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
15 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 ...
0
votes
0answers
11 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}) = ...
-2
votes
3answers
27 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
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0answers
17 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
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2answers
38 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
32 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 ...
1
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0answers
28 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
2answers
37 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 ...
4
votes
3answers
123 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 & ...
2
votes
0answers
23 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 ...
1
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1answer
16 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} ...
1
vote
1answer
19 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, ...
0
votes
1answer
17 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
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2answers
21 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 ...
3
votes
2answers
45 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
18 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
19 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 ...
1
vote
2answers
28 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
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0answers
22 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 ...
0
votes
1answer
17 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 & ...
0
votes
0answers
15 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 ...
4
votes
3answers
92 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 ...
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votes
1answer
39 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
2answers
44 views

Does there exist a steady state vector of this Markov Matrix?

Does there exist a steady state vector of Markov Matrix $$P=\begin{bmatrix} \frac{1}{2} & \frac{1}{3}\\ \frac{1}{2} & \frac{2}{3} \end{bmatrix}$$ Initially I was not sure whether to answer ...
0
votes
1answer
42 views

What is the determinant of the sum of a diagonal matrix and a matrix of ones?

Given a square matrix, all elements outside of the main diagonal being equal to $1,$ what is its determinant?
1
vote
1answer
35 views

How to find a eigenvector with a repeated eigenvalue?

The eigenvalues of my matrix are $x_1= 1$ and $x_2=3$ I get an eigenvector $V = t~[ 4~~~~~~ 3 ~~~~~1 ]^T $ but how can I diagonalize the matrix if I have the same column repeated twice. Should I ...
2
votes
1answer
54 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!
0
votes
3answers
60 views

If a matrix A square is 0, does it follow that A = 0? [duplicate]

Let A be a square matrix. If $A^2 = 0$, then it follows that $A = 0$. Is there a counterexample for this? If there isn't, what kinds of explanation can I make to justify this statement?
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0answers
9 views

Transitivity of a Boolean Matrix?

I'm wondering if there's an easy way of visually telling if a boolean matrix has transitivity? The question in particular is: ...
0
votes
0answers
3 views

Link between the cofactors of two related symmetric positive-definite matrices

Let $S = \left( s_{ij} \right)_{i,j\in\left\{1...d\right\}^2}$ be a symmetric positive-definite matrix. Let $\Sigma = \left( \sigma_{ij} \right)_{i,j\in\left\{1...d\right\}^2}$ be a symmetric ...
-3
votes
1answer
23 views

Prove whether the statement is true or sometimes false. [on hold]

Prove whether the statement is true or sometimes false. If matrix A has row of zeros, does adj(A) have it also?
0
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0answers
10 views

there is a unitary $U \in {M_m}$ such that $X = YU$. Why $X$ and $Y$ have the same range?

Let $X,Y \in {M_{n \times m}}$ have orthonormal column. Also there is a unitary $U \in {M_m}$ such that $X = YU$. Why $X$ and $Y$ have the same range(column space) ?
0
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0answers
14 views

Triangulation of matrices

Suppose that $A$ is some triangularizable matrix in $M_n(\mathbb R)$. The usual approach I know of to find a triangular matrix similar to it is to find bases for all the eigenspaces, then find their ...
0
votes
0answers
8 views

Fast modular trace of matrix exponentiation using Fermat's little theorem for matrix

The question might be related to http://stackoverflow.com/questions/12268516/matrix-exponentiation-using-fermats-theorem but is slightly different as I only concentrate on the trace of the matrix. I ...
-1
votes
0answers
13 views

inequalities for $tr(AB)$ , where A and B symmetry, positively definite matrix

Let $A$ and $B$ be two symmetry, positively definite $n\times n$ matrix with positive eigenvalue $a_1,...,a_n$ and $b_1,...,b_n$ respectively. What's the relationship between them and $tr(AB)$? Are ...
0
votes
1answer
11 views

$X$ and $Y$ have the same range(column space). Why there is a unitary $U \in {M_m}$ such that $X = YU$?

