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
12 views

Random projection onto orthonormal bases

Given an arbitrary N dimensional vector of length $L$, and a $M$ dimensional orthonormal basis chosen uniformly at random with $M<N$, what is the CDF of the length of the projected vector?
1
vote
1answer
16 views

For which values of $a$ does the matrix can be diagnolized?

Given $$A=\begin{pmatrix} 2 & 0 & 0\\ a & 2& 0\\ a+3 & a &-1 \end{pmatrix}$$ For which values of $a$ can $A$ be diagonal? I found that $p_A(x)=(x-2)^2(x+1)$ and tried to ...
0
votes
1answer
10 views

Solving inequality(limit)

Can someone explain how we get from $(x - 3) < \varepsilon/8$ and $x < 4$ to: $(x-3)(x+3) < (\varepsilon/8)(4+3) = (7\varepsilon)/8$
0
votes
0answers
6 views

Each bilinear form induces a unique bilinear form from the dual space

Let $V$ be a finite dimensional vector space over a field $K$. Let $\gamma: V \times V \to K$ be a nondegenerate bilinear form. I now want to show that there exists one and only one bilinear form ...
3
votes
3answers
43 views

Why is cross product not commutative?

Why, conceptually, is the cross product not commutative? Obviously I could simply take a look at the formula for computing cross product from vector components to prove this, but I'm interested in why ...
0
votes
0answers
8 views
0
votes
0answers
10 views

finding equality with subspaces direct sum

assume that $U_1 \cap U = \{0\}$ and $U_2 \cap U = \{0\}$ $U_1 \oplus U = U_2 \oplus U$? I thought that it's correct because I could find a counterexample.
0
votes
1answer
19 views

What is $\max(\operatorname{Re} \{ \frac{x^* Ax}{x^* x}:0 \ne x \in C^n\} )$?

Let $A = \left( \begin{array}{*{20}{c}} 1&2\\ 0&1 \end{array} \right)$. What is $\max\left(\operatorname{Re} \left\{ \dfrac{x^* Ax}{x^* x}:0 \ne x \in C^n\right\} \right)$?
0
votes
1answer
14 views

Why is the transformation the geometrically projection ? Where and along what is projected?

Find eigenvalues and eigenvectors of the matrix in standard basis corresponding matrix
0
votes
4answers
32 views

Linear Dependent Span

$\{x \cos x, x, \cos x \}$ is a subspace of $V$. I need to find if it's a linear dependent or linear independent. So I thought that its dependent since $x \cos x$ is multiplication of $x$ and $\cos ...
0
votes
1answer
14 views

Similarity of a specific block matrix

Let $A$ and $B$ be square matrices of same dimension. I considered $n$-by-$n$ block matrices of the form \begin{align*} \begin{pmatrix} A & & \\ & \ddots & \\ & & A ...
1
vote
1answer
26 views

What are the facts used in each step of this proof?

What are the facts used in each step of this proof ? Suppose that $A\in F^{nm}$ and $B\in F^{ml}$ $$\begin{align}rank A + rank B &= rank\begin{bmatrix}0 & A\\B & 0\\ \end{bmatrix}\\ ...
-1
votes
1answer
14 views

an inequality for the projection on the intersection of 2 subspaces

Can someone give an inequality, bounding the distance of a point from its projection on the intersection of two subspaces by a function of both the distances of the point from the insividual ...
0
votes
0answers
13 views

Hermitian, orthogonal, unit trace matrix bases.

Consider the vector space of Hermitian matrices acting on a finite dimensional vector space, equipped with the Hilbert-Schmidt norm. I'm interested in matrix bases that satisfy three properties: 1) ...
-1
votes
0answers
20 views

Orthogonal Projections Composition iff Statement

Given m , n ⊂ V a finite dimensional inner product vector space, Prove that for Pm, Pn orthogonal projections onto m, n accordingly, PmPn is an orthogonal projection ⇔ PmPn = PnPm and in this case ...
0
votes
0answers
11 views

find Jordan form

Determine the jordan form of $A = \begin{pmatrix} 1 & 2 & 3\\ 0 & 4 & 5\\ 0 & 0 & 4 \end{pmatrix} $ First, I find the characteristic polynomial. $C_A(x)=(x-1)(x-4)^2$. ...
1
vote
4answers
23 views

Does an arbitrary matrix $X \in M_{n \times p}$ have a SVD?

I have proven, as below, that if $X \in M_{n \times n}$ is symmetric, then it has a SVD. $D(\lambda_i) = \text{Diag}(\lambda_i)$ is a diagonal matrix with entries $\lambda_1, \lambda_2, \dots$. ...
0
votes
0answers
11 views

upper bound for the sum of trace related to product of two matrices?

