Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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154 views

Spectrum of doubly stochastic matrices

Let $M$ be a doubly stochastic matrix in which every entry is strictly positive. Prove that for any eigenvalue $\lambda$ we have $\lambda \neq 1 \Longrightarrow |\lambda|< 1$ and the geometric and ...
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2answers
94 views

Suppose R is a commutative ring, $A \in R_n$, and the homomorphism $f: R^n \to R^n$ defined by $f(b) = Ab$ is surjective. Show $f$ is an isomorphism.

How would I go about showing this? Suppose $R$ is a commutative ring, $A \in R_n$, and the homomorphism $f: R^n \to R^n$ defined by $f(b) = Ab$ is surjective. Show $f$ is an isomorphism. (Edit: ...
5
votes
3answers
376 views

prove that $\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$

If $A$ is a $m \times n$ matrix and $B$ a $n \times k$ matrix, prove that $$\text{rank}(AB)\ge\text{rank}(A)+\text{rank}(B)-n.$$ Also show when equality occurs.
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0answers
37 views

How to prove the existence of a factorization with the given precision for ID?

Let us decompose matrix $A \in \mathbb{R}^{m\times n}$ as the multiplication of matrices $B \in \mathbb{R}^{m\times k}$ and $P \in \mathbb{R}^{k\times n}$ where some subset of the columns of $P$ make ...
3
votes
1answer
66 views

How to be sure that the $k$th largest singular value is at least 1 of a matrix containing a k-by-k identity

In section 8.4 of the report of ID software, it says that the $k$th largest singular value of a $k \times n$ matrix $P$ is at least 1 if some subset of its columns makes up a $k\times k$ identity. I ...
4
votes
1answer
314 views

Largest eigenvalue of a $A^T A$ matrix?

I have a large real matrix A of size $40K\times 400K$, is there an efficient way to calculate the largest eigenvalue of $A^T A$ (size $400K\times 400K$)? Thanks.
0
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1answer
129 views

find linear transformation by given Images

Here is the question. I need to determine if there is linear transformation according to the given information. If there is I need to write it as $T(x,y,z)=(a,b,c)$. I don't know the algorithm for ...
0
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1answer
85 views

How to model a system for tracking a person using kalman filter?

I need to model a system for human motion. The following link shows for to build a system for a plane. I am currently reading the documentation for a kalman filter library ...
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3answers
230 views

Will $V+V^{\perp}$ always span $\mathbb{R}^n$?

Let's say we have a subspace $V$, that is a subset of $\mathbb{R}^n$. Does $V + V^{\perp}$ always span $\mathbb{R}^n$?
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1answer
96 views

How to solve this matrix?

at the moment I am a little bit confused. Here is the matrix I am trying to solve $$ \left( \begin{array}{cc|c} 5 & -1& 12 \\ -1 & 2& 12 \end{array} \right) $$ I tried ...
1
vote
1answer
225 views

Diagonalizable unitarily Schur factorization

Let $A$ be $n x n$ matrix. What exactly is the difference between unitarily diagonalizable and diagonalizable matrix $A$? Can that be that it is diagonalizable but not unitarily diagonalizable? What ...
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2answers
325 views

Invertability of submatrix?

If I have a matrix $A \in R^{(m \times n)}$ with $m \leq n$. All rows in matrix a are linearly independent and therefore $A$ has a full row rank. I can decompose matrix $A$ such that $A = [B|N]$ with ...
1
vote
1answer
47 views

Matrix column independence?

If I have a matrix $A \in R^{(m \times n)}$ with $m \leq n$. All rows in the matrix are linearly independent. Does it hold that I can select any $m$ columns from $A$ and they will also be linearly ...
4
votes
2answers
106 views

How to prove randomly-generated high-dimensional $0$-$1$ vectors are probably independent?

Today our teacher told us that if you randomly generated ten 100-dimensional $0$-$1$ vectors, it's very unlikely that they are dependent in $\mathbb{R} ^ {100}$. To be specific, every entry has equal ...
3
votes
2answers
81 views

Find vectors from inner products

The problem is given all the inner products of N N-dimensional vector, can we work out the vector set? For example given a N-by-N matrix $K$ $\{K\}_{ij} = v_i^Tv_j$ for all $i,j \in \{1,...,N\}$ Is ...
0
votes
1answer
182 views

Proving the group of homomorphisms is isomorphic to matrices

I'm trying to understand this theorem: If $f, g: R^n \to R^m$ are given by the matrices $A, B \in R_{m,n}$ then $f + g$ is given by $A + B$. Thus, $\operatorname{Hom}_R (R^n, R^m)$ and $R_{m,n}$ are ...
2
votes
1answer
60 views

For a non-square matrix $X$, what conditions must be satisfied so that $X^t\cdot X$ results in the identity matix?

