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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Is the maximum of the eigenvalues of any symmetric positive?

Let $A$ be a symmetric matrix having dimension $n \times n \;, \; \; n\geq 2$. If one wants to pick the maximum of its eigenvalues, will the value be positive? Suppose A was an adjacency matrix, ...
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Evaluating a limit involving the power of specially structured matrix

Let $k\times k$ right-stochastic matrix $A$ be defined as follows: $$A=\left[\begin{array}{cccccc} p & 0 & 0 & \cdots & 0 & 1-p\\ 1 & 0 & 0 & \cdots & 0 & 0 \\ ...
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Finding a solution vector to linear system of equations with lowest hamming weight efficiently

I'm trying to solve a linear system of equations modulo 2. After performing gaussian elimination, I can get a solution of the form $v + c_1 \cdot n_1 + c_2 \cdot n_2 + \cdots + c_k \cdot n_k, c_i ...
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1answer
14 views

Is a symmetric diagonal matrix in which every entry is non-negative positive semidefinite?

Let $A$ be a symmetric diagonal matrix in which $(A)_{ii} \geq 0$. Should one conclude that this matrix is positive semidefinite?
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32 views

Linear Systems: Exponentials of a Matrix

I have a rather odd question to some, but one that has stumped me for a good few minutes on a homework assignment that states: For each matrix, find the eigenvalues of $\text{exp}{(A)}$, ...
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6 views

Properties of Linear Transformations

The book I am using is Differential Equations and Dynamical Systems by Lawrence Parko. Seeking to confirm my attempt at proving the following. Use the lemma in this section to show that if $T$ ...
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10 views

Linear Systems and Linear Transformation

I want to confirm my attempt to see if I am on the right track. The question is as follows. Show that the operator norm of a inear transformation $T$ on $\mathbb{R}^n$ satisfies ...
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1answer
53 views

What exactly is an integral kernel?

I am not sure if I have seen integral transforms in the right way, but given a transform like fourier transform - its actually a basis transformation right ? $$ F(y) = \int K(x,y) f(x) \text{d}x $$ ...
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1answer
26 views

How to solve the following system of equation?

Given two equations: $2mx+6y =1$ and $4x -(1-m)y = -16$ Find the value of $m$, such that the system has no solution? My attempt: From the first: $2mx = 1-6y$ Then $x= (1-6y)/(2m)$ Substitute ...
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2answers
33 views

Real Linear vs. Complex Linear

I recently started a new math course and got hung up on a particular problem from the book "Linear Algebra Done Wrong". Specifically, problem 1.3.6 (c). I am an engineer, and I believe I simply lack ...
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0answers
43 views

sign of eigenvalue [on hold]

Let L is a linear operator, we say it is stable if it has negative spectrum, why it is equivalent with there exists some $‎\varepsilon‎>0$ such that $$\langle Lh,h \rangle ...
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1answer
30 views

Confusion regarding dimension of a vector space

The dimension of a vector space is the number of elements in the basis for that vector space. If we look at $\mathbb R^n$, then we say that the dimension of $\mathbb R^n$ is $n$. So every element in ...
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Finding the Requested Value of Composition Functions?

I seem to still be having difficulty with understanding Composition Functions in Calculus and wondered if someone might be able to help explain in a way that will make the light bulb "turn on"? For ...
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2answers
37 views

Matrices such that $\det(p(A)-p(0))=p(\det A)-p(0)$ for all polynomial $p$

Question: Find all $2\times 2$ matrix $A$ such that $$\det(p(A)-p(0))=p(\det A)-p(0)$$ for all polynomial $p$. The zero matrix works since both sides are obviously zero. But I cannot find any ...
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4answers
320 views

is it true every left inverse of a matrix is also right inverse of it?

I am wondering that, consider there are $m$ linear equations with $n$ unknowns. We can represent it as $AX=B$. Let $L$ is the left inverse of $A$ therefore $LA=I$. Again from $AX=B$, we get $LAX=LB$ ...
2
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1answer
9 views

Maximum of a Rayleigh quotient with non-Euclidean inner product

It's well known that, for a real and symmetric matrix $A$, $$ \max_v \frac { (Av,v) } {(v,v)} = \lambda_{\max}(A). $$ Now I'm looking at generalized Rayleigh quotients of the form $$ R = \max_v ...
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1answer
16 views

Simple exercise regarding space spanned by two vectors

Consider the vectors $X_1=(1,3,2)$ and $X_2=(-2,4,3)$ in $\mathbb R^3$. Show that the set spanned by $X_1$ and $X_2$ is given by $\{(Y_1,Y_2,Y_3):Y_1-7Y_2+10Y_3=0\}$.
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If w is a 2-vector in char not 2, such that $w \wedge w \not = 0$, why is w decomposable?

