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 ...

learn more… | top users | synonyms

0
votes
0answers
3 views

Prove $T|_{V_\lambda}$ is diagonalizable

Let $V$, an $n$-dimensional vector space and let $T, S:V\to V$, two diagonalizable linear operators. Show that if $TS=ST$ then every $V_\lambda$ of $S$ is $T$-invariant and the restriction, ...
0
votes
1answer
12 views

Linear transformation: Change of basis

I am given the following linear transformation $L$: $A=\begin{bmatrix}1&2\\0&3\end{bmatrix} \in \Bbb R^{2 \times 2}$ $L: \space \Bbb R^{2 \times 2} \longrightarrow \Bbb R^{2 \times 2}; ...
0
votes
0answers
7 views

properties of the solution to a non-homogeneous matrix equation with a non-singular M-matrix

I have a matrix equation Ax=b; where A is a 4x4 non-singular A matrix (A has negative off-diagonal and positive diagonal entries). b is a strictly positive vector. Let x=[x1 x2 x3 x4] be the solution ...
3
votes
1answer
22 views

Understanding a simple proof about minimal polynomials

Let $T \colon V\to V $ be a linear operator, where $V$ is a vector space over $F$. Suppose that the minimal polynomial $M(t)$ of $T$ can be factored into the product of two coprime and monic ...
0
votes
0answers
15 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) = ...
1
vote
1answer
35 views

Are $A$ and $A^\top$ similar?

Let $K$ be a field and $A$ a square matrix with entries in $K$. Then A and $A^\top$ have the same characteristic polynomial. What do we know about similarity? Do you have an example where $A$ and ...
0
votes
2answers
26 views

Simple matrix derivative identity

Is the following correct, and is there some kind of similar identity when $x$ and $y$ are matrices? For $A \in \mathbb{R}^{n \times n}$, $\nabla_A x^T A y = x y^T$. And my proof: ...
4
votes
3answers
45 views

$A$ is a symmetric postivie definite matrix. Prove that $A^k$ is also a positive deinite

Let $A\in M_n(\mathbb{R})$, a symmetric positive-definite matrix. Prove that for every $k\in\mathbb{N}$, $A^k$ is also positive definite. So since $A\in M_n(\mathbb{R})$ is symmetric and positive ...
2
votes
1answer
31 views

$\left\| A \right\| \le \varepsilon \Rightarrow \left\| {\mathop A\limits^{\_\_} } \right\| \le \varepsilon$

Suppose $A \in {C^{n \times n}}$ $\left\| A \right\| \le \varepsilon$ such that $\left\| . \right\|$ is matrix norm subordinate to the euclidean vector norm. Is this true that $\left\| {\mathop ...
5
votes
0answers
21 views

Is There a Basis Free Definition of the Pfaffian

$\DeclareMathOperator{\pf}{pf}$ I recently came across a delightful fact that: The determinant of a $2n\times 2n$ skew-symmetric matrix is a the square of a certain polynomial called the pfaffian. I ...
1
vote
1answer
20 views

Adjoint and Adjugate are same or different?

The notions of adjoint and adjugate, which I saw, are as follows: (1) Let $T:V\rightarrow W$ be a linear map. Then there is a corresponding linear map between the duals of these spaces: ...
4
votes
1answer
31 views

Largest eigenvalue of a Hermitian matrix

I have two positive semi-definite Hermitian matrices $\mathbf{R}_1, \mathbf{R}_2 \in \mathbb{C}^{M \times M}$. They are in fact covariance matrices. I'm interested in the largest eigenvalue (or ...
1
vote
1answer
33 views

Proof for $\mathbf{M}$ unitary if $||\mathbf{M}\mathbf{v}|| = ||\mathbf{v}||$

Let $\mathbf{M} \in \mathbb{F}^{n, n}$. Then $||\mathbf{M}\mathbf{v}|| = ||\mathbf{v}||$($\mathbf{v} \in \mathbb{F}^n$) implies that $\mathbf{M}$ is unitary. My question is, how to prove this ...
1
vote
5answers
43 views

Is it possible to solve for $m$ in a linear equation without knowing $b$?

Suppose you know certain points on a line say $(5,2)$ up to $(8,10)$ but you don't know exactly where the $y$ intercept would be being somewhere down there at like $-25$ area. How would you solve for ...
0
votes
1answer
51 views

Which matrices diagonalizes a diagonal matrix? [on hold]

I think the answer is the set of all diagonal matrices but I am not sure. Can anyone give the answer with a proof?
2
votes
4answers
137 views

Do row operations change the column space of a matrix?

I know that (i) row operations do not change the row space (ii) column operations do not change the column space and (iii) row rank = column rank (but this is sort of unrelated, I think). But, ...
0
votes
2answers
60 views

Prove that similar matrices have the same nullity.