If $X$ is an $N\times M$ real valued matrix (with $N < M$), the product of its transpose with itself ($X^t\cdot X$) results in a square $M\times M$ matrix. Is there some simple property that $X$ ...
5
votes
1answer
171 views

Complexity of a quadratic program

I have a quadratic program: $$\displaystyle\min_{\mathbf{X}} (\mathbf{X^TQX +C^TX}) \quad{} \text{subject to} \quad{} \mathbf{A X \leq Y}$$ $\mathbf{Q}$ is positive definite and is $N \times N$, ...
5
votes
2answers
448 views

Norm inequality for sum and difference of positive-definite matrices

If $X_{1}$ and $X_{2}$ are positive definite matrices, how to show that $\left\Vert X_{1}-X_{2}\right\Vert \le\left\Vert X_{1}+X_{2}\right\Vert$ for the spectral norm? and how about for the nuclear ...
5
votes
4answers
154 views

Should $x=-2$ be included as an answer for $\frac{x^2+8x+12}{x^2+5x+6}>0$?

$$\frac{x^2+8x+12}{x^2+5x+6}>0$$ First of all while solving inequalities I need to check domain so in this case $$x^2+5x+6\neq0$$ $$x\neq-2,\ x\neq-3$$ Later on ...
2
votes
1answer
358 views

Determine a formula for a dual basis.

Let $\beta= \{ (2,1),(3,1) \} $ be an ordered basis for $\Bbb R^2$. Suppose that the dual basis of $\beta$ is given by $\beta^*= \{f_1,f_2 \} $ To explicitly determine a formula for $f_1$ we need to ...
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2answers
546 views

Perron-Frobenius theorem

In the proof of the Perron-Frobenius theorem why can we take a strictly positive eigenvector corresponding to the eigenvalue $1$? Before that, why can we even take a non-negative eigenvector? Books ...
6
votes
1answer
119 views

eigenvalues of $C=\begin{bmatrix}−I &-I\\L&0\end{bmatrix}$

Consider the following matrix $$C=\begin{bmatrix}−I &-I\\L&0\end{bmatrix}$$ where for $L$ we have: $$L\mathbf{1}=0$$ $$\mathbf{1}^TL=0$$ $$\text{rank}(L)=\dim(L)-1$$ $$L+L^T\geq 0$$ zero is a ...
7
votes
2answers
173 views

Difference between Kernel for Linear Maps and Group Homomorphisms

Suppose I am given $G_1,H_1$ as groups and $f_1: G_1 \to H_1 $ a group homomorphism. Then $$\ker f_1 := \{g_1 \in G_1 : f_1(g_1) = id_{H_1}\} \tag{1}$$ Suppose I am given $G_2, H_2$ as vector spaces ...
3
votes
1answer
81 views

properties of $\begin{bmatrix}-A& -B^T\\ -B &0\end{bmatrix}$

Consider the following matrix $$C=\begin{bmatrix}-A& -B^T\\ -B &0\end{bmatrix}$$ where $A>0$ and B is a matrix such that the diagonal entries of B are all zero and the rest of the entries ...
1
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0answers
109 views

fixed point spectral radius

We have the following stationary matrix iteration $$x_{k+1} = Mx_k + c$$ where $M$ is nxn matrix and $c$ is a vector. Let $r(M)$ denote the spectral radius of $M$. Show that spectral radius ...
2
votes
2answers
206 views

least square problem normal equations

Can you give an example which shows that loss of information can occur in forming the normal equations. How is accuracy improved using iterative improvement? Thank you
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1answer
2k views

Matrix equation solver (Mathematica) [closed]

I have matrix $$A = \left( \begin{array}{ccc} -1 & 1 & 1 \\ 2 & 3 & -1 \\ -2 & 3 & 3 \end{array} \right)$$ and matrix $$B = \left( \begin{array}{ccc} -1 & 2 & -1 \\ ...
0
votes
2answers
74 views

Is it true that for any square matrix of real numbers A, there exists a square matrix B, such that AB is a symmetric matrix?

Is it true that for any square matrix of real numbers $A$, there exists a square matrix $B$, such that $AB$ is a symmetric matrix? This is obviously true if $A$ is invertible, but how about if $A$ is ...
1
vote
2answers
72 views

How to find the homogeneous equation of non-homogeneous equation?

I have homework and I don't understand the request. this is the task: (I'm translating from Hebrew, so I'm sorry for unclear details, if there are): Solve the following linear equations , and ...
18
votes
3answers
306 views

Bound on nilpotency index of endomorphisms

Let $A$ be a Noetherian ring (commutative with $1$) and $M$ a finitely generated $A$-module. I want to show that there exists a bound $n$ such that for every nilpotent endomorphism $T : M \to M$ we ...
2
votes
2answers
50 views

The relation between unknown (specific example)

I have a certain problem that I've managed to convert to a matrix problem. I have 3 unknown variables and the problem is defined by a 3x3 matrtix and a 3x1 vector. From the nature of the problem the ...
2
votes
2answers
148 views

How to calculate the rotation of a vector?

So, let's say I have vector $\vec{ab}$ and vector $\vec{ac}$. How do I calculate the amount of rotation from $b$ to $c$? Note, this is in a 3D space, of course...
1
vote
1answer
551 views

Lower bound on norm of product of two matrices

Let $\vert \vert . \vert \vert$ be the 2-norm. Since this norm is submultiplicative, we know that for any two square matrices $A, B \in \mathbb{R}^{n \times n}$, $$ \vert \vert A B \vert \vert \leq ...
2
votes
1answer
1k views

Do eigenvectors always form a basis?

