For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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0answers
55 views

Can I replace the {1,3}-inverse by the Moore-Penrose inverse for Minimum Norm Solution?

Suppose the equations $Ax=b$ are compatible, and the Moore-Penrose inverse of $A$ is known as $A^{\dagger}$. In order to calculate the Minimum Norm Solution $$x = A^{(1,4)}b$$ I take the advantage of ...
2
votes
3answers
322 views

Vector multiplication. Difference between scaler and dot product?

We just started a new class where the first topic is briefly talking about vectors and vector multiplication. All tying this into neural networks. I am a bit behind with the understanding of what the ...
0
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2answers
179 views

Why do Matrices work the way they do?

You get taught about matrices and how they work but nobody ever tells you WHY they work in the way that they do. What was the idea that sparked the creation of matrices?
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1answer
85 views

Guess the formula of a matrix

Given a matrix $A$ of size $2\times2$ . $A^2$, $A^3$,$A^4$,and $A^5$ are calculated as seen above. It is required that : Based on your calculation above, Guess a formula for $A^{2n}$ and ...
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1answer
40 views

relationship between matrix and adjoint of a matrix and orthonormal system

The problem is: Let $A$ be a $m\times n$ matrix. Show that if $A^*A=I$, the $n\times n$ identity matrix ($A^*$ is the adjoint), then the columns $g_1,\cdots,g_n$ in $A$ constitute an orthonormal ...
2
votes
2answers
114 views

Internet problem solving contest question

I am trying to solve a problem from the IPSC http://ipsc.ksp.sk/2001/real/problems/f.html It basically asks to compute the following recursion. ...
0
votes
1answer
67 views

Module, End(M), acts

What it means for a module $M$ and the endomorphism ring $\text{End}(M)$ that $\text{End}(M)$ acts diagonally on $M^n$, where $n$ is a positive integer? Also, if $G=(r_{ij})$ is a matrix with ...
1
vote
1answer
49 views

Matrices - is this matrix in reduced row echelon form?

I have a matrix: $$\begin{bmatrix}1&0&0\\0&0&0\\0&0&1\end{bmatrix}$$ Im just wondering if it is or not...
1
vote
1answer
194 views

Solving an equation involving a trace: find $X$ in $M=\mathrm{Tr}[CX]$

So, I have this algebra problem: I have an equation of the form $$M_{ij}=\sum_{A,B}C_{ij}^{AB} X^{AB} \equiv \mathrm{Tr}\big[C_{ij}X\big]$$ where upper upper-case letters are some kind of ...
8
votes
1answer
167 views

Linear combination of matrices

Let $A, B, C, D$ be four linearly independent symmetric 3 x 3-matrices over $\mathbb K$. Show that some linear combination of these matrices is a matrix of rank 1. I know it is supposed to be a ...
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2answers
93 views

Invariant Factors of the Zero Linear Transformation

Show that the zero linear transformation has invariant factors (and elementary divisors):$$q_1=x,q_2=x,\cdots,q_n=x$$ Here is my idea so far. If we have the zero linear transformation defined on ...
2
votes
1answer
302 views

Gauss-Jordan Elimination to solve for variables

I have the following linear system: $$x + 2y - 3z = 4$$ $$3x - y + 5z = 2$$ $$4x + y + (s^2 - 14)z = s+2$$ Im trying to solve for $s$ to figure out how many solutions it has (if any). I know how to ...
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3answers
177 views

Matrices - Find the rank and determine if its invertible

Find the rank of $A = \begin{bmatrix}2&1&-4\\-4&-1&-6\\-2&2&-2\end{bmatrix}$ and explain why $A$ is not invertible. What I have done is: Guass-Jordan Elimination: ...
1
vote
1answer
43 views

Matrices - Find matrix E

Suppose $A = \begin{bmatrix}1&2&-1\\1&1&1\\1&-1&0\end{bmatrix}$ and $D = \begin{bmatrix}1&2&-1\\-3&-1&3\\2&1&-1\end{bmatrix}$. I need to find the matrix ...
0
votes
2answers
1k views

Write the following in the form of AX = B

Write the following system of equations in the form $AX = B$, and calculate the solution using the equation $X = A^{-1}B$. $$2x - 3y = - 1$$ $$-5x +5y = 20$$ I'm not the strongest at linear algebra ...
4
votes
3answers
1k views

Find $2\times 2$ matrices $A$ and $B$ such that $AB=0$ but $BA$ does not equals to $0$

Find $2\times 2$ matrices $A$ and $B$ such that $AB=0$ but $BA$ does not equals to $0$ (please show working and the concepts used) Thanks :)
1
vote
1answer
265 views

Alternating multilinear map and products

I was reviewing some school notes from many semesters ago and I came across a point which I wish to prove but can't. Let $F$ be a field (real or complex for example), and we say $\delta : ...
0
votes
0answers
75 views

Show that $H$ is a normal subgroup of $G$? [duplicate]

