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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0
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1answer
439 views

Linear Transformation Reflection Equation

Linear Transformation $$T(\vec{x}) =\left(\begin{array}{cccc} 0.6&0.8\\0.8&-0.6\\ \end{array}\;\begin{array}{c}\end{array}\right)\vec{x}$$ is a reflection about a line L. I need to find the ...
1
vote
2answers
146 views

Matrix with same image and kernel

Does exists a matrix A for which kernel of A is the same as the image of A? Answer is True. But I couldn't find the example. I think I saw it from somewhere but I can't find it. It was 2 by 2 ...
1
vote
1answer
64 views

Linear Algebra True False: Inconsistency

Q. The system $A\vec{x} = \vec{b}$ is inconsistent if and only if $rref(A)$ contains a row of zeros. Answer is False. But my answer was True because I was thinking of the following example. ...
4
votes
1answer
117 views

$M_n(D)$ has only finitely many right ideals if and only if $n = 1$ or $D$ is finite.

Let $D$ be a division ring. Then prove that $R = M_n(D)$ has only finitely many right ideals if and only if $n = 1$ or $D$ is finite. I know that the ideals of $M_n(D)$ are of the form $M_n(I)$, ...
1
vote
1answer
82 views

Matrix and Linear Algebra True or False problem.

Q. True or False: If matrix A is a reduced row-echelon form, then at least one of the entries in each column must be 1. It comes down to this question. Can I have the following as Reduced ...
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2answers
119 views

Linear Equations using matrix and variables on the line

Q. I need a find a system of linear equations with three unknown variables whose solutions are the points on the line through (1,1,1) and (3,5,0). $ \frac{x-1}{2} = \frac{y-1}{4} = \frac{z-1}{-1}$ ...
2
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1answer
77 views

Linear Equation unknown variables and number of equations

Q. True of False The four linear equations with Three unknown variables is always inconsistent? Is it true or false? I thought of this example $$\left(\begin{array}{ccc} ...
8
votes
1answer
204 views

Intersection of Algebraic Varieties.

Let $\mathbb K$ be an algebraically closed field. Consider the set $M_n(\mathbb K)$ of all matrices of order $n$. Identify the set $M_n(\mathbb K)$ with the affine space $\mathbb A^{n^2}_{\mathbb ...
5
votes
1answer
116 views

What is the isomorphism function in $M_m(M_n(\mathbb R))\cong M_{mn}(\mathbb R)$?

What is the isomorphism function in $M_m(M_n(\mathbb R))\cong M_{mn}(\mathbb R)$. I tried this $[[a_{ij}]_{kl}]\mapsto[a_{ijkl}]$ , but I couldn't prove all steps.
2
votes
2answers
69 views

A commutator problem

Let us consider $N \times N$ complex matrices, with $N >2$. Let D be a diagonal matrix, with $$D_{kk} = \sin \left(\frac{2\pi k}{N}\right), \space k = 0,..N - 1$$ I am looking for two ...
2
votes
1answer
50 views

how to prove $2^{n-1}|\det(A)$ where $A=[a_{ij}]\in M_n(\mathbb R)$ and $a_{ij}\in\{-1,1\} $

let $A=[a_{ij}]\in M_n(\mathbb R)$ such that $a_{ij}\in\{-1,1\} $ then how prove $$2^{n-1}|\ \det(A)$$ Thanks in advance
3
votes
1answer
111 views

Find the smallest square matrix in which some objects fit following some rules

I have to put some objects in a matrix. The data of these objects is given in another matrix in which each line contains an object, and the first column represents its width, and the second its ...
0
votes
1answer
1k views

Rank normal form of a matrix

There is a standard result in matrix theory that goes like this: Suppose $A$ is an $m\times n$ matrix of rank $r$, then there exist two non-singular matrices $E$ (of size $m\times m$) and $F$ (of size ...
2
votes
1answer
65 views

are normal subgroups of $SL(2,\mathbb{Z})$ also normal under the action of integer matrices in $GL(2, \mathbb{Q})$?