How do I approach this? I'm assuming it might have something to do with $B = P^{-1}AP$.
-4
votes
0answers
47 views

Practice Exam question need help! [on hold]

For vectors $f,g \in C[-\pi,\pi]$, we use the inner product $\langle f,g \rangle = \displaystyle \int_{-\pi}^{\pi} f(x)g(x)\,dx$. Then, $S=\{1/(2\pi)^{1/2},\sin(x)/\pi^{1/2}\}$ is an orthonormal set ...
2
votes
2answers
46 views

Finding Eigenvalue det(λI - A);

I want to know if what I'm doing to derive equation (2) from (M2) is correct or not; usually, before moving onto the next row in Guass-Jordan elimination we turn a_11 into a leading one or whatever ...
0
votes
1answer
33 views

Problem about dual of $W = V \oplus V'$

Let $V$ by finite dimensional, let $W = V \oplus V'$, and prove that the correspondence $(x,y) \rightarrow (y,x)$ is an isomorphism between $W$ and $W'$. (The direct sum is defined as the set of ...
1
vote
2answers
36 views

Help With Finding A Basis

I came up to the following matrix: $$\begin{pmatrix} 3 & 1& 3& -4\\ 0 & 0 & 0& 0\\ 0 & 0 & 0& 0\\ 0 & 0 & 0& 0\\ \end{pmatrix}$$ I know that ...
2
votes
2answers
44 views

Do linear operators that map one space into a different space have a Jordan canonical form?

I know that this answer is most likely "yes", and that, in the setting of matrices, all matrices are similar to its Jordan form, which is unique (up to the ordering of the Jordan blocks.) But what ...
0
votes
3answers
21 views

Finding Rank And Eignvalues Of Vectors Multiplication

Let $v=(3,1,3,-4)$ and $A=v^tv$, Find: the rank of $A$ $Null(A)$ eigenvectors and eigenvalues Is there a way to approach this without finding $A$ explicitly?
0
votes
1answer
12 views

Question about row operations and row-echelon form,

If I have a matrix, with, say, the first two columns consisting of all zeroes, then is the first entry of the third column, which is non-zero, my first pivot variable, so that when solving Ax=b, for ...
0
votes
2answers
20 views

Distributive property of scalar multiplication over scalar addition

I need help with a simple proof for the distributive property of scalar multiplication over scalar addition. Help with proving this definition: $(r + s) X = rX + rY$ I have to prove the truth of the ...
0
votes
1answer
22 views

how to calculate the projection of a vector onto a closed convex set? [duplicate]

suppose we have a vector $x \in \mathbb{R}^{n}$, and a closed convex set $C \in \mathbb{R}^{n}$. $C =\{x|Ax=b\}$ how to calculate the vector $y \in \mathbb{R}^{n}$, which is the projection of $x$ ...
1
vote
1answer
25 views

Proving space of skew-symmetric matrices is orthogonal complement of symmetric matrices

Problem: Prove that $\left\{ A \in \mathbb{R}^{n \times n} \mid A \text{ is symmetric}\right\}^{\bot} = \left\{ A \in \mathbb{R}^{n \times n} \mid A \ \text{is skew-symmetric}\right\}$ with $\langle ...
3
votes
2answers
35 views

Eigenvalues of Matrix Product.

Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their product? What about the special case when one of these matrices is a diagonal (positive) matrix? I ...
0
votes
3answers
55 views

Help with proof: $\mathrm{rank}(A+B) \le \mathrm{rank}(A) + \mathrm{rank}(B)$

The question is: If $A,B$ are any $m\times n$ matrices, prove that $\mathrm{rank}(A+B) \le \mathrm{rank}(A) + \mathrm{rank}(B)$. ($\mathrm{rank}(A)$ is the dimension of the column space of $A$, ...
0
votes
3answers
37 views

Eigenspace and $\ker(T)$

It seems like eigenspace and $\ker(T)$ are strongly connected, I have thought about some properties and I would like to make sure I got it right. for all matrix/transformation there is an Eigenspace ...
1
vote
0answers
15 views

reoder basis vectors to get 'more diagonal' representation of NxM matrix

I am trying to reorder the basis vectors of an NxM matrix in a way that leads to a representation with 'as much weight close to the diagonal as possible'. I would be happy, if instead of a matrix ...
1
vote
2answers
25 views

Finding orthonormal basis for a subspace $W$ of the Euclidean space $\mathbb{R}^3$.

Problem: Let $\mathbb{R}^3$ be an Euclidean space. Find an orthonormal basis for the subspace $W$ defined as $x + 2y-z = 0$. Attempt at solution: So this is a plane in $\mathbb{R}^3$, so I guess I ...
1
vote
2answers
41 views

Every vector space is isomorphic to the set of all finitely nonzero functions on some set

I am trying to prove the statement in the title, that Every vector space is isomorphic to the set of all finitely nonzero functions on some set. A finitely nonzero function from $X \rightarrow ...
1
vote
1answer
32 views

Using the Gram -schmidt procedure to find the orthonormal set (Linear Algebra)

(a) Construct an orthonormal basis of the space $R^3$ satisfying the requirment of the Gram-Schmidt prodcure from the basis $v_{1}=(-3,4,0)$ , $v_{2}=(5,10,-24)$ , $v_{3}=(0,0,1)$ (b) Given that ...
0
votes
2answers
81 views

Possible to express the diagonal matrix $D=\tiny\left(\begin{matrix}1&0&0\\0&2&0\\0&0&3\end{matrix}\right)$ as function of $3\times3$ identity matrix?

