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

2
votes
1answer
72 views

Are eigenspaces unique?

I have calculated an eigenspace of a matrix. It is 2 dimensional. I checked it with WolframAlpha, but in WolframAlpha's solution one basis vector in this eigenspace is different from my solution.
0
votes
1answer
58 views

If $E$ has eigenvalues $\lambda_1, \ldots, \lambda_n$, then $B \mapsto [E,B] = EB - BE$ has eigenvalues $\lambda_i - \lambda_j$.

Let $E \in M_n(C)$ be an $n \times n$ matrix with entries in a(n algebraically closed, characteristic 0) field $C$, with eigenvalues $\lambda_1, \ldots, \lambda_n$. Show that the commutator map ...
3
votes
2answers
93 views

Manipulating identities

I'm having some trouble deriving certain identities. If $$S(z) = \prod_{i=1}^n (z-z_i)$$ then how can I write $$\frac{1}{S(z)}\frac{d^2S}{dz^2} = \sum_{i=1}^n\frac{1}{z-z_i}\sum_{j\neq ...
3
votes
0answers
57 views

Row operations that change similarity class

Let $\mathbb{K}$ be a field and $A\in \mathcal{M}_{n\times n}(\mathbb{K})$ be a matrix. Which row operations on $A$ do not change its similarity class?
1
vote
2answers
328 views

Consistency of matrix norm: $||Ax||_2 \leq ||A||_{Frobenius}||x||_2$

I'm trying to show that $||Ax||_2 \leq ||A||_{F}||x||_2$ where $A$ is an n by n matrix, $x\in \mathbb R^n$, $||x||_2$ is the euclidean norm, and $||A||_F$ is the frobenius norm. I actually wrote ...
0
votes
1answer
24 views

Problem with decrreasing sequence of subspaces

Let $X$ be a finite dimensional vector space and $f: X\rightarrow X$-a linear mapping. We have then decreasing sequence of subspaces $(Im f^k)_{k\in \mathbb N}$. How to prove that if for some $n\in ...
1
vote
1answer
74 views

Cross Product in Levi-Civita Notation - The elementary basis vector's missing?

http://www.unl.edu.ar/ceneha/uploads/Cartesian_tensors_Index_notation_&_summation_convention.pdf avers: $1.$ $(a×b).(c×d) = \epsilon_{i jk}a_jb_k \quad e_{ilm}c_ld_m$ $2. \nabla × ...
1
vote
1answer
57 views

The trace as an integral over a sphere [duplicate]

Let $V$ be a real vector space of dimension $n$ and let $\langle \, \cdot\, , \,\cdot\, \rangle$ be an inner product on $V$. We can define a linear functional on the space of endomorphisms of $V$ by ...
1
vote
2answers
72 views

Nilpotent Mappings

Got completely confused with this nilpotent and JCF stuff, need some help. Matrix $A_{n\times n}$ is nilpotent of order K, $1\le k\le 4$ Need to find: a list of all possible dimensions of ...
1
vote
0answers
67 views

Square root of a positive-definite Markov matrix

Let $M$ be a positive definite Markov matrix, meaning that $M_{ij}\geq0$, $\sum_j{M_{ij}}=1$ and all eigenvalues of $M$ are positive. Given the spectral decomposition $M=V\Lambda V^{-1}$, the matrix ...
1
vote
0answers
28 views

Rank of a subtraction of matrix: equality vs inequality

Are there any cases in which the following inequality $$ \mathrm{rank}\left(A-B\right) \geq \lvert \mathrm{rank}(A) - \mathrm{rank}(B) \rvert. $$ converts into a equality? For some matrices, like $$ ...
3
votes
2answers
30 views

Algebra Question ( possible values) [closed]

If $$k = \dfrac{a}{b+c} = \dfrac{b}{a+c} = \dfrac{c}{b+a}$$ How many possible values of $k$ are there?
13
votes
4answers
316 views

Is it true that all matrices in $M_2(\mathbb R)$ is the sum of two squares?

I recently show that every polynomial with real coefficient and $P$ is always positive. is a sum of two squares of polynomials. These questions also appear often in arithmetic. What if we change ...
1
vote
1answer
189 views

the rank of a matrix and its inverse are always equal

I had a true or false quiz in a linear algebra course, one of the statements read the rank of a matrix and its inverse are ...
1
vote
1answer
41 views

What can I do to this expression to lose the summations?

I'm at the end of a past paper question and need to derive this answer: I am very close and have got to this by doing d/dx to the * equation: What can I do to get rid of these summation signs ...
2
votes
2answers
220 views

Powers of permutation matrices.