Let $X,Y \in {M_{n*m}}$ have orthonormal column. Also $X$ and $Y$ have the same range(column space). Why there is a unitary $U \in {M_m}$ such that $X = YU$?
0
votes
0answers
25 views

Solution of a general linear system of equations: 4-term n-equations

I have the following system of equations.... $$y_1 = c_{11} \cdot x_{11} + c_{12} \cdot x_{12} + c_{13} \cdot x_{13} + c_{14} \cdot x_{14}$$ $$y_2 = c_{21} \cdot x_{21} + c_{22} \cdot x_{22} + ...
0
votes
2answers
40 views

Dot and Cross Product Proof: $u \times (v \times w) = ( u \cdot w)v - (u \cdot v)w$

How do you prove that: $u \times (v \times w) = ( u \cdot w)v - (u \cdot v)w$ ? The textbook says as a hint to "first do it for $u=i,j$ and $k$; then write $u-xi+yj+zk$ but I am not sure what that ...
3
votes
2answers
31 views

Unitary Matrices and the Hermitian Adjoint

I saw in a definition for unitary matrices, that for a complex matrix being unitary if $M: \mathbb{C}^{n} \rightarrow \mathbb{C}^{n}$ is unitary, or: $\langle Mv, Mw \rangle = \langle v,w \rangle$ ...
2
votes
0answers
46 views

Restoring Bidiagonality to a Matrix in SVD Algorithms

Good Afternoon, I am implementing the Golub-Reinsch SVD algorithm and am having difficulty with a boundary case Given a bidiagonal matrix of the form: $$ \begin{bmatrix} b11 & ...
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votes
1answer
26 views

What would be the basic solution of this maximization problem? [on hold]

Maximize $P=40x_1+50x_2$ Subject to $x_1+6x_2 \leq 72$ $x_1+3x_2 \leq45$ $x_1, x_2 \geq0$
2
votes
2answers
50 views

What's the easiest way to find all $\alpha\in\mathbb{R}$ such that $\tiny\left(\begin{matrix}1&2\\2&\alpha\end{matrix}\right)$ is positive definite?

For which $\alpha\in\mathbb{R}$ is $$C:=\left(\begin{matrix}1&2\\2&\alpha\end{matrix}\right)$$ positive definite, positive semidefinite or indefinite? It seems to be a simple task, but for ...
6
votes
4answers
85 views

For matrices, if $AB=BA$, then does it follow that $B^{2}A=AB^{2}$?

Suppose $AB=BA$ ($A, B$ are $n\times n$ matrices). Does that mean $B^{2}A=AB^{2}$ ? I looked for counter cases and couldn't find any. I tried to prove this by multiplying both sides and comparing, but ...
1
vote
1answer
11 views

diagonalizing a matrix with random elements

Consider the matrix $A = \begin{pmatrix} cY & 0 \\ 2 & 1\end{pmatrix}$, where $c \in \mathbb{R}$ and $Y$ is a random variable that is uniformly distributed over $[0,1]$ (That is, $Y \sim ...
2
votes
1answer
39 views

Question about matrices?

I have been learning about matrices in my math class and I am confused as to how exactly they work. Take this example: $\left(\begin{array}{c c c c c | c} 1 & 4 & 1 & 0 & 0 & ...
1
vote
1answer
44 views

Finding Matrix of Linear Transformation from $R^2 \rightarrow R^2$

Let $T: R^2 \rightarrow R^2$ be given by: $$T(x_1,x_2) = (4x_1 -2x_2, 2x_1 +x_2)$$ And let $$B = \{(1,1), (-1,0)\}$$ be a basis for $R^2$. First, I write down the matrix of $$T = ...
0
votes
0answers
12 views

relation of dim kers of AB and B operators

I try to prove For any matrixes $A_{ms},B_{sn}$ $$\operatorname{rank}{A}+\operatorname{rank}{B}-s\leq\operatorname{rank}{AB}$$ First, as for any $X$ that $BX=0$ also $ABX=0$, that ...
2
votes
2answers
67 views

How do you find the determinant of this $(n-1)\times (n-1)$ matrix?

It's for a proof of Cayley's Formula, I know I'm being dumb and can't see it, how do I find the determinant of this $(n-1)\times (n-1)$ matrix where the diagonal entries are $n-1$ and the off diagonal ...
1
vote
1answer
18 views

A question about unitary block matrix

For $n,m \in \mathbb N$, let $M_{n,m}(\mathbb C)$ denote the set of complex $n \times m$ matrices and put $M_{n}(\mathbb C):=M_{n,n}(\mathbb C)$. For matrices $A \in M_{n}(\mathbb C), B \in ...
1
vote
0answers
12 views

Discrete fractional fourier transform matrix

I am trying to write a matlab code for some calculations based on Discrete fractional fourier transform. in this article: Optimal filtering in fractional Fourier. after equation (7) a notation Fa is ...