I want to see this as directly as possible, since I want to understand the variety of lines in P^3. I don't want to use the general convolution formula (given in Shaferevichs Varieties book). Any ...
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1answer
36 views

3 equations 4 unknowns.

I need to know what does this system describes please: $x+3t=2$ $y+t=-1$ $z+2t=1 $ Now subtracting equation (1) from (2) and (3) i.e (1)-(2)-(3) leads to $x-y-z=2$ hence the set of points of the ...
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0answers
27 views

Connection between Dirichlet series and integration?

For quiet sometime I've been working on an idea of mine (I think its not discovered at least, within my googling capabilities). I think I found a connection between the Dirichlet series and ...
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2answers
33 views

What is the matrix $\left[ DS(A) \right]$, which gives $\left[ DS(A) \right] H=AH+HA$?

In Hubbard's multivariable calculus book $DS(A):H \mapsto AH+HA$ is introduced as a linear transformation where $A$ is an $n \times n$ matrix, $S(A)=A^2$, and $D$ is the notation for derivative. It ...
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1answer
18 views

Linear Operators and Convergence

I am struggling with this question from my Differential Equations course: If $T$ is a linear transformation on $\mathbb{R}^n$ with $||T-I||<1$, prove that $T$ is invertible and that the ...
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1answer
17 views

Nonsingularity of submatrices

I'm still working on my question: Warm start of simplex algorithm after update of constraint matrix. While reading Schrijver's book "Theory of linear and integer programming" (reprint 1999) I ...
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1answer
12 views

Check if the equality holds

I have the following problem. For orthogonal $8\times 8$ matrix $M$ ($M\cdot M^{T} = 1$) check if the following equality holds $$ U = M^{T} \cdot \left( \begin{array}{cc} 1_{3\times 3} & 0\\ ...
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1answer
14 views

Riesz representative of gradient of $f(u) = u^*u$ in different inner products

This is a seeming "paradox" that has been bothering me for some time, as it (or other situations like it) show up often when computing gradients for numerical optimization on complex vector spaces. ...
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3answers
24 views

Checking whether the result is positive definite or positive semi-definite with two methods

Given, $$A = \begin{bmatrix} 1 &1 & 1\\ 1&1 & 1\\ 1& 1& 1 \end{bmatrix}.$$ I want to see if the matrix $A$ positive (negative) (semi-) definite. Using Method 1: ...
3
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0answers
17 views

Operator Norm of a Linear Transformation of a Matrix

The book I am using for the ODE course is Differential Equations and Dynamical Systems by Lawrence Perko. I am having a difficult time understanding what an operator norm of a linear transformation ...
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1answer
30 views

Prove that for a quadratic form $Q(x) = x^TAx$, $A$ is symmetric.

We want to show that $A=A^T$. Any matrix $A$ may be written as a sum of its symmetric and skew symmetric parts: $A = \frac{A+A^T}{2} + \frac{A-A^T}{2}$. Upon substitution, we obtain $Q(x) = x^T ...
2
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2answers
45 views

What is the good way to remember the signs of the rotational matrix?

Recall rotational matrix in (x,y) is given by: $R = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}$ For the life of me I cannot remember if the ...
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Transformations of Quadratic Forms to their Normal Forms

Assume we are given a quadratic form in one variable, $Q(x) = ax^2$, where $a \in \mathbb{C}$. We may construct a change of coordinates $x = cX$, where $c \in \mathbb{C}$ and is of our choosing. Upon ...
2
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4answers
32 views

Is the matrix $A$ positive (negative) (semi-) definite?

Given, $$A = \begin{bmatrix} 2 &-1 & -1\\ -1&2 & -1\\ -1& -1& 2 \end{bmatrix}.$$ I want to see if the matrix $A$ positive (negative) (semi-) definite. Define the ...
2
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2answers
15 views

volume of a $n$-d parallelepiped with sides given by the row vectors of a matrix $A$ is the product of the singular values of this matrix $A$

Why the volume of a $n$-dimensional parallelepiped with sides given by the row vectors of a matrix $A$ can be seen as the product of the singular values of this matrix $A$? I only know in ...
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Matrices and Linear Algebra- Determine if the list is linearly independent in the real vector space. [on hold]

1.Determine if the list $((3,2,0,1),\,(2,1,4,0),\,(0,-1,12,-2))$ is linearly independent in the real vector space $\mathbb R^4$. 2.In the real vector space $C(\mathbb R,\mathbb R)$ of all continuous ...
3
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2answers
69 views

Showing an equation has one positive root

Let $n\geq 2$ be an integer and $\beta > 0$. Consider the polynomial equation: $$p(x) = x^n + x^{n-1} - \beta = 0$$ Show the equation had exactly one positive root $p(\beta)$ Do I use the ...
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1answer
35 views

Balancing chemical equations using linear algebraic methods

I know there are already plenty of questions on this site regarding this topic but I am having difficulty with a particular chemical equation. I am trying to balance the following: $$ { C }_{ 2 }{ H ...
3
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1answer
32 views

Then prove that $\exists U$ subspace of $V$ such that $W_i \oplus U=V$ $\forall i=1,…,n$

Let $V$ be finite dimensional vector space over an infinite field. Let $W_1, W_2,...,W_k$ are subspaces of $V$ of same dimension. Then prove that $\exists U$ subspace of $V$ such that $W_i \oplus U=V$ ...
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0answers
45 views

How many $2\times3$ real matrices are needed to guarantee that at least one of them is a linear combinations of the others?