Let $P$ be a permutation matrix obtained by the identity matrix by switching 2 rows $n$ times, (with no two rows switched more than one time). How to show that $$P^{\ n+1} = I$$? Is it true that, ...
1
vote
0answers
29 views

Linear gradient equation in the plane

Observe the following equation: $V(y)=\frac{a}{2}y_1^2+\frac{b}{2}y_2^2+c y_1y_2$ a) Find the matrix $A$ which you find on the right hand side of the equation. b) Calculate the Trace, determinant, ...
3
votes
1answer
96 views

Eigenvalues of $ADA^T$

Consider a rectangular matrix $A\in\mathbb{R}^{M\times N}$ and a diagonal matrix $D\in\mathbb{R}^{N\times N}$. What can one say on the eigenvalues and eigenvectors of $ADA^T$? For example, if we ...
0
votes
2answers
54 views

System of 3 equations with 3 unknowns with 6 solutions

I was trying to solve this equation that I constructed $$x^3 - 6x^2 + 11x - 6 = 0$$ I know the solutions is 1, 2 and 3. But I wanted to see if I could solve the equation, so I messed around a bit and ...
0
votes
0answers
33 views

Change of basis help?

Question: Find the change of basis matrices P(C<---B) and P(B<---C) for the basis B={1, x, x^2} and C = {1+x, x+x^2, 1+x^2} of P2. Then find the coordinate vector of p(x) = 1+2x-x^2 with ...
1
vote
3answers
86 views

Proof of triangle inequality for $d(x,y)=\sqrt{\lvert x-y\rvert}$

There is this problem that says: show that $d(x,y)=\sqrt{\lvert x-y\rvert}$ is a distance function on $\mathbb{R}$, and I am unable to proof the triangle inequality for this? any suggestion I look ...
0
votes
1answer
30 views

Eigenvectors of linear transformation

Assume that a linear transformation $T$ has two eigenvectors $x$ and $y$ belonging to distinct eigenvalues $\lambda$ and $\mu$. If $ax + by$ is an eigenvector of $T$, prove that $a=0$ or $b=0$.
2
votes
2answers
48 views

Spectral Radius and Norm of multiplied vector

Let $\mathbf{A}$, $\mathbf{B}$ be square matrices of equal dimensions, $\mathbf{w}$ a vector of compatible dimensions and $\rho$ be the spectral radius operator. Does the following hold? If $\rho ...
2
votes
1answer
59 views

Relation between Tensor Product and Homomorphism [duplicate]

We know that there is a natural isomorphsm between $$V^*\otimes W \text{ and } Hom(V,W)$$ whenever either $V$ or $W$ is finite dimensional. (We also know that there always exists a linear map from ...
10
votes
3answers
233 views

Show that $A$ is symmetric, with $A \in M_n(\mathbb R)$

Let $A \in M_n(\mathbb R)$. Show that if $A(\,{}^t\!A)A$ is is symmetric, then $A$ is also symmetric. My attempt: If $A \in Gl_n(\mathbb{R})$, We have : ${}^t(A^{-1})=(\,{}^t\!A)^{-1}$ ...
0
votes
3answers
51 views

Eigenvalue of a linear transformation substituting $t+1$ for $t$ in polynomials.

Let $V$ be the linear space of all real polynomial $p(x)$ of degree $\leq n$. If $p \in V$, define $q=T(p)$ to mean that $q(t)=p(t+1)$ for all real $t$. Prove that $T$ has only the eigenvalue $1$. ...
0
votes
1answer
129 views

Is a positive definite matrix times a positive semidefinite matrix positive semidefinite?

I have a couple questions actually, couldn't get both of them in the title. First suppose I have a positive definite matrix H and a positive semidefinite matrix D. I believe that H*D would be a ...
2
votes
1answer
35 views

Unitary transformation between complete and orthonormal bases

I'm using the Dirac notation for vectors here, since this is a quantum mechanics question. Suppose the complete orthonormal bases $\{|\psi_n\rangle\}$ and $\{|\psi{'}_n\rangle\}$ are related by the ...
0
votes
0answers
30 views

Compound inequalities with separate variables

So I've got to find the set of elements for which the following inequality holds true: $$x-2<12\leq6-5x$$ I've only been taught to solve these kinds of inequalities with one x in the middle, so I ...
1
vote
1answer
186 views

Negative eigenvalue of doubly stochastic nonnegative matrices

I am studying n by n doubly-stochastic entry-wise positive matrices. I was wondering if there are any necessary and sufficient conditions for the existence of a negative eigenvalue for such a ...
3
votes
2answers
340 views

Conditions for a matrix to be invertible

Let $n \geq m$ and let $C$ be a $n \times m$ full rank matrix, that is $rank(C) =m$. Considering that $D$ is a diagonal positive semidefinite matrix, under which conditions is the $ m \times m$ matrix ...
1
vote
1answer
95 views

What does it mean when a matrix is to the (-1/2) power?