Suppose we have a $n \times n $ matrix over $\Bbb R$. Is it necessary that we should have $n$ linearly independent eigenvectors associated with eigenvalues so that they form a basis? Can you give ...
4
votes
4answers
249 views

$A$ and $B$ are $3\times 3$ real matrices such that $\operatorname{rank}(AB)=1$, then $\operatorname{rank}(BA$) can not be which of the following?

I was thinking about the problem that says: If $A$ and $B$ are $3\times 3$ real matrices such that $\operatorname{rank}(AB)=1$, then $\operatorname{rank}(BA)$ can not be which of the following? ...
3
votes
2answers
63 views

computing with unitary matrices

I am currently working on a problem and I am stuck with the following issue. For $A \in GL(n)$ and $B \in U(n)$ I am hoping that it is true that $$ A(B-A)^{-1}B = B(B-A)^{-1}A $$ My question is ...
5
votes
1answer
199 views

trace of the matrix $I + M + M^2$ is

Let $ \alpha = e^{\frac{2\pi \iota}{5}}$ and the matrix $$ M= \begin{pmatrix}1 & \alpha & \alpha^2 & \alpha^3 & \alpha^4\\ 0 & \alpha & \alpha^2 & \alpha^3 & ...
2
votes
1answer
89 views

Approximating a function with a convex function

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous, differentiable function. Is there a known algorithm that fits $f$ with $g$, which is an order-$n$ polynomial that is convex, in the least ...
5
votes
1answer
121 views

Laplacians, Diagonal Perturbations

Setup: Consider a Laplacian (or Kirchoff) matrix $L = L^T \in \mathbb{R}^{n \times n}$ corresponding to a weighted, undirected and connected graph. That is, a matrix with $L_{ij} \leq 0$ for $i\neq j$ ...
2
votes
2answers
81 views

Sufficient condition for a matrix to be hermitian

Let $A\in \mathcal{M_{n\times n}}(\mathbb{C})$ be a matriz. Does $x^*Ax>0$ for all $x\in \mathbb{C}^{n\times 1}$ such that $x\neq 0_{\mathbb{C}^{n\times 1}}$ implie that $A$ is hermitian? Please ...
4
votes
3answers
322 views

Image of function definition notation

In my Linear Algebra and Geometry textbook, it defines the image of a linear transformation $T$ as: $$\operatorname{Im}\, (T) := \{\; w \in W : \; w=Tv \;\;\text{ for some } v \in V \} $$ As far as ...
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0answers
170 views

Jacobi method for determining the canonical form

$f : \mathbb R^3 \to \mathbb R, f(x_1, x_2, x_3) = 3x_1^2 - x_2^2 - 2x_3^2 - 4x_1x_2 - 2x_1x_3 + 6x_2x_3$ I am trying to find $f$'s canonical form using Jacobi's method and I don't know how to ...
1
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0answers
94 views

solve system of linear equation $AX=b$ and define determinate for this matrix

Assume that matrix $A$ define in this form $$A=[a_{ijk}] , a_{ijk} \in F$$ ($F$ is arbitrary field ). The size of $A$ is $m \times n \times k$ ($m$ is number of row and $n$ is number of column and ...
3
votes
3answers
254 views

for a $3 \times 3$ matrix A ,value of $ A^{50} $ is

I f $$A= \begin{pmatrix}1& 0 & 0 \\ 1 & 0 & 1\\ 0 & 1 & 0 \end{pmatrix}$$ then $ A^{50} $ is $$ \begin{pmatrix}1& 0 & 0 \\ 50 & 1 & 0\\ 50 & 0 & 1 ...
3
votes
4answers
138 views

Let $A$ be a $2\times2$ real square matrix of rank $1$. If $A$ is not diagonalizable, then which of the following is true

Let $A$ be a $2\times2$ real square matrix of rank $1$. If $A$ is not diagonalizable, then which of the following is true. (a) $A$ is nilpotent (b) $A$ is not nilpotent (c) the characteristic ...
2
votes
1answer
162 views

Do solutions for these matrix equations always exist?

Suppose I have a matrix: $$A \in \mathbb{R}^{n \times m}$$ and another one (same size): $$W \in \mathbb{R}^{n \times m}$$ When is it possible to find a square matrix $L$ such that: $$L\cdot ...
2
votes
0answers
110 views

DFT shift theorem generalizations?

The DFT shift theorem implies that any circular shift in the input space is equivalent to a phase change in the frequency domain, while the absolute values are preserved. $$ ...
5
votes
1answer
604 views

Positive definite matrix must be Hermitian

Is there a simple way to show that a positive definite matrix must be Hermitian? I feel there is a long drawn out proof of this to be had by taking unit vectors and applying the positive definiteness ...
3
votes
3answers
934 views

Intuitive proof of multivariable changing of variables formula (jacobian) without using mapping and/or measure theory?

iWhat is a intuitive proof of multivariable changing of variables formula (jacobian) without using mapping and/or measure theory? I was thinking that textbooks make the proofs over complicate. If ...