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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Invertibe matrix is a transition matrix?

It is true that all transition matrices are invertible, but does the converse hold: All invertible matrices are transition matrices? I'm asking with regard to matrices over a field, but more general ...
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33 views

Is there any simple way of finding a matrix which commutes with a given (say, more complicated) matrix?

Suppose I want to find the eigenvectors and eigenvalues of a hermitian matrix $A$, but $A$ is big and ugly. Is there an easy way to find another, nicer, hermitian matrix $B$, such that $AB=BA$ and so ...
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55 views

Basis of the matrices with only non diagonalizable matrices

Is it possible to find a basis of $M_n(\mathbb{R})$ that only has non diagonalisable matrices ? I'm looking for a rather easy example, or a proof of the (non-)existence.
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34 views

Proving that Mx is an eigenvector of B with λ as the eigenvalue given that $B=MAM^{-1}$ and $Ax=λx$

The square matrix A has λ as an eigenvalue with corresponding eigenvector x. The non-singular matrix M is of the same order as A. Show that Mx is an eigenvector of the matrix B, where $B = ...
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112 views

Unnecessary Elements in the Tensor Product?

For vector spaces $U, V$ there exits a unique (up to isomorphism) vector space, denoted by $U \otimes V$, and a bilinear map $\eta : U \times V \to U \otimes V$ such that for every bilinear map $\xi : ...
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379 views

Why does Givens rotation avoid iteration and Jacobi rotation doesn't in case of reducing a symmetric matrix to tridiagonal?

I am currently implementing symmetric matrix reduction to tridiagonal. I read that Givens rotation avoids iteration when it is used for reducing a matrix to tridiagonal whereas Jacobi rotation is ...
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89 views

Can Cayley-Menger Determinant Be Negative?

Cayley-Menger determinant is used to calculate the area of a triangle, volume of a tetrahedron etc. Can be seen here. My question is; If given only positive numbers, can Cayley-Menger determinant ...
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25 views

subsapce of f(T)V T -invariant

On a vector space $X$, choose a nonzero element $v \in X$ and a linear map $T : V \to V$. $f(T)v$ is the space generated by $v, T(v), T^2(v),\dots$ I think any subspace of $f(T)v$ is also ...
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61 views

Schur's Lemma: Is the isormorphism between two irreducible spaces unique?

Suppose $V_1 \neq V_2$ are two irreducible representations of the finite group G. Then Schur's Lemma says that any G-invariant map between them is either 0 or an Isormorphism. I understand that if ...
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34 views

How to determine the Jordan form and give a Jordan base for a matrix?

given is $\begin{pmatrix} 3&0&-1&0&0 \\ 1&3&0&1&0 \\ 0&0&3&0&0 \\ 0&0&0&3&0 \\ 0&0&0&0&-3 \end{pmatrix}$ I have to ...
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109 views

Skew symmetric Matrix - Commutative property

If A and B are two odd size skew symmetric matrices(for example $3 \times 3 $). Let us say $X=AB,Y=BA$ Question What is the general relationship between X and Y? Can we write Y using X?
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206 views

Different Definitions of Tensor product, Halmos, Formal Sums, Universal Property

In the classic Finite-Dimensional Vector Spaces by P. Halmos he defines the Tensor product as The tensor product $U \otimes V$ of two finite-dimensional vector spaces $U$ and $V$ (over the same ...
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121 views

Compute a 4 4 matrix M such that MA is the row-reduced echelon form of A.

Compute a 4 X 4 matrix M such that MA is the row-reduced echelon form of A. (Hint: M can be written as a product of elementary matrices.) A:= ...
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18 views

Linear map problem

Given that $T$ is a linear map $T:\Bbb R^3 \to \Bbb R^2$ and that $$ T\left(\matrix{1\\0\\0}\right) = \left(\matrix{1\\0}\right)\qquad T\left(\matrix{1\\1\\0}\right) = \left(\matrix{0\\-3}\right) ...
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105 views

How to find maximum of an inverse of a matrix?

If there is a square $~n\times n~$ matrix $~H~$ where ALL the elements of $~H_{i,j}~$ are variables between two bounds, $~H_{i,j})_{min}~$ and $~H_{i,j})_{max}~$. Is there any relation to maximize ...
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Regarding Linear Subspaces over a Finite Field… TFAE:

Let $V=\mathbb{F}^n$, for a finite field $\mathbb{F}$. Prove the equivalence of the following statements: There is a linear subspace $C$ of $V$ with the property that every vector $v$ of ...
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533 views

Which of the subsets of $\mathbb{R^{3\times 3}}$ are subspaces of $\mathbb{R^{3\times 3}}$?

The invertible $3 \times 3$ matrices The $3\times 3$ matrices whose entries are all integers The $3\times 3$ matrices with all zeros in the third row The non-invertible $3\times 3$ matrices The ...
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583 views

Equation of a curved line that passes through 3 points?

