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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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
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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
169 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
443 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 ...
1
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2answers
541 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
108 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
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1answer
545 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
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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
248 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
88 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
320 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 ...
0
votes
0answers
167 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
252 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
137 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
600 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
925 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 ...
2
votes
2answers
267 views

What are matrix coefficients in linear algebra?

What are matrix coefficients in linear algebra? And what does it mean "integer matrix coefficients"?
5
votes
2answers
555 views

Is every square traceless matrix unitarily similar to a zero-diagonal matrix?

This question asks for the symmetric case, but after consideration I believe that any complex square matrix with zero trace is unitarily similar to a matrix with zero diagonal. This answer to another ...
1
vote
1answer
268 views

$T,U$ self-adjoint, $U$ positive definite, then $TU$ has only real eigenvalues

Let $V$ be a finite dimensional inner product space over $\Bbb R$. Let $T$ and $U$ be linear self-adjoint operator on $V$. Assume $U$ is positive definite. Show all the eigenvalues of $TU$ are real. ...
0
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1answer
42 views

Can you plot a line without knowing the constant value?

A simple example : $Z=3x_1 + 2x_2$ Z is a constant with no given value. $x_1, x_2 >0 $ $ x_1 \leq 5$ $ x_2 \leq 8$ The lecturer said that the slope is $-3/2$ , which I understand. However, ...
2
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4answers
266 views

Special determinant formula for a specific matrix

How to show that the determinant of the following $(n\times n)$ matrix $$ \begin{pmatrix} 5 & 2 & 0 &0&0&\cdots & 0\\ 2 & 5 & 2 & ...
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2answers
201 views

What can you say about these matrices.

Let $A$ and $B$ be two $n\times n$ matrices over $\mathbb{R}$ such that $AB=BA=0$ and $A+B$ is invertible then: (1) Is there any relation between rank$(A)$ and rank$(B)$? (2) Is $A-B$ invertible? ...
0
votes
2answers
508 views

What is the rank of the cofactor matrix of a given matrix?

Let $A = (a_{ij}) ∈M_n(\mathbb{R})$; $n≥3$. Let $B = (b_{ij})$ be the matrix of its co- factors, i.e. $b_{ij}$ is the cofactor of the entry $a_{ij}$ in $A$. What is the rank of $B$ when a. the rank of ...
0
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0answers
56 views

Studying a nonlinear system under constraints in linear fashion

Suppose that $a = aBc$ where $a$ and $c$ are vectors and $B$ is some matrix that changes as time "continuously" goes on - making this system dynamical system. But suppose that at any time, if $B$ is ...
2
votes
3answers
359 views

Any suggestions for abstract algebra-multilinear algebra books?

I want to read a little about these: The characteristic polynomial and minimal polynomial of a $T \in\mathrm{End}(V)$, or given a matrix $A$, finding the Jordan form and when can I say it is ...
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4answers
1k views

Is a linear combination linearly independent?

I am a bit confused... Linear combination means $$F(X)=af(x_1)+bf(x_2) + \cdots$$ and linearly independent means $$af(x_1)+bf(x_2) + \cdots=0$$ where $a=b=\cdots=0$ My question: is a linear ...
1
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2answers
89 views

Matrices congruence

Let $A, B \in \mathbb{C^{n^2}}$ be hermitian matrices. If $A$ is positive-definite, then there exists an invertible matrix $S\in \mathbb{C}^{n^2}$ such that $S^*AS=I_n$ and $S^*BS$ is a diagonal ...
4
votes
1answer
335 views

Algorithm to find the basis of intersection of subspaces without gaussian elimination.

Is there an algorithm to find the basis of intersection of subspaces $A_1$ and $A_2$, if we have the bases of subspaces $A_1$ and $A_2$, without using Gaussian elimination? Thanks.
1
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3answers
87 views

positive definite matrix interval

Hi could you help me with following $$ \begin{pmatrix} 1 & a & a \\ a & 1 & a \\ a & a & 1 \end{pmatrix} $$ is a $3 \times 3$ matrix. Find the largest interval for a such ...
2
votes
1answer
174 views

Dependency of vectors in eigenspace corresponding with eigenvalue zero

The eigenspace corresponding with the eigenvalue zero is the same as the null space of the original matrix. All vectors in the null space are linearly independent so the eigenvectors of zero are also ...
1
vote
0answers
66 views

What is the best way of introducing singular value decomposition (SVD) in a linear algebra course?

Why is it so important? Are there any applications which have a real impact?