Given A and B positive definite matrices The inequality $\sqrt{4tr(AB)}$ $\leq$ $tr(A+B)$ is lower bound for tr(A+B) is there another inequality for the upper bound, i.e. ?? ≥ tr(A+B)?
0
votes
2answers
57 views

Findinf the remaining eigenvalues of a $3\times 3$ matrix

Let $n$ be a fixed natural number. We wish to compute the eigenvalues of the matrix below: $$\begin{bmatrix} a&b&c\\ d&e&f\\ g&h&i \end{bmatrix}$$ We have a condition on the ...
2
votes
3answers
162 views

Help determining whether a transformation is linear or not

I have the following transformation: $T : \Bbb R^3 → \Bbb R^2 , T (x_1 , x_2 , x_3 ) = (x_1 − x_2 , 2x_2 )$ I need to determine whether it's a linear transformation or not. I understand that ...
0
votes
2answers
32 views

Find the standard matrix for the transformation

Find the standard matrix for the transformation that projects $$ \begin{bmatrix} 1\\ 2 \\ \end{bmatrix} \text{ to } \begin{bmatrix}2\\ 4 \\ \end{bmatrix} $$ and $$ \begin{bmatrix} 1\\ 1 \\ ...
0
votes
1answer
36 views

Prove that if $A,B\in M_n(\mathbb{F})$ are $(n-1)$-nilpotent then they are similar.

If $A,B\in M_n(\mathbb{F})$ are $n-1$ nilpotent, prove they are similar. Can I say that, since their minimal polynomial is $X^{n-1}$ they are similar? I know that If $A,B$ are similar, they have ...
0
votes
3answers
36 views

How to find the basis of the following vector spaces?

I'm trying, in vain, to find the basis of the following vector spaces: (a) $W = \{x = (x_1 , x_2 , x_3 ) ∈ \Bbb R^3 : x_1 − 2x_2 + x_3 = 0, 2x_1 − 3x_2 + x_3 = 0\}$ (b) $W = \{x = (x_1 , x_2 ...
7
votes
2answers
85 views

$A^2=A^*A$.Why $A$ is Hermitian matrix?

Let $A$ be $n \times n$ matrix and $A^2=A^*A$. Why is $A$ a Hermitian matrix?
1
vote
1answer
33 views

Spectrum of the matrix $A=(a_{ij})$ where $a_{ij}=i+j$

What is the spectrum of the matrix $A=(a_{ij})_{n\times n}$ where $a_{ij}=i+j$ for any $n$. Also, what are the eigenvectors corresponding to their eigenvalues? Progress. This matrix is definitely ...
7
votes
1answer
30 views

A question about the vector space spanned by shifts of a given function

Let $E$ be the space of continuous real functions and $f\in E$ Let $T_t$ denote the shift operator: $T_t(f)(x)=f(t+x)$ Let $T(f)$ be the linear span of the set $\{T_t(f) \;|\; t\in ...
2
votes
1answer
34 views

Why does ${\lambda _{\max }}(A) = \max \{ \frac{1}{{{x^*}x}}:{x^*}Ax = 1\} $?

Let $A \in {M_n}$ be hermitian and suppose that at least one eigenvalue of $A$ is positive ($\lambda $ is eigenvalue of $A$). Why does ${\lambda _{\max }}(A) = \max \{ \frac{1}{{{x^*}x}}:{x^*}Ax = 1\} ...
2
votes
0answers
31 views

Second order derivation of Quadratic form

I would like to find the second order derivative of a Quadratic form. Assume we have a random complex column vector $x$ and a real constant value $C$. I am interested in computing the following: $$ ...
2
votes
0answers
31 views

Is there an easier way to show that a kite has perpendicular diagonals using scalar products?

I want to use scalar products to prove that a kite has perpendicular diagonals. My attempt : Let $a,b,c,d$ vectors with $a+b+c+d=0$ and $a^2=d^2$ and $b^2=c^2$ Then, we get ...
1
vote
0answers
20 views

Number of two dimensional sub spaces of a vector space over a finite field.

Let {$e_1,e_2,e_3,e_4$} br a basis of $4$-dimensional vector space over a finite field with p elements. The number of $2$-dimensional subspaces of $V$ not containing $e_4$ and not contained in ...
0
votes
0answers
14 views

Circulant matrix - Eigen decomposition in matlab [on hold]

$A = FD{F^{H}}$ , where $A$ is a circulant matrix, $F$ is normalized FFT matrix (unitary) and $F^{H}$ is the conjugate transpose (Hermitian) of the $F$ matrix. $D$ is the diagonal matrix which ...
0
votes
0answers
20 views

Linear Algebra L2 minimization

Im really confused about how to solve this question or even what its asking. Any help would be much appreciated! Let A be an m x n real matrix ($m \gt n$). Let x* be the minimizer of $||Ax - b||^2 + ...
0
votes
2answers
39 views

Help understanding the range and kernel of a linear transformation

I'm having some trouble understanding the Range and Kernel of a linear transformation. The definition goes as follows: Let $T:V \longrightarrow W$ be a linear transformation. Define the sets ...
0
votes
0answers
12 views

congruent matrices

If the rank of a matrix A is 1, the matrix is row equivalent to one with only one row different to 0. But not necessary the matrix A is congruent to one wuth only one row different to 0/ How to find a ...
1
vote
2answers
35 views

Straight line equation is linear or not?