I'm reading a machine learning paper that uses a form of matrix normalization called symmetric divisive; given a matrix A and a diagonal matrix D derived from A, we define $$N=D^{-1/2}AD^{-1/2}$$ I am ...
3
votes
1answer
112 views

Lower bound on the smallest eigenvalue

Recently I encountered a lower bound on the minimum eigenvalue of positive Hermitian matrices in (Lower bound on smallest eigenvalue of (symmetric positive-definite) matrix). The lower bound is stated ...
-1
votes
1answer
25 views

Determinant of complex matrices

Suppose A1 and A2 real matrices, A = A1+iA2 and A*=A1-iA2. If det(A) = a + ib, how can I prove that det(A*)=a-ib?
1
vote
1answer
41 views

Find a general formula for x_k

The sequence $x_k$... is defined by $x_0 = 0, x_1 = 2$, and $x_{k+2} = 6x_{k+1}−13x_k$ for $k≥0$. Find a general formula for $x_k$. I actually came here because I found a solution on here for a ...
0
votes
2answers
49 views

basis for subspace of set of transformations

Given $S$ a subspace of the set of linear transformations $T : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $T(x) = ax$ for some scalar $a$, how do you find the basis for $S$?
1
vote
2answers
127 views

Example: Sum of non-commuting matrices is not normal

I'm trying to find an example of two non-commuting normal matrices, such that their sum is not normal. I know that unitary, orthogonal, hermitian and symmetric matrices are all normal. I figure it ...
1
vote
1answer
3k views

Find an orthogonal matrix P and a diagonal matrix D such that $P^TAP$=D

The symmetric matrix $A$ below has distinct eigenvalues $−6, −12$ and $−18$. Find an orthogonal matrix $P$ and a diagonal matrix $D$ such that $P^TAP=D$ $$ A = \left[ \begin{array}{ccc} -13 ...
0
votes
1answer
39 views

Show that the norms $ | p | _1 $ and $ | p | _2$ are not equivalent

Let P be a vector space of polynomials with real coefficients. Show that the norms $ | p | _1 $ and $ | p | _2$ are not equivalent, where $|p|_1$=max$ \{|p(t)|$; $0\leq t \leq 1 \}$ and $|p|_2$ = max ...
1
vote
2answers
363 views

Express the vector as a sum of two vectors

Express the vector $\vec{u}$ below as a sum of two vectors $\vec{u}_1$ and $\vec{u}_2$, where $\vec{u}_1$ is parallel to the vector $\vec{v}$ given below, and $\vec{u}_2$ is perpendicular to ...
1
vote
2answers
333 views

Find the closest point p in S to the point w, given

NOTE: Nobody showed me how to do this before. I AM DESPERATE for a step by step solution. Please help!! Let $S$ be the subspace of $\mathbb R^3$ spanned by vectors $u$ and $v$. Find the closest point ...
1
vote
1answer
54 views

A question about the proof of the transpose of the product being equal to the product of the transpose in reverse order.

Let $A, B$ be matrices of a size such that $AB$ is defined. Then $(AB)^t = B^t A^t$ where $t$ denotes the transpose. Proof: Let $A = (a_{ij})$ and $B = (b_{jk})$. Then $AB = C = (c_{ik})$ where ...
3
votes
3answers
660 views

Prove that $AB=BA$ if $A, B$ are diagonal matrices

Could you confirm my proof? A fixed Proof (Confirm please): Let $A, B$ be two diagonal matrices of order $n$. Then, both $AB,BA$ are defined and are of the same order $n$ (i.e. sizes match). Also, ...
0
votes
1answer
102 views

orthogonal triangular decomposition and ordinary least squares

I have just come across orthogonal triangular decomposition whilst looking at ordinary least squares regression. I'm not quite sure how this is being used though to find a solution. In my example I ...
0
votes
1answer
37 views

how to complete arbitrary basis knowing 2 orthonormal vectors of Rd (d > 2)

In a paper the following statement is used: "To construct the matrix B, complete the vectors (y, x) to an arbitrary basis of Rd and then apply the Gram-Schmidt orthonormalisation". assume we know x ...
0
votes
5answers
54 views

How to solve this homogeneous system, with a missing column?

Find the solution set of triplets $(x,y,z)$ that fulfil this system using Gauss-Jordan: $$\begin {cases} -x + 2z = 0\\ 3x - 6z = 0\\2x - 4z = 0\end {cases}$$ First of all, I don't see any ...
1
vote
2answers
86 views

Obtaining $A$ from $A$$A^t$$A$

Let the matrix $B$=$AA^TA$ be given to us where A is a mxn real matrix.Than how can we obtain $A$ from $B$ ? Can we do the same thing if A is a complex matrix ? I have no idea how to do this ...
0
votes
2answers
90 views

Dense Countable basis on Hilbert space

Let say that I have a $H$ hilbert space and linear independent countable set $\beta =\{ \beta_1 , \beta_2, \beta_3... \}$ such that $span(\beta)$ is dense set in H. does $span(\beta-\beta_1) =span( ...
0
votes
1answer
22 views

Characterization of linear transformation with same Kernel and range

Can we characterize class of all linear transformation T:V -> V for which Kernel(T) = Range(T)?
1
vote
0answers
50 views

Using Gauss-Jordan for an infinite-solutions system

I'm starting to get the hang of this Gauss-Jordan stuff - well, I have never done a system with infinite solutions, so I decided to try this one. You can scroll to the bottom instead to see my doubts ...