I have a screen wherein the upper-leftmost part is at x,y coordinate (0,0). Then I have a curved line that passes through 3 points: (132, 201), (295, 661) and (644, 1085). Now, say I want to find 7 ...
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54 views

Solving $Ax=B$: what's wrong with this linear algebra argument?

With $K>L$ and assuming that we are working with real variables, suppose that $B$ is $K\times 1$ and $A$ is $K\times L$ with full column rank. I'm trying to find $x$ ($L\times 1$) satisfying: $$ ...
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71 views

Standard basis for Complex vector space

What will be the standard basis of $\mathbb{C}^3$ or in general how can I find the standard basis for $\mathbb{C}^n$ ? Note: $\mathbb{C}$ is complex vector space
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Limit of solution of linear system of ODEs as $t\to \infty$

I am completely stuck on the following problem: Consider the linear system: $x'(t)=A(t)x(t)$ where $A(t)$ is an $n$ by $n$ matrix. Assume that $\lim_{t\to \infty}A(t)=B$. Suppose that each eigenvalue ...
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Help me understand my linear algebra book

If $(v_1,...,v_m)$ is a list of vectors in $V$, then each $v_j$ is a linear com- bination of $(v_1,...,v_m)$ (to show this, set $a_j = 1$ and let the other a’s in $a_1v_1 +···+a_mv_m$ equal 0.Thus ...
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101 views

Sign pattern symmetric matrices

I am interested in sign pattern symmetric real matrices ($a_{ij} a_{ji} \ge 0$ for all $i \ne j$). I have seen a published proof that such sign-symmetric matrices cannot have purely imaginary ...
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55 views

Basis and dimension of the null space and range

The linear map $T : \Bbb R^{n\times n}\to \Bbb R^{n\times n}$ is defined by the formula $$T(A) = \frac12(A+A^T)\;.$$ How do I find a basis of the null space of $T$ and determine its dimension? and ...
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19 views

Questions about a special tensor transformation

Suppose tensor $U_{i\alpha\beta}$ with dimension $M*N*N$ satisfy following condition: $$U_{i\beta\alpha}=W^1_{\alpha\alpha'}W^2_{\beta\beta'}U_{i\alpha'\beta'}$$ where $W^1$ and $W^2$ are $N*N$ ...
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41 views

Prove True or false

if the rref of a has a row of 0', then the set of row vectors of a is linearly dependent. Please help me prove or give a counterexample
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33 views

Orthonormal Basis and Matrix of a Linear Operator Proof.

Let B = $\{v_1, \dots, v_n\}$ be an orthonormal basis for Rn Let p = $[v_1, \dots, v_n$]. Prove that for any x, we have that the B-matrix of x is equal to the tranpose of P times x. I am unsure as to ...
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45 views

Determinant Question

This is not a homework question. Let $A$ be an $n\times{n}$ matrix with entries in the field $F$ such that each entry is relatively prime to any other entry. What is the number of elementary matrices ...
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question about vector spaces and linear algebra

LEt $V = \{ (x,y) : x,y \in \mathbb{C} \} $. The addition and scalar multiplication are the usual ones. We know $V$ is a vector space over $\mathbb{C}$ with dimension $2$. My question: Why does $V$ is ...
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So why isn't $\Bbb R^n = \oplus _{n = 1}^{m}\Bbb R^n$

I thought that direct sum means each component of $V = \oplus U_i$ can be decomposed into elements of $U_i$. But if $U_i$ is replaced by the whole space, doesn't it mean the everything else in the ...
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22 views

Irreducibility of markov chains

Let $A=(a(x,y))_{x,y\in X}$ be a finite irreduzibel nonnegative matrix. Let $b,c >0$, and $\alpha=a^{(b+c)}(x,x)>0$. So $a^{(n(b+c))}(x,x)\ge \alpha^n$. And therefore $\lim ...
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Mapping entire cartesian plane to a line with 2x2 matrix

I have a homework question that I have been staring at helplessly for far too long, and if anyone could point me in the right direction that would be great. Matrix A = \begin{bmatrix} a & b ...
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1answer
35 views

Define a “rotation of u” by $R_{A}u\doteq u\circ A $ with A an orthogonal $n\times n$ matrix and “$\circ$” means composition.

Show that $\Delta (u\circ A )=(\Delta u)\circ A$. (Note: $u(x)\in C^{m\geqslant 2} (\mathbb{R}^{n}) $) I tried to treat u as merely a number in $\mathbb{R}^{n}$ and then take the Laplacian to each ...
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72 views

How to determine if a set is a subspace of the vector space of all complex $2\times 2$ matrices?