Ie, if $\Gamma\subseteq\text{SL}_2(\mathbb{Z})$ is a normal subgroup, and $\alpha\in\text{GL}_2(\mathbb{Q})\cap M_2(\mathbb{Z})$, then is $\alpha\Gamma = \Gamma\alpha$? (if necessary we can assume ...
-1
votes
1answer
148 views

by finding the eigenvalues and eigenvectors the evaluate the following.

so the question is : by finding the eigenvalues and eigenvectors of the matrix $$ P=\begin{bmatrix}1&6\\0&-2\end{bmatrix}\qquad\text{evaluate }P^{20}\begin{bmatrix}-2\\1\end{bmatrix} $$ I ...
4
votes
1answer
814 views

what does it mean for a matrix to be greater than another?

I am reading these notes on viscosity solutions, here is a theorem: Let us assume $u\in C^2$ is a classical solution of $F(x,u,Du,D^2u)=0$, $x\in \Omega$ then $u$ is a viscosity solution whenever ...
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vote
1answer
343 views

How do I write this $t^2$ as a linear combination of polynomials in the basis?

I have this homework problem that says "In $\mathbb P_2$ find the change of coordinates matrix from the basis $\mathcal B=\{1-3t^2, 2+t-5t^2,1+2t\}$ to the standard basis. Then write $t^2$ as a linear ...
0
votes
1answer
67 views

$[T]^{\gamma}_{\beta}=[v]_{\gamma}$ with $\beta=\{1\}$ a basis for $F$

Let $V$ be a finite-dimensional vector space over $F$ with basis $\gamma$ and let $v\in V$. Find a linear map $T:F \rightarrow V$ such that $[T]^{\gamma}_{\beta}=[v]_{\gamma}$, where $\beta = \{1\}$ ...
2
votes
1answer
3k views

How Does One Find A Basis For The Orthogonal Complement of W given W?

I've been doing some work in Linear Algebra for my course at school. I just want to be clear about how to find the orthogonal complement of a subspace. The basis for the subspace, W, is shown below, ...
8
votes
1answer
87 views

*-homomorphisms $M_n(\mathbb{C})\rightarrow M_m(\mathbb{C})$

I've heard that every *-homomorphism $\phi:M_m(\mathbb{C})\rightarrow M_n(\mathbb{C})$ is unitarily equivalent to some *-homomorphism of the form $A\in ...
0
votes
1answer
38 views

Equal Shape: Recovering an Isomorphism Between $M_{3\times 2}(F)$ and $P_5(F)$

I'm asked to find an isomorphism between $M_{3\times 2}(F)$ and $P_5(F)$, but what does it mean for a $3\times 2$ matrix to have an inverse?
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6answers
4k views

If $A^2$ is invertible, then $A$ is also invertible?

True or False: If $A^2$ is invertible, then $A$ is also invertible. ($A$ is a matrix here.) The answer is true. I was trying to come up with an example that makes this false. But I couldn't. ...
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2answers
78 views

Number of Solutions in Linear System

$$\left(\begin{array}{cccc} 1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&0 \end{array}\;\middle\vert\;\begin{array}{c}2\\3\\4\\0\end{array}\right)$$ This is a $4$ ...
2
votes
1answer
95 views

elementary but confounding question about integer matrices (related to hecke operators)

Let $\Gamma(N)$ denote the kernel of the reduction map $\text{SL}_2(\mathbb{Z})\rightarrow\text{SL}_2(\mathbb{Z}/N\mathbb{Z})$ Let $p$ be a prime that is $1$ mod $N$, and let $M$ be the set of ...
2
votes
1answer
535 views