The only thing I know is that $$\left(\begin{array}{ccc}1&0&1\\0&1&1\end{array}\right)$$ Seems to have a column to be linear combinations of the others.
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0answers
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Linear Algebra help needed [on hold]

It's been a while since I've taken linear algebra and I am trying to figure out the problem below. I am not sure how to start. Thank you for any and all help. I am not sure how to For any real ...
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0answers
27 views

How to prove the following

Let $\mathbf{A}\in\mathbb{R}^{p\times n} (n\ge p)$ be a positive definite symmetric matrix having a Wishart distribution with mean $\mathbf{0}$ and covariance $\boldsymbol\Sigma\otimes \mathbf{I}$. ...
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2answers
23 views

Proving the existence of an inverse of a matrix. (Linear algebra)

Suppose that $A$ has no inverse. Prove that there exists a vector $b$ such that $Ax = b$ has no solution My try Proving by contradiction , Assume that for all vector $b$, $Ax = b$ have at least one ...
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1answer
46 views

Proof for linearity on tensor products

Theorem: Let $U$ and $V$ be vector spaces. Let $\mathbf{u}^* \in U^*$. Define $\mathbf{f} : U \otimes V \to V$: $$\mathbf{f}\left(\sum_{r} \mathbf{u}_r \otimes \mathbf{v}_r\right) = \sum_{r} ...
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1answer
14 views

Choleski decomposition of a positive matrix

Let us consider a matrix $\boldsymbol{F}$. We consider its Choleski decomposition, $ \boldsymbol{F} = \boldsymbol{M} \boldsymbol{M}^T $. We know that $\boldsymbol{F}$ needs to be positive definite. ...
3
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1answer
42 views

Determinant structure of symplectic matrix

I want to show that if $\lambda$ is a real eigenvalue of a symplectic matrix $A$ then its char poly is of the form $\det(A-\mu id) = (\lambda-\mu)(\frac{1}{\lambda}-\mu) \det(\hat{A}- \mu id) $ where ...
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3answers
28 views

Determining the intersection of kernel and image.

I was posed the following question: If $T$ is a linear operator on a finite dimensional vector space $V$ such that rank of $T$ = rank of $T^2$. I'm supposed to show that the kernel and image of $T$ ...
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0answers
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I need to write equations for 2 rates of change. [on hold]

here is my problem, Write the following as an equation. x / y 1 / 3 2 / 12 3 / 27 4 / 36 5 / 51
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0answers
19 views

Finding a variable in the determinant of sum of matrices [on hold]

I Don't Know how to earn P that is scalar from Below Formula : R = log2(abs(det(I + P * H*H'))) Everything is known except P. P is scalar and positive. I is an Identity NxN , H is complex NxN ...
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1answer
63 views

$(1)$ $a\boxplus b=a+b-ab$ for all $a,b \in F$. $(2)$ $a\boxdot b=1-t^{\log_t{(1-a)}\log_t{(1-b)}}$ for all $a,b\in F$

Let $1<t\in \mathbb{R}$ and let $F=\{a\in \mathbb{R}: a<1\}$. Define $\boxplus$ and $\boxdot$ on $F$ as follows: $a \boxplus b=a+b-ab$ for all $a,b \in F$. $a \boxdot b=1-t^{\log_t ...
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1answer
29 views

Measuring the effect of a linear transformation on the result

I have an unknown vector $x\in\mathbb{R}^n$, a known orthogonal matrix $\Phi\in\mathbb{R}^{n\times n}$, a known matrix $A\in \mathbb{R}^{m\times n} (m \le n)$, and a known vector $b\in \mathbb{R}^m$ ...
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2answers
17 views

eigenvalues of cycle graph and its complement graph

I am trying to find the eigenvalue of cycle graph and its complement. How to simplify.Suppose $\omega^{1}+\omega^{n-1}=2cos (2\pi/n) $, then, $\omega^{\frac{n-1}{2}}+\omega^{\frac{n+1}{2}}=?$ Is it ...
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0answers
21 views

Need help about field decision; mathematics or physics? Who can be good at these? [on hold]

First of all, you may want to delete this question because it is not an mathematical question, but this question can be an opening door to thousands of mathematical question. Hello everybody, I need ...