Let $\mathbb M(n;\mathbb R)$ denote the set of all real matrices (identified with $\mathbb R^{n^2}$ and endowed with its usual topology) and $GL(n;\mathbb R)$ denote the group of all invertible ...
2
votes
3answers
201 views

Cayley-Hamilton Theorem

I am trying to prove that all strictly upper triangular $n \times n$ matrices $A$, are nilpotent such that $A^n=0$. I am having trouble proving: $A$'s eigenvalues are all zero implies that ...
1
vote
1answer
30 views

Find a matrix A

Find a matrix $A$ such that $(A - 3\mathcal{I}_2)^{-1}$ = $\begin{bmatrix}1&2\\3&4\end{bmatrix}$ I dont understand what the question is asking and how to solve it! Any ideas?
0
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3answers
235 views

Matrices - Find the value(s) of constant k

Find the values of the constant $k$ such that $(k$A$)^T(k$A$) = 28$, where: $$A = \begin{bmatrix}-1\\2\\-3\end{bmatrix}$$ Actually, I got no idea how to solve this. how do i solve this? Can you ...
0
votes
1answer
121 views

max induced norm of matrix

I have to prove that matrix norm $||A||_\infty$ induced by vector norm $||x||_\infty = \smash{\displaystyle\max_{1 \leq k \leq n}} |x_k|$ where $x_k$ is k-th element of vector can be disribed by ...
0
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3answers
55 views

Matrices - Prove A and B are symmetric 2 x 2

Prove that if $A$ and $B$ are both symmetric 2x2 matrices, then $A$ + $2B$ is also a symmetric matrix. The problem I have with this question is proving. How do I prove that? All I understand to do is ...
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1answer
66 views

convexity of a Hessian matrix.

Suppose I have $f(x_{1},x_{2}) = x_{1}^2 + x_{2}^2, S = \mathbb{R}^2$. How do I determine whether the function is concave or convex based off of the Hessian of what is above? I know the Hessian is ...
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votes
2answers
79 views

$n^{th}$ root of a matrix.

What conditions do I need on a matrix $A$ in order to know an $n^{th}$ root exists. In other words there is a matrix $B$ such that $B^n=A$ for $n \in \mathbb{Z}^+$.
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2answers
120 views

Matrices - Find x, y and z

I have two matrices $A$ and $B$ and I'm trying to figure out what $x$, $y$, and $z$ are. $$\begin{bmatrix}x+2y&x\\-x+y&2x-y\end{bmatrix} = ...
0
votes
1answer
64 views

Finding the transformation matrix of this linear map.

I've being doing several exercises and none was of this kind, which I can't figure out: Let $V$ and $W$ be vector spaces with basis $B=\{\vec{v_1},\vec{v_2},\vec{v_3}\}$ and ...
4
votes
1answer
75 views

Jordan's decomposition

I have a matrix $A\in R^{n,n}$. $A= \begin{bmatrix} 1&0&-2&0&0&\dots&0\\ 0&1&0&-6&0&\dots&0\\ 0&0&1&0&-12&\dots&0\\ ...
1
vote
2answers
85 views

A question about diagonalizable matrices

Let $A$ be a square matrix such that $A \ne0$, but $A^k=0$ for some integer $k \gt1$. show that $A$ is not diagonalizable. Could somebody give me some hints?Many thanks
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1answer
535 views

inverse of quadratic matrix form

I have an expression of the form: $ACA′$ where C is an invertible, symmetric and positive definite matrix. I'm trying to figure out if the expression above is invertible (or what additional ...
0
votes
1answer
43 views

Which of the following subsets are dense in the given spaces?

The sets of trignometric polynomials in thespce of continous functions on $[-\pi,\pi]$ which are 2$\pi$ periodic (with the sup norm topology). The subset of $C^{\infty}$ function with compact support ...
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vote
1answer
25 views

Continuity of the mapping of a lattice in $\mathbb{R}^{d}$ onto the length of the shortest vector

I have the following problem: Let $X_{d}:=\{\mathbb{Z}^{d}g;g\in\operatorname{SL}_{d}(\mathbb{R})\}$ and define $\lambda:X_{d}\to(0,\infty)$ by: $$ \lambda(\Lambda):=\min\left\{r>0;\Lambda\cap ...
0
votes
1answer
164 views

How to write a linear map as a matrix with respect to a given canonical basis

I am asked to write a linear map as a matrix with respect to a given canonical basis. The basis is $b = \left \{ \begin{bmatrix} 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right \} ...
2
votes
3answers
81 views

Why can matrix exponentiation be done by squaring?

Matrix multiplication is not communative: A*B != B*A Then why can matrix exponentiation be done by squaring? I have tried searching for special cases where this rule did not apply, but from what I've ...
1
vote
1answer
56 views

Is it true that positive definite matrices generates all the symmetric matrices?