Commuting in Matrix Exponential

Let $A$, and $B$ be commuting $n\times n$ matrices, i.e. $A.B = B.A$. Let \begin{equation} \exp(A) = \sum_{i=0}^\infty\frac{1}{i!} A^i \end{equation} show that $\exp(A+B) = \exp(A).\exp(B)$.
1
vote
2answers
288 views

How to prove the existence of solution of a non linear system of equations

Writing the ortogonality condition for any element of O(n), I've arrived to: If we take n=2, we know that $\Lambda\Lambda^{T}=\mathbb{I}$, so: $$\begin{pmatrix} x & y \\ z & t \end{pmatrix} ...
2
votes
3answers
2k views

Is there any standard notation for specifying dimension of a matrix after the matrix symbol?

I want to explicitly specify dimension of matrices in some expressions, something like $$\boldsymbol{A}_{m \times n} \boldsymbol{B}_{n \times m} = \boldsymbol{C}_{m \times m} \, .$$ Is there any ...
0
votes
1answer
217 views

Augmented Matrix with a constant in 'A'

I have an augmented matrix defined: $$\left[\begin{array}{ccc|c} 1& 0& 2& 1\\ 0& 1& -1& 2\\ 1& -2& k+4& 5 \end{array}\right]$$ ...
1
vote
1answer
244 views

Question related to diagonally dominant matrix

A matrix is said to be positive if each entry in the matrix is positive. If $A$ is real, irreducible, diagonally dominant (or strictly dominant matrix) and has positive diagonal and non-positive ...
0
votes
1answer
55 views

Stability of a matrix

Suppose the hermitian part $H$ of a complex matrix $A$ be defined by $H=\frac{A+A^\ast}{2}$ and the skew hermitian part $S$ by $S=\frac{A-A^\ast}{2}$. If the hermitian part $H$ of $A$ is negative ...
1
vote
1answer
47 views

Combining elimination matrices

I am trying to combine several elimination steps into one matrix: more specifically I try to come up with a 3 by 3 matrix that first subtracts row 1 from row 2, subtract row 1 from row 3 and then ...
0
votes
2answers
234 views

Gram-Schmidt verifying orthonormal basis

Gram-Schmidt If I have an orthonormal basis, how do I verify that they are indeed orthonormal? I have Q, R and A is it enough to times Q` by Q to give me I? or A=QR? Edit: Let's say I ...
0
votes
1answer
488 views

Finding upper triangular matrix

I have this question, and im not sure I know how to solve it. "Find an upper triangular $U$ (not diagonal) with $U^2 = I$ which gives $U=U^{-1}$". Anybody who can help me getting the first steps of ...
0
votes
1answer
95 views

Invertibility of a monotone matrix.

I have a question regarding monotone matrix. How to prove that monotone matrix is invertible?
2
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3answers
120 views

Revisited: How is $\phi:{\cal{L}}(V,W)\rightarrow M_{m\times n}(F)$ an isomophism of vector spaces?

I'm told in lecture that if $V,W$ are vector spaces over $F$ and ${\cal{L}}(V,W)$ is the vector space of all linear maps $V\rightarrow W$ and ${\scr{B}}$ and ${\scr{C}}$ are bases for $V$ and $W$ ...
4
votes
1answer
51 views

Sequence of matrices, equivalent conditions

I am trying to prove that: Let $B$ be a square matrix. The following conditions are equivalent: $\lim\limits_{k\rightarrow\infty}B^k = 0$ $\lim\limits_{k\rightarrow\infty}B^kv = 0$ for ...
3
votes
0answers
70 views

How to check whether it is possible to solve this problem?

We have a matrix with numbers. We can add $1$ to any selected element and this action adds $1$ to elements according to some function, which I'll call the $X$-function. For instance, $1$ could be ...
0
votes
2answers
61 views

What is $[T]^{\scr{C}}_{\scr{B}}$?

