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
votes
1answer
16 views

Calculate matrix $X$ in expression $X + B = (A-B)X$

I have to calculate matrix $X$ in expression $X + B = (A-B)X$. $$ A=\left[ \begin{array} k1 & -2 & 3\\ 2 & 4 &0\\ -1 & 2 & 1\\ \end{array} ...
0
votes
0answers
10 views

Relationship between eigenvectors of two matrices

Suppose I have matrix $A$ given by $X^{-1} diag(W - iY, W + iY) X$ and Matrix $B = W + iY$. Let $v$ be an eigenvector of $A$. How can I relate $v$ to some eigenvector of $B$?
6
votes
1answer
27 views

Show that $Y$ is invertible

Let X be a $40\times40$ matrix such that $X^3 = 2I$. I want to show that $Y= X^2 -2X + 2I$ is invertible as well. I tried working with the equations to see if I can get Y as a product of matrices ...
1
vote
1answer
41 views

Find spectrum for matrix $A$

Let $A = \left[ \begin{array}{*{20}{c}} 0&b&0&0&0&0\\ c&0&b&0&0&0\\ 0&c&0&b&0&0\\ 0&0&c&0&b&0\\ ...
2
votes
4answers
81 views

Logic behind finding a $ (2 \times 2) $-matrix $ A $ such that $ A^{2} = - \mathsf{I} $.

I know the following matrix "$A$" results in the negative identity matrix when you take $A*A$ (same for $B*B$, where $B=-A$): $$A=\begin{pmatrix}0 & -1\cr 1 & 0\end{pmatrix}$$ However, I am ...
1
vote
1answer
36 views

find spectrum matrix A

Let $A = \left[ \begin{array}{*{20}{c}} 0&b&0&0&0&0\\ c&0&b&0&0&0\\ 0&c&0&b&0&0\\ 0&0&c&0&b&0\\ ...
1
vote
0answers
13 views

Skew-symmetric non-degenerate bilinear form

If we do symplectic linear algebra on a finite-dimensional vector space $V$, then what does $$\omega(v,w) \neq 0$$ or $$\omega(v,w) = 0$$ actually tell us about the vectors $v,w$? ($\omega$ is the ...
5
votes
4answers
156 views

Any hint about solving this monster determinant?

I'm asked to solve the following determinant: $$|A|= \begin{vmatrix} 1 &2 &3 &\cdots &{n-1} &n\\ 2 &3 &4 &\cdots &n &1\\ \vdots &\vdots &\vdots & ...
0
votes
0answers
17 views

Show convergence of Power method

Given a symmetric positive definite matrix $A_0 \in R^{n \text{x} n}$ with Cholesky decomposition $A_0 = LL^T$. How can I show that $A_k$ converges to $diag(\lambda_1, ..., \lambda_n)$ where $A_k$ is ...
1
vote
1answer
79 views

How to prove that the rank and the nullity of similarity invariants are the same?

Given matrix $A$ and $P^{-1}AP$ how do prove that the $\mathrm{rank}(A)$ and $\mathrm{rank}\left(P^{-1}AP\right)$ are the same? Also, how do you prove that the $\mathrm{nullity}(A)$ and ...
1
vote
4answers
25 views

Basis of a $2\times2$ matrix

How would I find the basis for an arbitrary matrix W such that: $$ W =\left\{ \begin{pmatrix} a & b \\ c & a +b +c\end{pmatrix} \ \big| \ \ a ,b ,c \in \mathbb{R} \right\} $$
1
vote
1answer
26 views

convergence of singular values

I jus want to know how to show that if a matrix X converge to Y ( with respect to any matrix norm) then the ith singular value of X converge to the ith singular value of Y. Thank you
-1
votes
0answers
21 views

How do I fit this piece of code on one line in Latex [migrated]

I am trying to have three matrices on one line as i) A=, ii) B= and ii) C=. I tried \nopagebreak, \noindent just after item. Instead I always get the Roman numeral on one line, a comma on the next ...
0
votes
2answers
38 views

Cofactors and determinant

Anyone can explain to me why $\det(C)=\det(A)^{n-1}$ where $A$ is $n$-by-$n$ matrix and $C$ is the matrix of cofactors of $A$. I have been thinking, anyone can help? Thanks!
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, ...
1
vote
1answer
25 views

How to find a “flag base” to an endomorphism?

I found several exercises that ask me to find a flag base for a given matrix, for example: $$ A=\left( \begin{array}{ccc} -1 & 1 & 0 \\ 2 & 2 & 4 \\ -1 & -2 & -3 \end{array} ...
0
votes
0answers
17 views

Equating eigenvalues of Hermitian matrix and correlating symmetric/antisymmetric matricies

I have a matrix $AH$ which is created by adding $AS$ and $i*AA$, which are the symmetric and antisymmetric components of the matrix $A$ So $AS=(A+A')/2$ $AA=(A-A')/2$ $AH=AS+i*AA$ AH has ...
0
votes
1answer
13 views

matrix sampling and its rank preservation

Assuming matrix $X\in R^{m\times n}$ is row orthogonal of rank $m$. Then, if I construct a new matrix $Y\in R^{m\times t}$, whose columns are directly sampled from $X$ with or without replacement ...
0
votes
1answer
26 views

Reflection matrix in $ \mathbb{R}^{3} $.

