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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A question about linear transformations mapping straight lines to straight lines

Obviously, a linear transformation over a space maps all straight lines to other straight lines. My question is: is the converse true? That is, if we're looking at a space after some transformation ...
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80 views

How do I find 2x2 orthonormal diagonalizing matrices using only trigonometry?

I have a matrix $A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ (where all values are known), and I eventually want to diagonalize it into: $$ A=UDV^T $$ for orthonormal U and V. If I ...
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relation between isomorphism induced by Kahler form and any other (non-degenerate) two-form

Let $V$ be an euclidean vector space, and $\omega \in \wedge^2 V^*$ be non degenerate, i.e. the induced homomorphism $\tilde\omega: V \to V^*$ is bijective. What is the relation between the two ...
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Derivation of Steepest Descent Direction used in Line Search Methods

In the numerical optimization text I am reading, the Steepest Descent Direction was derived by considering $$ \min_{||p||_2\leq 1} p^T\nabla f(x_k) $$ This resulted in $$ p_k=-\frac{\nabla f(x_k)}...
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55 views

Prove that $a,b,c$ are the sides of a triangle

$a,b,c\in\mathbb R_{>0}$ are such that $$\begin{cases}a^2x+b^2y+c^2z=1\\xy+yz+zx=1\end{cases}$$ has a unique solution $(x,y,z)\in\mathbb R^{3}$. Prove that $a,b,c$ are the sides of a triangle. ...
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99 views

Computing intersection of vector spaces spanned by two lists

Assume that I'm given two lists of vectors $l_1$ and $l_2$, where all the vectors have equal dimension. I want to compute a basis for the intersection of their spans. What is the easiest setup for ...
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70 views

Multiplicity as roots of the minimal polynomial

Let $V\neq\{0\}$ be a finite-dimensional vector space over a field $F$ and let $\alpha \in \text{End}(V)$. Suppose that $\lambda$ is an eigenvalue of $\alpha$ with multiplicity $r$ as a root of the ...
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287 views

Gram-Schmidt orthogonalization basis for Continuous functions

Use Gram-Schmidt orthogonalization process to find an orthogonal basis of the subspace of $\mathbb{R}^4$ spanned by $v_1=(1,1,0,0),v_2=(1,1,1,0) \,and, v_3=(1,1,1,1)$ with respect to the (standard) ...
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1answer
96 views

Hamel Basis Exercise Proof Clarification.

While looking up something else on stack exchange, I ran across this question An exercise about a Hamel basis and it intrigued me. The answer was provided by Jonathan Golan (http://math....
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Linear algebra (Coordinates)

Question: Find the coordinates of $x=(1,0,0)$ in relation to base $$B=\{(1,1,1),(-1,1,0),(1,0,-1)\}.$$ I tried: $a,b,c\in R$ such that $$a(1,1,1)+b(-1,1,0)+c(1,0,-1)=(1,0,0)=x$$ but I'm not sure ...
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22 views

Probablistic bound for $\|RR^TM\|$ for uniformly random orthonormal matrix $R$

I am stuck on a finding a probablistic bound on a nonstandard random matrix. I looked around on the internet and couldn't find any results. This could be because I don't know the key words or because ...
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1answer
78 views

Algorithm to determine if integer matrix is similar to symmetric integer matrix with nonnegative entries

Let $A\in M_n(\mathbb{C})$ be a matrix with integer entries (treated as a matrix over the complex numbers). Is there an efficient way to check if $A$ is similar to a symmetric matrix with nonnegative ...
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1answer
123 views

Partition of a Matrix

In Linear Algebra, we have been taught that the partition of a matrix $A$ consists of matrices,or blocks. In other words, its elements are matrices. This same, partitioned matrix, however is said to ...
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266 views

Vector space basis change: is this “index-free” notation correct?

There are already quite a number of questions on basis change in a vector space. Nevertheless, to fully grasp the underlying idea I made up the following notation and I have some doubts on it (note: ...
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26 views

Find solution to matrix sandwich product [duplicate]

For any two $n \times n$ real symmetric and positive definite matrices $B$ and $C$, is it always possible to find a third real symmetric and positive definite matrix $A$ such that $ABA=C$? If not, ...
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Solving for matrices [T] and [T]', and the transition matrix Q given a basis.