I read somewhere that for the linearity the equation should pass through the origin in this regard the equation of straight line y=mx+c is linear or not?
2
votes
1answer
63 views

Matrix and field extension

It is given that $F\subset K$ are fields. $A$ is a matrix of size $n\times n$ over $K$. I need to prove that there exist $c_1,\ldots,c_k\in K$, linearly independent over $F$, and matrices ...
1
vote
1answer
15 views

derivative of gradient involving inverse of matrices

I need to take three partial derivatives of this squared mahanalobis distance with respect to these three matrices: $Q, A,$ and $S$ $$(x+Ab)^T(A^TQA+S)^{-1}(x + Ab)$$ $x$ and $b$ are vectors of ...
1
vote
1answer
36 views

Do positive-definite matrices always have real eigen values?

Do positive-definite matrices always have real eigenvalues? I tried looking for examples of matrices without real eigenvalues (they would have even dimensions). But the examples I tend to see all ...
0
votes
0answers
9 views

Exponential Demand Periodic Review

I have exponentially distributed demand data and I am trying to find a formula for an 'order up to level (OUL)' periodic review ordering policy. We are not using a re order point for this policy. ...
0
votes
0answers
26 views

A inquality in matrix norm [duplicate]

Let $A,I \in {M_n}$($I$ is identity matrix) and $\left| {\left\| . \right\|} \right|$ is matrix norm.Suppose $\left| {\left\| A \right\|} \right| < 1$ and $\left| {\left\| I \right\|} \right| \ge ...
0
votes
2answers
23 views

Prove every isometry on an odd- dimensional real product space has 1 or -1 as an eigenvalue.

This is a question from Axler. I was hoping for some help. It seems easy to understand, but I don't know where to go about on proving this.
0
votes
0answers
20 views

Questions about the position of a matrix and a vector

I got quite confused by position of a matrix and a vector. For example, the definition of a range space put a matrix in front of a vector, like $R(A) = \{ Ax | x∈ R^n \}$; However for a linear ...
1
vote
0answers
26 views

Positive linear functionals on the space of positive semidefinite matrices

Suppose $f: \mathbb{R}^{n \times n} \rightarrow \mathbb{R}$ is a linear functional with the property that $f(A) \geq 0$ whenever $A$ is positive semidefinite. Is it true that there exist vectors $v_1, ...
0
votes
1answer
40 views

Is there a function over $\mathbb{Z}_p$ that is never linear?

Let $p$ be a prime. I wonder if there is a function $f$ that satisfies the following rule: Whenever $$z_1 + \dots + z_c \equiv cx \mod p$$ (where $1 < c < p$, and $z_j, x \in \mathbb{Z}_p$) ...
0
votes
1answer
17 views

Clustering of vectors via inner product relationship

This might be an odd question, but suppose I have a lot of vectors $a_i\in\mathbb{R}^{3}$ (not necessarily unit) and for some unit vector $u\in\mathbb{R}^{3}$ I find $$ \sum_{i=1}^m ...
1
vote
1answer
21 views

How to find the span for a linear transformation?

I'm learning Linear Transformations and I understand what is a linear transformation. Now I'm trying to look at an example question and I'm not really sure how the span is found. The question goes as ...
1
vote
4answers
39 views

Prove that $\|v \|^2= |\langle v, e_1 \rangle |^2 + \cdots + | \langle v, e_m\rangle |^2$

Suppose $(e_1,\cdots, e_m)$ is an orthonormal basis in $V$. Let $v \in V$ . Prove that $\|v\|^2= |\langle v, e_1 \rangle |^2 + \cdots + | \langle v, e_m\rangle |^2$ Let $v\in V$ and ...
0
votes
0answers
26 views

What is wrong with my solution to this problem?

The base $ABCD$ of the figure has area $9$. The point $M$ divides the segment $AB$ on ratio $2$ and the edge $BF$ of length $2$ forms an angle of $60º$. Calculate $[CM,CB,BF]$, knowing that ...
0
votes
0answers
5 views

How many binary vectors of weight 3 can you have before their span contains one of weight 2?

In other words, I am looking for the smallest $k$ for which the following is always true: Let $v_i \in \mathbb{F}_2^n$ for $i = 1\ldots k$ be distinct vectors of Hamming weight 3, that is, vectors ...
3
votes
0answers
14 views

Behavior of MGF of Quadratic Combination of Dependent Multivariate Gaussians

Sorry if the formatting is poor, this is my first time asking a question. I'm investigating how squared gaussians behave, using the techniques provided here, which are giving me inconsistent results. ...