrix $\mathbf D = \text{diag}(\lambda _1, ...
0
votes
1answer
35 views

Hermitian matrices and their eigenvalues

Let $C=A+B$ where $A$ and $B$ are two hermitian matrices can I prove that $\lambda_{i,C}=\lambda_{i,A}+\lambda_{i,B}$ iff $x_{i,A}=x_{i,B}$? Where $x_i$ is the eigenvector related to eigenvalue ...
2
votes
1answer
116 views

Does $GL(n,K)$ act transitively on $1$-dim subspaces of $K$

If we let $K$ be a field and $GL(n,K)$ act by right multiplication on the $1$-dim subspaces of $K^n$. Then if we take $\langle v_1 \rangle, \ldots \langle v_n \rangle \in K^n$ distinct and $\langle ...
1
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2answers
65 views

If $T$ is a linear transformation on $V$,then which is true?

Let $V$ be a vector space with finite dimension $n$ and $T:V\longrightarrow V$ is a linear transformation such that $T^{2}=0$. Then $rank(T)\leq\frac{n}{2}$ $n(T)\leq\frac{n}{2}$ $rank(T)\geq n(T)$ ...
1
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0answers
72 views

Eigenvalues of sum of two particular matrices

Let $A$ be a matrix with real eigenvalues, its maximum eigenvalue is $0$ and it has sum for rows equals to zero. Let $B$ be a matrix $\mathrm{diag}([1\,0\, ...\, 0])$ and let $I$ be the identity ...
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0answers
36 views

eigenvalues of a symmetric matrix

I diagonalized an arbitrary 3x3 symmetric matrix using two SO matrices at respective angles and the diagonal elements are nothing but eigenvalues. I separately found eigenvalues with the help of ...
0
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0answers
176 views

Homology out of Smith normal form: simultaneous or independent diagonalization?

Let $R$ be a PID and $R^m\overset{A}{\longrightarrow} R^n\overset{B}{\longrightarrow} R^o$ matrices with $BA=0$ and Smith normal forms ...
2
votes
1answer
65 views

general expression for isomorphism of tensor product

(I am still waiting for an answer to the following question. Thank you.) While I was reading postings relating to tensors, I came across the following explanation from Tensors as matrices vs. Tensors ...
0
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2answers
779 views

Find tangent vector to surface given a point on the surface and its normal vector (for a sphere)

I need to know how to find a tangent vector to a point on the surface of a sphere if I am given the point P and the normal vector at that point N. I know that there are many possible tangent vectors ...
1
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4answers
80 views

Linear maps using Tensor Product

While I was reading some posts (Definition of a tensor for a manifold, and Tensors as matrices vs. Tensors as multi-linear maps), I encountered the following explanation: "To give a linear map $V ...
0
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1answer
76 views

If A = BC and B is invertible, then how does reducing “B to I” also reduce “A to C”?

If $A = B*C$, where $B$ is an inverse, use row-ops to reduces "$B$ to $I$" also shows that it will reduce "$A$ .. $C$". Big-Hint: Represent the row operations by a sequence of elementary matrices.
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votes
2answers
92 views

Checking whether an operator is self-adjoint. Problem with domain of an operator.

I want to check whether the position operator $A$, where $Af(x)=xf(x)$ , is self-adjoint. For this to be true it has to be Hermitian and also the domains of it and its adjoint must be equal. The ...
1
vote
1answer
32 views

Adjoint of linear operator is linear.

I'm trying to show that in a unitary space, the adjoint $A^{*}$ of a linear operator $A$ is itself linear. I'm using the definition which defines the adjoint through the inner product. $$ ...
1
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1answer
35 views

Moving between different ellipse representations

I have a representation of an ellipse that is the affine transform of the unit ball, $\|Ax + b\| <= 1$. My question is, how can I change this ellipse representation? I would like to have it in ...
-3
votes
1answer
30 views

Algebra & Re-arrangement [duplicate]

I need to make this: $K(K+1)(2K+7) + 6(k+1)[(k+1)+2]/6$ Equal to this: $(K+1)[(k+1)+1][(2(k+1)+7]/6$ By using algebra and re-arrangement. From the initial equation there should be just one more ...
1
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0answers
39 views

Why is the diagonal part of a matrix,A, a polynomial in A?