Is possible to express the diagonal matrix $$D=\left(\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix}\right)$$ as function of $3\times3$ identity matrix? For ...
-2
votes
0answers
20 views

A qustion in matrix polynomial [on hold]

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
0
votes
3answers
25 views

Size of a triangle using determinant [duplicate]

find the size of a triangle using (determinant) with the following points: $(x_1,y_1)=(1,-2)$ $(x_2,y_2)=(-4,-2)$ $(x_3,y_3)=(-5,-1)$ How should I place those points in the ...
2
votes
2answers
47 views

Prove there's a unitary linear operator

Let $u, v\in V$, where $V$ is a finite dimensional vector-space, such that $\|u\|=\|v\|$. Prove there's a unitary linear operator such that $T(u) = v$ So if there's such unitary linear operator, it ...
0
votes
1answer
18 views

How to prove this “local invertibility” theorem for bounded linear operators?

The theorem states that, suppose $X,Y$ are complete normed vector spaces, if $\mathscr A_0\in \mathscr L(X;Y)$ is invertible (i.e., $\exists \mathscr A_0^{-1}\in\mathscr L(Y;X)$ s.t. $(\mathscr ...
4
votes
2answers
83 views

An inequality for the dimension of the sum of subspaces

The answer with the most of upvotes on MO is this answer on $\dim(U+V+W)$. Question: 1. Is it nonetheless true that every three vector subspaces $U$, $V$ and $W$ of a vector space $M$ satisfy $$ ...
2
votes
1answer
17 views

if $E,F$, two bases are orthonormal then $T$ is unitary.

Let $T:V\to V$ and two bases of $V$: $E = \{v_1, \ldots, v_n \}$ and $F = \{T(v_1), \ldots, T(v_n)\}$. Prove: $E,F$ are orthonormal implies $T$ is unitary. So basically we want to prove ...
0
votes
0answers
14 views

Adjugate matrix-Almost Inverase

I Came across the following statement: "If the inverse of the original matrix exists, the adjoint is "almost" that inverse". What does it mean? Moreover, intuitivly, what does the property ...
0
votes
2answers
41 views

Normalizing a basis

Let the basis $B = \{1,x,x^2\}$ which is orthogonal. Now, I've seen the following: $$\|1\| = \sqrt {\langle 1,1\rangle} = \sqrt {4\cdot 1\cdot 1} = 2 $$ $$\|x\| = \sqrt {\langle x,x\rangle} = ...
3
votes
2answers
245 views

If two matrices have the same trace and determinant, do their powers have the same trace?

Let $A,B$ be two $2 \times 2$ matrices over some finite field $\mathbb{F}_q$, such that they have the same trace and determinant. Does this imply that tr $A^k$ = tr $B^k$ for any integer $k$? I've ...
1
vote
0answers
45 views

How can I use the Bullet-Physics's ray-cast normal to calculate angles for a object to lay on a surface?

[Give the normal of a surface in XYZ format, how do I calculate rotations (also in XYZ format) needed to set an object parallel to the surface?] I have a collision library that uses the bullet ...
0
votes
2answers
24 views

Proving the orthogonality of an inner product space (Linear Algebra)

Prove that any orthogonal set $S$ consisting of non zero vectors is linearly independent. My try By contradiction we assume that the orthogonal set $S$ consisting of non zero vectors is linearly ...
0
votes
1answer
43 views

Calculation of $trace(L^THL)$, L is lower triangular, H is symmetric.

I am working on a problem where I had to find the following expression: $$ l = Tr({P'HP})$$ I already modified my model formulation using cholesky decomposition for PSD matrices and came up with ...
1
vote
3answers
47 views

Finding dimension of subspace

I know that any polynomial in subspace $W$ must have $(x-1)$ as factor so that $p(1)=0$ But I don't understand how $p'(2)=0$ can be incorporated. Thankful for any kind of help.
1
vote
1answer
23 views

How to denote a tensor in terms of matrices product?

How to write a tensor in terms of a product of matrices? For example, I have $a \times b$ matrix $F$, and I want to create a 3D $a \times a \times a$ tensor $T$, where $T_{i,j,k} = \sum_{m=1}^{b} ...
3
votes
1answer
50 views

A result about commuting matrices in $ M(n, \mathbb{C} ) $

Let $ A $ be a matrix in $ M(n, \mathbb{C} ) $ and let $ A^{*} $ be its Hermitian adjoint. Suppose that the matrices $ A $ and $ AA^{*}-A^{*}A $ commute. Show that $ AA^{*} = A^{*}A $. Here is a ...