Is it true that positive definite matrices generates all the symmetric matrices in $M_n(\mathbb{R})$? And is it true that the set of nonsingular symmetric matrices generates all the symmetric ...
4
votes
2answers
188 views

List of connected Lie subgroups of $\mathrm{SL}(2,\mathbb{C})$.

I am not familiar with the theory of Lie groups, so I am having a hard time finding all the connected closed real Lie subgroups of $\mathrm{SL}(2, \mathbb{C})$ up to conjugation. One can find the ...
3
votes
2answers
140 views

Proving algebraically that $\mathbb RP ^3\cong SO(3,\mathbb R)$

I am giving a simple introductory course on algebraic geometry and I plan to mention that $$\mathbb RP ^3\cong SO(3,\mathbb R).$$ I know a rather simple proof of this using the fact that $\mathbb ...
1
vote
1answer
274 views

How to find a function mapping matrix indices?

There are several posts dealing with this kind of topic, but I so much would like to know, how to tackle such little problems. This seems so logical and still for me it's so hard to crack. Given a n x ...
0
votes
1answer
51 views

Equivalence of Lattices

Let $\Gamma=\{mw_1+nw_2:m,n\in\mathbb{Z}\}$ and $\Gamma'=\{mw_1'+nw_2':m,n\in\mathbb{Z}\}$. Show that $\Gamma=\Gamma'$ if and only if there exists a matrix $A\in SL(2,\mathbb{Z})$ such that $\left( ...
0
votes
1answer
48 views

Can anyone tell me how to do multiple rank update of a non square matrix?

I had seen the rank one update for pseudo inverse in book "Generalized inverses of linear transformation" by S.L. Campbell. But i want to do rank 3 or rank 4 update in my existing matrix. Please tell ...
0
votes
2answers
858 views

Given the following vector $X$, find a non-zero square matrix $A$ such that $AX=0$:

Given the following vector X, find a non-zero square matrix $A$ such that $AX=0$: So this problem stumped me and I've resorted to stack exchange. I need to find $A$, when I have a vertical vector ...
0
votes
2answers
156 views

Solving homogeneous systems using Gaussian elimination

I have a system of equations: $$2x_1 + 6x_2 - 4x_3 = 0$$ $$3x_1 + x_2 + 7x_3 = 0$$ $$4x_1 - x_2 + 2x_3 = 0$$ I have tried to solve it, but I'm stuck at this part: ...
2
votes
2answers
120 views

Understanding Gauss-Jordan elimination

I have a following system: $$x_1 + x_2 - x_3 = 5$$ $$2x_1 + 2x_2 - 4x_3 = 6$$ $$x_1 + x_2 - 2x_3 = 3$$ I dont understand how to solve this system using Gauss-Jordan elimination. I was told it I had ...
1
vote
3answers
287 views

Matrices - Understanding row echelon form and reduced echelon form

I have the following two matrices: 1) $$\begin{bmatrix}1&0&0\\ 0&1&1\\ 0&0&0\\0&0&0 \end{bmatrix}$$ I believe this matrix is in the form of reduced row echelon form ...
2
votes
1answer
95 views

Nilpotent matrix in $\mathbb R$

We can prove that $A \in \mathbb C^{n,n}$ is nilpotent ($\exists m\ A^m=0$) if $p_A(t)=t^n$, where $p_A$ is the characteristic polynomial of matrix $A$. What if $A \in R^{n,n}$? Proof that ...
0
votes
1answer
29 views

Fundamental Matrix with Sums

Let $$\Phi(t)=\begin{bmatrix} x_{11}(t) & x_{12}(t)\\ x_{21}(t) & x_{22}(t) \end{bmatrix} $$ be a fundamental matrix for $$x'=A(t)$$ where $$A=\begin{bmatrix} a_{11}(t) & a_{12}(t)\\ ...
0
votes
1answer
59 views

Integrating a Matrix in differential-equations

Find $$\int_0^t A(s)ds$$ if $$A(t)=\begin{pmatrix}\sin(t),\cos(t)\\ -\sin(t),\cos(t)\end{pmatrix}$$ I'm a little confused with the format of the question because it asks me to integrate with respect ...
0
votes
2answers
445 views

Raising $e$ to the power of a matrix

Does there exist a definition for matrix exponentiation? If we have, say, an integer, one can define $A^B$ as follows: $$\prod_{n = 1}^B A$$ We can define exponentials of fractions as a power of a ...
0
votes
1answer
42 views

finding this linear transformation

i am following this guide: http://www.calpoly.edu/~brichert/teaching/oldclass/f2002217/handouts/goof.pdf my question is to find the linaer transformation that adheres to $T(1,1,1) = (1,1,1)$ ...
0
votes
1answer
54 views

can one matrix have 2 inverses?

can this matrix have 2 inverse? $$\left(\begin{matrix}-0.173796&0.003408&0.170298&0.052286&-0.000000&-0.000000\\ ...