I'm sure this is pretty standard but I haven't been able to find it any books. In the Jordan decomposition of a matrix $A=D+N$ into a diagonal, $D$ and nilpotent $N$, why is it that $D= p(A)$ where ...
2
votes
1answer
98 views

Domain when dividing two functions

Let's say we have two function $f(x) = \sqrt{x-3}$ and $g(x) = \sqrt{16-x^2}$, when finding the domain of $\frac{f}{g}$ do you find the domain of $\frac{\sqrt{3-x}}{\sqrt{16-x^2}}$ so that $x$ is an ...
3
votes
1answer
669 views

Dimension of the vector space of homogeneous polynomials

Let $k[X_0, X_1, \ldots, X_n]_d$, or briefly $k[X]_d$, be the $k$-vector space whose elements are the zero polynomial and homogeneous polynomials of degree $d\geq 1$. I found the following formula for ...
1
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1answer
201 views

$2\times 2$ matrices over complex numbers

I am trying to solve this problem. If $A$ is a $2 \times 2$ matrix with complex entries, then $A$ is similar over $\Bbb C$ to a matrix of one of the two types $$ M= \left[ {\begin{array}{cc} a ...
1
vote
1answer
42 views

Finding $\max_{||x||_2=1} \min_i |(Ax)_i|$

Let us define for $x \in \mathbb{R}^n$ $$M(x)=\min_i|x_i|$$ Is there a way to solve the following optimization problem: $$\max_{||x||_2=1}M(Ax)$$ for a given $A$?
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5answers
102 views

multiplying a matrix by a row vector

Is multiplying a matrix by a row vector the same as multiplying it by a column vector? Or are there any differences between the two?
1
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1answer
47 views

$T$, $S$ are lineary dependent $\Leftrightarrow$ $[T]_B$, $[S]_B$ are lineary dependent.

EDIT: I see in the comments that my question is not clear enough so I will explain: if I want to check whether $T$, $S$ are linearly independent or not, I will just pick an easy to work with, basis ...
1
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1answer
7k views

Finding the coordinate vector relative to a given basis

Given the basis $\beta = \{(1,-1,3),(-3,4,9),(2,-2,4)\}$ and $x = (8, -9, 6)$, I am to find the corresponding coordinate vector $[x]_\beta$. I claim that the coordinate vectors entries $x_1,x_2,x_3$ ...
0
votes
2answers
46 views

Distorted Unitary matrices

Let $U$ be an unitary and $D$ be a diagonal matrix. We know that for all vectors $v$ on the sphere $Uv$ is on the sphere and, $$\langle Uv,Uv\rangle=\langle v,v\rangle.$$ What are the vectors $v$ on ...
2
votes
0answers
57 views

Exterior power of a space of maps $(\mathbb{K}^T)$

We are given a set $T \neq \emptyset, \ \ p \ge 1, \ \ p_i : T \rightarrow \mathbb{K}$ Could you help me prove that if $ \phi: (\mathbb{K}^T)^p \ni (f_1, ..., f_p) \rightarrow \rho \in ...
-1
votes
0answers
34 views

when is a symmetric matrix triangularizable?

suppose we have a matrix 4x4 with one variable and values of that variable can be both real complex numbers when is the matrix tringularizable (excluding the cases when it is only diagonilizable)?
3
votes
1answer
270 views

Absolute values in linear programming

Suppose I have an objective function in my LP as follows $max$ $|x|$ Based on some googling, I have found there are two ways to convert this into a standard LP. Method 1. $|x|$ = $ x^+ + x^-$ $x ...
2
votes
0answers
28 views

Motivation behind Binet form and its generalization to arbitrary coefficients

Binet form (http://www.proofwiki.org/wiki/Binet_Form) gives a closed-form solution to $n^{th}$ term in a series with recurrence relation as below $U_{n}=m\cdot U_{m-1}+U_{m-2}$ I have two questions ...
0
votes
1answer
97 views

Explain the convexity by looking the hessian matrix of a function

The hessian matrix of a function is given by, $$ H = \begin{bmatrix} a & b & c \\[0.3em] b & b & 0\\[0.3em] c & 0 & c \end{bmatrix} $$ where, ...
1
vote
4answers
102 views

Reducing the System of linear equations

\begin{align*} x+2y-3z&=4 \\ 3x-y+5z&=2 \\ 4x+y+(k^2-14)z&=k+2 \end{align*} I started doing the matrix of the system: $$ \begin{pmatrix} 1 & 2 & -3 & 4 \\ 3 & -1 & 5 ...
3
votes
2answers
54 views

Determinant algebra

If $A$ and $B$ are $4 \times 4$ matrices with $\det(A) = −2$, $\det(B) = 3$, what is $\det(A+B)$? At first I approached the problem that $\det(A+B) = \det(A) + \det(B)$ but this general rule would ...
0
votes
1answer
27 views

Finding solution in polar form, raised to a power

The problem asks, find an equation equal to: z^3 = ( 1 + sqrt(3)*i ) where i is the square root of negative one. I tried approaching this problem first by ...
2
votes
1answer
46 views

Is it true: $||A||_2 = \min\{ ||A||_1 ,||A||_3,||A||_4,\ldots \ldots, ||A||_{\infty},\|A\|_F\} $?