I must determine if a each of the following is a subspace of the vector space consisting of all complex $2\times 2$ matrices. All matrices with real diagonals. All matrices for which the sum of the ...
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44 views

Showing $\alpha$ is diagonalisable iff the characteristic of the field does not divide the order of $\alpha$.

Let $\alpha$ be a linear operator on a finite-dimensional vector space $V$ over a field $F$. Suppose that $F$ is algebraically closed, and that $\alpha^n=id_{V}$ for some positive integer $n$ while ...
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Given the frame $B=\{(1,1,0),(0,1,1),(1,1,1),(0,0,1),(0,1,-1)\}$, find (if possible) a (i) $(2,1)$ surgery,

Given the frame $B=\{(1,1,0),(0,1,1),(1,1,1),(0,0,1),(0,1,-1)\}$, find (if possible) a (i) $(2,1)$ surgery, and a $(1,2)$ surgery that produce tight frames. A frame is tight if and and only if the ...
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Linear Algebra Subspace test

I'm currently studying Subspace tests in my linear Algebra module at uni, but am struggling to understand it, can anyone explain how to conduct a SubSpace test?
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82 views

finding if a linear transformation exists, and proving it.

We just started the topic of linear transformations and I have this hw question that I just don't understand. Does there exist a non-trivial linear transformation, represented by some 2x2 matrix, ...
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89 views

Elementary row operations in matrices

This is really such a lovely math community, I am working on some differential equations hw and my teacher didn't teach this topic yet so I am a little confused. My first question is pertaining to ...
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129 views

Well-defined $\xi$-weighted (Euclidean) norm

Suppose $\xi$ is a vector, that is used for $\parallel z\parallel_\xi$ calculation. Should every element of $\xi$ be positive, $\xi(i)>0$?
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Find a polynomial $f(Z)$ of degree less than 2 such that $e^{tA}=f(A)$

Let $A=\begin{pmatrix}3&-2\\2&-2\end{pmatrix}$. As the question says I need a polynomial $f(Z)$ of degree less than 2 such that $e^{tA}=f(A)$. Should my polynomial just be the first 2 terms ...
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1answer
72 views

Non-standard quadratic matrix equation

I have an equation that looks like the following: $$ A\cdot\mathrm{diag}(x)\cdot x + B\cdot x + c = 0 $$ where $A, B, C \in \mathbb{R}^{n \times n}$ and $x, c \in \mathbb{R}^n$. $ x $ is unknown. ...
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Am I correct about these being subspaces

I am self learning (today is my Day 1) Linear Algebra today using "Linear Algebra Done the Right way" and I have just finished the first chapter. Since the exercises don't have answers and I have ...
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46 views

Prove that this statement about A and B is true.

$A,B \in \mathbb{R}^{2}$, If $AB - BA = A^2$ Prove that $ (B - A)^{2014} = B^{2013}(B-2014A)$
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45 views

for every frame $B$ with $k$ vector in $\mathbb{R}^2$ such that $B \cup v$ is a tight frame.

Prove the following: for every frame $B$ with $k$ vector in $\mathbb{R}^2$ such that $B \cup v$ is a tight frame. Is the same statement true in $\mathbb{R}^3$? Through the discussion provided in ...
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12 views

Proving $\forall x \in E\setminus\{0_E\}, (x,u(x),…,u^{n-1}(x))$ is a basis of $E$

Suppose that $u$ is an endomorphism of a vector space $E$ and $\dim({E})=n \ge2$. And supposing that $E$ is the only sub-space not equal to zero, and stable by $u$. How to prove that: $\forall x ...
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24 views

Showing that this is a subspace iff a constant is zero

If $b ∈ F$, then ${[(x_1, x_2, x_3, x_4) ∈ \mathbb{F^4} : x_3 = 5x_4 + b]}$ is a subspace of $\mathbb{F^4}$ if and only if $b = 0$, as you should verify. The first example in my book ...
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20 views

Find couples of frames $B\neq B_1$ with 5 elements in $\mathbb{R}^3$ which are (i) PRR equivalent, (ii) similar, and (iii) unitary equivalent

Find couples of frames $B\neq B_1$ with 5 elements in $\mathbb{R}^3$ which are (i) PRR equivalent, (ii) similar, and (iii) unitary equivalent. Assume that the frames do not contain the vector 0 or ...
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386 views

Prove that the product of two positive semidefinite and symmetric matrices has non-negative eigenvalues

How can I prove the following fact: If $A$ and $B$ are two positive semi-definite and symmetric matrices then all eigenvalues of $AB$ are non-negative.
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83 views

How to find nonnegative solutions of a linear system?

I have a $M$ equation and $N$ variables like this : $ \begin{bmatrix} 3 & 0 & 1 & 0 & -1 & -3 & 2\\ 1 & 2 & 0 & 4 & 0 & 0 & -1\\ 1 & 1 & 0 ...