I need help in understanding how they got the transformation matrix $ Q_{L} $ from Theorem 2 and $ P_{M} $ at the bottom of the page. They skipped some steps and I find it confusing. Any help would ...
1
vote
0answers
12 views

Can independence of a system and a vector be establish if there is only cross-indepedence?

Say that I have the following linear system: $$[A a'] \begin{bmatrix} x \\ x' \\ \end{bmatrix} =Ax + a'x' $$ I want to know when this system is zero if and only if ...
0
votes
1answer
24 views

Linear operators proof, projection and reflection matrices

I am trying to understand two parts from the picture below in my textbook, but I dont understand how they arrived at it. I am trying to understand the proof below and how they got $P_L(\vec{v}) = ...
0
votes
0answers
16 views

About lower bounds on the size of irreducible representations of subgroups of symmetric groups.

Is there a subgroup $G_n$ of $S_n$ (one $G_n$ for each $S_n$) increasing in size such that their permutation representations are such that the smallest non-trivial irreducible size in them is ...
-14
votes
0answers
73 views

Exist the proof of Goldbach's Conjecture… is it correct? [on hold]

Every even integer > 2 is the sum of two prime numbers & equivalent Each odd integer > 5 is the sum of three prime numbers USING THE SIEVE OF ERATOSTHENES ...
0
votes
0answers
20 views

How to find spectral radius of ${0,1}$ and ${0,1,-1}$ matrices?

[this is kind of a continuation of this question ] It seems that the following is true, Among $n=3$ dimension symmetric matrices over $\{0,1\}$ which have $d=7$ ones the maximum spectral radius is ...
1
vote
0answers
22 views

Maximal ideals of the ring of matrices

Let K be a field. We consider $K^n$ as a left module of $M_{n, n}(K)$, the ring of matrices of size $n$ over $K$. 1) For any $M_{n, n}(K)$ module homomorphism $ 0 ≠ \phi: M_{n, n}(K) \to K^n$, show ...
-1
votes
3answers
41 views

Determine 9 variables by 3 equations with approximation

I have an equation in the form of Q*d=z, where Q is 3by3 matrix of variables, and d and z are vectors of 3 known numbers. What would be the best way to compute all 9 elements of matrix Q, provided ...
1
vote
2answers
56 views

An (open?) problem about a sequence of nested sub-matrices and their determinant

I had an idea. Let us start with an example. Consider the matrix $$ A = \left[ \begin{array}{ccc} 1 & 1 & 0 \\ 1 & 1 & 1 \\ 1 & 0 & 1 \end{array} \right] $$ It is invertible, ...
0
votes
1answer
33 views

How to find the dimension of the given vector space

Let $L=\{p(B)|\ p\ \text{is a polynomial with real coefficients}\},$ where $B =\begin{pmatrix} 0 & 1 &0\\0 & 0&1\\ 1&0&0\end{pmatrix}.$ Then the dimension $\;d\;$ of the vector ...
0
votes
1answer
452 views

Calculating eigenvectors and eigenvalues of a 2x2 complex matrix

I've previously asked elsewhere, http://stackoverflow.com/questions/21118820/non-trivial-eigenvectors-of-a-22-matrix-in-code, how to calculate the eigenvectors and eigenvalues of a 2x2 matrix in a ...
0
votes
2answers
43 views

Process of finding the eigenvalues of a 3x3 matrix

I'm trying to find the eigenvalues of a 3x3 matrix in order to eventually find an orthogonal matrix $Q$ and diagonal matrix $D$ such that $Q^TAQ = D$, where $A$ is a symmetric matrix, however I'm not ...
2
votes
0answers
47 views

Argument for the zero vector not being defined as an eigenvector

Two days ago, my lecturer of Advanced Numerical Methods gave a review on the topic about eigenvalues and eigenvectors. Just as the lecturer presented the definition of eigenvalues and eigenvectors, a ...
0
votes
1answer
349 views

Diagonally Dominant Matrix Preserved after Gaussian Elimination (with a modification)

Prove or disprove: If a matrix has the property $0 \neq |a_{ii}| \geq \sum_{\substack{j=1 \\ j \neq i}} |a_{ij}| $ then Gaussian Elimination (without pivoting) will preserve this property. I assume ...
1
vote
1answer
22 views

Transformation and matrices

Two sequences $y_t$ and $z_t $ satisfy $$y_t = ay_{t-1} + bz_{t-1}$$ $$z_t = cy_{t-1} + dz_{t-1}$$ Where $a = 6$, $b = -20$, $c = -17$ and $d = -12$. From the two given equations above, ...
2
votes
0answers
33 views

Let A be an m × n matrix, and b an m × 1 vector, both with integer entries.