Problem Part 1: I've already proved that T is a linear transformation, but I need to verify that I solved the other parts of the problem correctly. Here's my attempt to find [T] and [T]' below: ...
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Constrained Quadratic Optimization(Reproducing Kernel)

I am attempting to use a constrained quadratic optimization to find the coefficients of a reproducing kernel. The problem is as follows: $y(t)=\sum_{i=0}^J\alpha_iK(t, t_i)$ $Q(\alpha)= \alpha^tK\...
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299 views

Consistent Augmented Matrix

Well, the linear system which at least has one solution is called "consistent" linear system. Find an equation involving g, h, and k that makes this augmented matrix correspond to a consistent system: ...
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1answer
39 views

Vector space property proof verification

My problem is from Linear Algebra by Friedberg, Insel, Spence: In any vector space $V$, show that $(a+b)(x+y)=ax+ay+bx+by$ for any $x,y$ in $V$, and any $a,b$ in F. My approach is: $$\begin{...
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47 views

Question about a definition

There was a definition on my notebook. But sadly I cant read (...) part. What do we call $w_1,w_2,w_3...w_k$? Let V be a vector space on field F and $w_1,w_2, w_3..$ are subspaces of V. for any ...
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289 views

When is the solution to a n initial value problem matrix differential equation invertible?

Suppose $A (t,s)$ a $n\times n$ matrix is the solution of the initial value problem below, where $B_s$ is also an $n\times n$ matrix, invertible for all $s$: $$\dfrac{d A(t,s)}{ds} = B_s A(t,s)$$ $$ ...
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1answer
75 views

Density of Pythagorean triples

We define a Pythagorean triple as a triple $<a,b,c>$ such that $a,b,c\in \mathbb N$ and $a^2+b^2=c^2$. In order to avoid duplicates, we say that a triple $<a,b,c>$ is legit iff $b>a$. ...
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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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244 views

Vector Spaces: canonical basis for the usual vector spaces

I'm looking for a listing of the canonical basis for the most common vector spaces. Some standard basis are not so obvious. For instance, the basis for vector space $\Bbb C^2$ is $\{ (1,0),(i,0),(0,1)...
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66 views

Eigenvalues of ad (Adjoint action) in semisimple lie algebra?

Suppose $V=V_0\oplus V_1$ be a $Z_2$-graded semi-simple lie algebra and, $\xi\in V_1$. The maps $ad_\xi \circ ad_\xi :V_0\longrightarrow V_0$ and $ad_\xi \circ ad_\xi :V_1\longrightarrow V_1$ are ...
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Counting the operations of a problem

I have a square matrix $A\in\mathbb{R}^{n\times n}$, it has a LU decomposition. $L$ and $U$ are triangular and $L$ has ones on the main diagonal. I'm counting the number of operations for $(U^TL^TLU)...
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2answers
73 views

Transpose map in $M(2,\mathbb{R})$

Let $T$ the transpose map $T(A)=A^t$ for $A\in M(2,\mathbb{R})$. I want to find a basis such that $T$ is diagonal. I considered $T$ as a map from $R^4\rightarrow R^4$ where $T$ can be represented by $$...
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54 views

Existence of a subspace with a certain property

I am having trouble solving this problem.I have started solving the problem , so far my guesses for the subspace U were the intersection of V and complement of KerT , but i was soon able to come up ...
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1answer
74 views

Role of metric in the matrix representation of Hermitian adjoint

I'm working through Jeevanjee's "An Introduction to Tensors and Group Theory for Physicists", and while trying to prove that the matrix representation $M(A^\dagger)$ of a Hermitian adjoint $A^\dagger$ ...
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138 views

Definite Integral involving matrices

We have a definite integral of the form given below $ f(t) = \int_0^1 e^{\alpha X(t)} \frac{dX(t)}{dt} e^{(1-\alpha) X(t)}\,d\alpha \tag 1$ Given Data in the question $X(t)$ is a skew-...
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Any idea how to linearize this equation? $X^2-Y^2=aZ+bZ^2$

The intention is to linearize this equation $X^2-Y^2=aZ+bZ^2$ into something which looks like $Z=mX+nY+c$ so that a graph of $Z$ against $X$ or $Y$ can be plotted. X,Y,Z are variables while a,b,c are ...
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2answers
46 views

Equality in the sequence of increasing ranges.

Suppose $T \in L(V)$. Let $n = \dim V$. Prove that $\text{rangeT}^n = \text{rangeT}^{n+1} = \text{rangeT}^{n+2} = \dots$ I need help finishing this proof. This is what I have so far: First prove ...
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54 views

Show that $(Au,Bv)=(u,A^tBv)$

Let $ A, B $ be matrices of order $ n $, and $ \vec{u}, \vec{v} $ vectors from euclidean space $ \mathbb{R}^n $, then $ (Au,Bv) = (u,A^tBv) $ pd. $(\cdot ,\cdot)$ is my notation for inner product, ...
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1answer
125 views

Convergence of square root operators

Let $Q_n$ and $Q$ be compact positive and symmetric operators. Let $A_n = {Q_n}^{\frac12}$ and $A=Q^{\frac12}$. Given $Q_n$ converges to $Q$ w.r.t. operator norm. Does $A_n$ converges to $A$? Thanks.
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Consider a symmetric matrix $X$ with eigendecomposition $X=UVU^T$, how to call $\sum_{v_{k,k}>0}v_{k,k}u_ku_k^T$?