What does it mean for $[T]^{\scr{C}}_{\scr{B}}\in M_{m\times n}(F)$ to be a matrix of $T$ in basis $\scr{B}$ in $\scr{C}$?
1
vote
1answer
79 views

Show that $P = Q^2$

Suppose $P$ is a positive semi-definite $n\times n$ matrix. Show that there exists a unique positive semi-definite matrix $Q$ such that $P = Q^2$. In class we've been going over singular value ...
1
vote
0answers
803 views

Does a Symmetric Matrix with main diagonal zero is classified into a separate type of its own? And does it have a particular name?

For example, I have a Matrix as shown below. Does this Matrix belong to a particular type. I am CS student and not familiar with types of Matrices. I am researching to know the particular Matrix type ...
5
votes
2answers
639 views

Matrix Calculus in Least-Square method

In the proof of matrix solution of Least Square Method, I see some matrix calculus, which I have no clue. Can anyone explain to me or recommend me a good link to study this sort of matrix calculus? ...
2
votes
1answer
1k views

What's the trick for proving one eigenvalue of orthogonal matrix is $-1$ if the determinant is $-1$?

Obviously, the magnitude of the orthogonal matrix is 1, which is easy to prove.. However, I wonder how can one prove that the eigenvalue of an orthogonal matrix is $-1$, if the determinant of this ...
0
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2answers
42 views

Finding vector of co-ordinates

How do we find a co-ordinate vector in Algebra? For example, given: $$ \begin{align*} \left(\begin{matrix} 2 & -3 \\ 0 & -4 \end{matrix}\right) & \left(\begin{matrix} v_1 \\ v_2 ...
4
votes
2answers
77 views

Does there exist an $n\times n$ real matrix $A$ exist such that $e^{e^{A}} - I_n$ is singular?

I have one doubt whether an $n\times n$ real matrix $A$ exist such that $e^{e^{A}} - I_n$ is singular? I think I have to show that 1 is the eigenvalue of $e^{e^{A}}$ in case answer is yes. But I am ...
2
votes
1answer
41 views

Find the basis for $\text{Im} \, ψ$ of a matrix transformation

Let $\psi\colon\mathrm{Mat}_{ 2\times 2 }(\mathbb R) \to \mathrm{Mat}_{ 2\times 2 }(\mathbb R)$ be defined by $$\psi\colon \pmatrix{a&b\\c&d}\to\pmatrix{a+b&a-c\\a+c&b-c}.$$ Find ...
2
votes
1answer
2k views

Rotation Matrices - Rotating a point on a graph

I'm trying to understand how rotation matrices work in Linear Algebra... I don't think I'm visualizing it correctly though... I'd like to rotate a point (-2, 1) around a graph... the point (-2, 1) ...
0
votes
1answer
77 views

proof about deteminant of a complex linear transformation

say I have a linear space $V$ over $\Bbb C$ and a linear transformation $T:V \to V$ such that $T=A+iB$ where $A,B \in \Bbb R^{n \times n}$ I proved already that $T_\Bbb R = \begin{pmatrix} A & -B ...
0
votes
1answer
55 views

Question on matrix stability and definiteness

Is it true that $AX + XA^*$ is necessarily negative definite if $A$ is stable and $X$ is positive definite. If so, give a proof. If not produce a counter example. I am unable to solve it.
0
votes
1answer
53 views

Question related to non-negative matrix

I am facing a problem in this question. Let $y$ be fixed. Prove that $x \geq y$ implies $Ax \geq y$ for all $x$ if and only if $A \geq 0$ and $Ay \geq y$. A≥0 implies each entry in the matrix A ...
2
votes
2answers
428 views

Find the basis for kernel of a matrix transformation

$Let \ ψ\ :{Mat }_{ 2x2 }(ℝ)\ →\ { Mat }_{ 2x2 }(ℝ)\ be\ defined\ by$ $ ψ : \pmatrix{a&b\\c&d}→\pmatrix{a+b&a-c\\a+c&b-c}$ . Find basis for ker ψ I'm not sure how to do it for a ...