Let $A$ be an $m \times n$ matrix, and $b$ an $m \times 1$ vector, both with integer entries. If $Ax = b$ has a solution over $ \mathbb Z/p \mathbb Z $ for every prime $p,$ is a real solution ...
0
votes
1answer
20 views

Similar matrices represent an operator relative to different bases

I need to prove the following Let $A,C$ be two similar matrices over the field $\mathbb{F}$. Define $T_A : \mathbb{F}^n_{\text{col}} \to \mathbb{F}^n_{\text{col}}$ as $T_A(x) = Ax$. ...
1
vote
2answers
30 views

Prove that $\det(A) \cdot v \, A^{-1} = \det(A+uv) \cdot v \, (A+uv)^{-1}$.

Let $A$ be a $n \times n$ matrix, $u$ a $n \times 1$ matrix and $v$ a $1 \times n$ matrix. If $A$ and $(A+uv)$ are invertible, prove that $$ \det(A) \cdot v \, A^{-1} = \det(A+uv) \cdot v \, ...
1
vote
1answer
671 views

Proof that Gauss-Jordan elimination works

Gauss-Jordan elimination is a technique that can be used to calculate the inverse of matrices (if they are invertible). It can also be used to solve simultaneous linear equations. However, after a ...
1
vote
2answers
25 views

Why does the Gaussian-Jordan elimination works when finding the inverse matrix?

In order to find the inverse matrix $A^{-1}$, one can apply Gaussian-Jordan elimination to the augmented matrix $$(A \mid I)$$ to obtain $$(I \mid C),$$ where $C$ is indeed $A^{-1}$. However, I fail ...
4
votes
2answers
703 views

Irreducible Representations of Matrix Algebras

I am currently reading the book Spin Geometry by Lawson/Michelsohn to understand Dirac Operators and related topics. At some point it uses representation theory to classify Clifford Algebras. In ...
3
votes
2answers
47 views

Show that there exists a $3 × 3$ invertible matrix $M$ with entries in $\mathbb{Z}/2\mathbb{Z}$ such that $M^7 = I_3$.

Show that there exists a $3 × 3$ invertible matrix $M$ (which is not the identity matrix) with entries in the field $\mathbb{Z}/2\mathbb{Z}$ such that $M^7 = $Identity matrix. All I could do was use ...
0
votes
1answer
19 views

Can we represent the curl as a multiplication by skew-symmetric matrix?

Considering that two vectors $A \times B$ = $\hat A* B$, where $\hat A$ is a skew symmetric matrix containing elements of $A$ Can we then write the curl $\nabla \times A$ as $\partial \vec r *A$ ...
0
votes
1answer
26 views

Cholesky factorization exist?

Is there a theorem or a way to show that if I have a real and symmetric positive definite matrix $A$ and its Cholesky factorization is $A = LL^T$ then $B = L^TL$ is also positive definite? Or in other ...
0
votes
0answers
19 views

The $2$-norm of a Hermitian matrix does not exceed its $1$-norm

How to prove that the $2$-norm of a Hermitian matrix does not exceed its $1$-norm? In wiki, I see $2$-norm of matrix $A$ is $\le \sqrt{\|A\|_1\|A\|_\infty}$. But I don't know how to prove that ...
1
vote
1answer
610 views

Frobenius Inequality Rank

I was looking for an answer for this problem in terms of matrices, but I really don't know how to prove this result. The proposition says that: Let $A\in M_{m\times k}(\mathbb{C})$, $B\in M_{k\times ...
0
votes
1answer
18 views

Determinant of a matrix with specific main diagonal

Determine the determinant of the following matrix: $$A = \begin{pmatrix}1+a_1 &1 &\cdots &1 \\ 1 &1 +a_2&\ddots& \vdots \\ \vdots & \ddots &\ddots&1 \\ 1 & ...
1
vote
0answers
25 views

Spectral Radius of a Block Matrix

I have real matrix $P$ obtained from numerical solution (FEM) of a physical problem, as \begin{equation} P=P_1+P_2= \begin{bmatrix} A_{2n \times 2n}&B_{2n \times n}\\C_{n \times 2n}&D_{n ...
4
votes
3answers
94 views

Show that if $AA^t = A^tA$, then $A=A^t$

Suppose $A$ is a matrix with non-negative real entries. If $A^tA = AA^t$, show that $A=A^t$. My proof says: $AA^t = A^tA = (AA^t)^t$. I can't seem to get to the point of $A=A^t$ Edit: What if $A$ is ...
-1
votes
1answer
41 views

How to prove the equality of two matrix expressions

I am new to linear algebra and my question maybe too simple. I have a n-by-m matrix $D$ that its columns have unit L2 norm. Let $D_a$ be a sub-matrix of $D$ composed by some columns of $D$. I need to ...
0
votes
1answer
20 views

How to prove that the column sum for a markov matrix is 1?

As is the topic, it is obvious and easy to explain in non-math language but how do I mathematically prove it?
2
votes
1answer
55 views

Largest eigenvalues of AA' and A'A [on hold]

Prove that for every real matrix $A$, the largest eigenvalue of $A'A$ equals the largest eigenvalue of $AA'$ (where ' means transpose). Thanks!