Consider a symmetric matrix $X$ with eigendecomposition $X=UVU^T$ How do people call $\sum_{v_{k,k}>0}v_{k,k}u_ku_k^T$? Sum of positive components of $X$? The positive semi definite part of $X$?
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1answer
68 views

Find a linear operator given the kernel

"Find a linear operator $T:\mathbb{R}^3\to\mathbb{R}^3$ so that the kernel is generated by $(1,2,-1)$ and $(1,-1,0)$." It's been a while since I've worked with linear algebra, but from memory I know ...
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73 views

Techniques to find matrix inverses of general classes of matrices?

Suppose you're given some general description of an $n\times n$ matrix, and asked to find its inverse. By "general description" I mean that the matrix can be described in one or more sentences, and ...
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1answer
89 views

Existence of solution for matrix equation $ (I - \alpha A) \bar{x}=\bar{b}$

This is my first question in here and I would be really thankful if someone could help me with understanding the matter. I am solving a matrix equation $(I-\alpha A) \bar{x} = \bar{b}$ for a positive ...
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1answer
98 views

Recurrence - using power series

Could you help me in solving this recursion( a closed form ) using power series $\mu(n)=\mu(n−1)k_0+(n−1)\mu(n−2) k_1 \tag 1$, where $k_0,k_1$ are constants $\mu(0)=3,\mu(1)=5$ HINT: We can think ...
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47 views

Least number not being the determinant of a set of matrices

Let n > 1 be a natural number and u < v integers. How can I determine the least natural number not being the determinant of some n x n - matrix with integers in the range u..v without calculating ...
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1answer
57 views

Find a reduced echelon basis from a reduced echelon matrix.

The reduced row matrix was this ---> $\begin{pmatrix}1&2&0&1&0\\0&0&1&3&0\\0&0&0&0&1\\0&0&0&0&0&\end{pmatrix} = 0$ So i computed ...
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1answer
179 views

Prove vertices of a simplex are affinely independent

I'm given that the definition of a simplex $T$ is $x \in\mathbb R^n$ such that $x$ satisfies $n+1$ linear inequalities: $(u_k, x) \lt c_k$ for $k = 1,\ldots,n+1$ (i.e. $T$ is the intersection of $n+1$ ...
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Strange phenomena in determinants of matrix of determinants.

In my research, my computations are giving rise to the following strange phenomena: Let $$D=\begin{bmatrix}x_1^p & x_2^p & x_{3}^p\\ x_{1}^q & x_{2}^q & x_{3}^q\\ x_{1}^r & x_{2}^...
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64 views

Analytic Geometry: One sheeted hyperboloid

Good afternoon! I have a question about analytic geometry. I don't actually know if the answer is quite simple, and I missed something while revising, or if it is actually more complicated than I ...
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Finding gradient of an objective as a PDE

I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me? Assume that we have an optimization ...
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2answers
92 views

Line parallel to a plane and have 45 degrees between another

I need to find a direction vector for a line parallel to a plane $x+y+z = 0$ and that have $45$ degrees with the plane $x-y = 0$ So, i've assumed the vector $\vec V_r = (a,b,c)$ and since it is ...
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1answer
107 views

Canonical embedding into dual space?

How would one go about proving that there is no embedding of a vector space into it's dual that is independent of a choice of basis? Thanks
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144 views

Maximum determinant of a $m\times m$ - matrix with entries $1..n$

I want to find the maximal possible determinant of a $ m\times m$ - matrix A with entries $1..n$. Conjecture 1 : The maximum possible determinant can be achieved by a matrix only ...
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2answers
127 views

If $\|Tv\|=\|T^*v\|$ for all $v\in V$, then $T$ is a normal operator

I have solved a question but I am not sure the last step of the question. If someone can verify it that would be great. Let $V$ be a finite dimensional vector space with complex inner product. Let ...
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1answer
88 views

Is the following Eigenvalue inequality holds or not?

Can anyone help me with the following problem? Suppose $u=(u_1,u_2,...u_n)^T$, $e=(1,1,...1)^T$, and we have $u\geq e$. Now for any symmetric matrix $A\in S^n$ with $diag(A)=0$, can we claim the ...