While running one algorithm , I observed the following peculiar relationship (at-least to me). I am not quite sure whether it is true in general, but I could not succeeded either in producing any ...
0
votes
1answer
52 views

How to determine if two lines in 3D intersect? [closed]

I've seen literally dozens of "line segment" intersect solutions from my trip around the Internet, but that's not ideal for my situation. Given a single point on each line and a vector ...
4
votes
1answer
94 views

Complex square matrix with distinct eigenvalues

Is there a simple way to show that if $A$ is a complex square matrix with distinct eigenvalues ​​then $A$ is similar to a matrix whose all entries are nonzero.
2
votes
1answer
919 views

Show that if $A$ is invertible and $AB = AC$, then $B = C$.

Question: Show that if $A$ is invertible and $AB = AC$, then $B = C$. My work: My thought process: If I can find the inverse of $A$, then I can show A is invertible. I will prove by example. $A$ is ...
1
vote
1answer
33 views

How to get coordinates of some area

I have a rectangle and I divide it into 8 triangle with same size. Top left corner is origin. I want to check that if a point is inside the black area or not. Lets say point's x coordinate is pointX ...
1
vote
3answers
840 views

Show square matrix, then matrix is invertible

Question: Show that if a square matrix A satisfies the equation A^2 + 2A + I = 0, then A must be invertible. My work: Based on the section I read, I will treat I to be an identity matrix, which is a ...
1
vote
2answers
237 views

(generalized) eigenvectors

$\DeclareMathOperator{\rank}{rank}$ First off I'm sorry I'm still not able to make of use the built in formula expressions, I don't have time to learn it now, I'll do it before my next question. I ...
1
vote
1answer
63 views

Question regarding Sheldon Axler's proof that every operator on a complex vector space has an eigenvalue

For reference, Axler's proof was copied in full in this question. In the beginning of that proof, Axler chooses the nonzero vector $v$ without loss of generality. This made me doubt his proof ...
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0answers
258 views

Block Diagonalisation of 4x4 Matrix

I'm attempting to find a 4x4 matrix, P, that will convert my matrices, $A = \begin{bmatrix}1&1&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix}$ and, ...
1
vote
6answers
48 views

If $k$ cannot equal $0$ and $A$ is as given below, what is $A$ inverse?

$$ A = \pmatrix{ 1 & 0 & 2k\\ 0 & 1 & k\\ 0 & 0 & k } $$ I don't know what method to use to solve this problem as I haven't encountered and variable before when solving for ...
0
votes
2answers
52 views

Better way to show Hermitian 2x2 matrix is positive definie

Is there a more elegant way to show that a $2\times2$ matrix $A$ is positive definite when $Tr(A)=1$ and $Tr(A^2) \leq 1$ and $A = A^\dagger$. I find the proof below clumsy and long, and it doesn't ...
1
vote
1answer
142 views

How to find a positive semi-definite linear combination?

Suppose we are given two explicit symmetric matrices $X$ and $Y$ and we'd like to find a non-zero real linear combination $aX+bY$ that is positive semi-definite (if possible). Is there a way to go ...
2
votes
1answer
113 views

Prove that $DT = I_v$, $TD \neq I_v$, where $D$ = differentiation operator and $T$ is integration

Let $V$ be the linear space of all real polys $p(x)$. Let $D$ denote the differentiation operator, and let $T$ the integration operator that maps each polynomial $p$ onto the polynomial $q$ given by ...
0
votes
1answer
41 views

A basis of this vector space?

I am looking for a basis of the set of solutions of $u_{n+2}=u_{n+1}+u_{n}$... Is there some easy basis? I know that all solutions are determined by $u_0, u_1$ but I don't know how to find a basis. ...
0
votes
1answer
38 views

The Geometry of a Linear Transformation

Consider a square matrix of full rank (these assumptions are made for the sake of simplicity). This matrix expresses in coordinates a Linear Mapping that sends the unit sphere to a hyperellipsoid on ...
2
votes
2answers
616 views

Show linear system have no, only one or many solutions

Under what conditions on a and b will the following linear system have no solutions, one solution, infinitely many solutions? first row: 2x − 3y = a second row: 4x − 6y = b My work: I wrote the ...
0
votes
2answers
53 views

$AB=I_{n \times n}$ and $CA=I_{m \times m}$ prove that $m=n$

Let $A$ be an $m \times n$, $AB=I_{n \times n}$, $CA=I_{m \times m}$, prove that $n=m$. Is using inverse matrix is a valid solution?
9
votes
4answers
306 views

$\hom(V,W)$ is canonic isomorph to $\hom(W^*, V^*)$

Introduction My Semester just started and we have a new Professor for Linear Algebra II (replacing our former Professor). Apparently we are behind our schedule and thus we only had a brief ...