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Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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2answers
30 views

Geometric justification of a rotation matrix

From S.L Linear Algebra: We can define a rotation in terms of matrices. Indeed, we call a linear map $L: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ a rotation if its associated matrix can be ...
0
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1answer
20 views

algebraic expression of Matrix product [on hold]

Suppose $M = X^T \Delta X$, where $X$ and $\Delta$ are $P \times P$ matrices and $\Delta$ is symmetric. Can anyone give a simple algebraic expression of the matrix $M$?
2
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2answers
20 views

$A\in\mathbb R^{n\times n}$ has a single eigenvalue over $\mathbb C$, then $A$ is diagonalizable $\iff$ $A$ is a scalar matrix

So while playing with matrices a little, I found something that I think is true but I don't know how to prove in the general case. I tried to formalize it and this is the result: Let $A\in \mathbb ...
0
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1answer
8 views

Possible combinations of placing 14 different colors in a matrix 8x8

I have a matrix *8*x*8* where i can place color bricks. I can place bricks everywhere except on all four edges - So I have 60 different places where I can place bricks. I have 14 different colors of ...
0
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1answer
20 views

Can you find the z component in this system?

I will use uppercase letters to denote known variables and lowercase letters to denote unkown ones. Assume I have a vector $<X', Y', z'>$, and an invertible matrix $T$, and a vector $<x,y,z&...
0
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1answer
37 views

Does this special identity matrix have a name?

recently I stumbled upon the problem of defining a diagonal matrix whose elements are identity matrices of $dim = n$, where $n$ is the column/row index. For example, for $n=3$: $\mathbb{I}_3 = \left[{...
-1
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1answer
17 views

Is power distributable in product of two matrices? [on hold]

given two matrices A and B, is the following equality correct? $(AB)^n$=$A^n$$B^n$ for $n\in$ $\Bbb R$
0
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1answer
23 views

Matrix equation with several X

Solving matrix equation $A^2X-B^T = 3X$ (to find X), I'm trying to do next thing: $A^2X-B^T = 3X$ $A^2X-3X = B^T$ $(A^2-3)X = B^T$ Can we do it in that way and, if yes, what should we do with $(A^2 ...
4
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3answers
3k views

Proofs of Determinants of Block matrices [duplicate]

I know that there are three important results when taking the Determinants of Block matrices $$\begin{align}\det \begin{bmatrix} A & B \\ 0 & D \end{bmatrix} &= \det(A) \cdot \det(D) \ \ ...
0
votes
0answers
6 views

The Effect of Adding an Edge on the Laplacian of a Weighted Digraph

Let $G$ be a weighted digraph with Laplacian $L:=D-A$, where $D$ is the degree matrix and $A$ is the incidence matrix. Is there any result on the behavior of the eigenvalues of $L$ when we add an edge ...
0
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1answer
18 views

Find all units in the ring Z[i] = { a+bi : a,b ϵ Z } [duplicate]

Find all units in the ring $ Z[i] $= { $a+bi$ : $a,b$ ϵ $Z$ }. I faced a similar problem to find all the invertible matrices in $Z$. I concluded the solution must be all matrices of det ($\pm1$). I ...
1
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0answers
22 views

minimise the trace of a matrix over all column permutations

I have a 10x10 positive symmetric matrix, I need to find the optimal permutation of the columns in order to minimise the trace. I can't try all permutations because that would be a 10! problem. Any ...
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votes
1answer
23 views

If $\omega$ is non real cube roots of unity , then the eigenvalues of the matrix

If $\omega$ is a non-real cube root of unity, then what are the eigenvalues of the matrix $$\left[ \begin{matrix} 1 & 1 & 1\\ 1 & \omega & \omega^{2} \\ 1 & \omega^{2} & ...
0
votes
1answer
24 views

Find $\text{card}(T_n( \mathbb R ) \cap O(n))$.

Let $T_n( \mathbb R )$ be the set of upper triangular matrices of size $n$. Let $O(n)$ be the set of general orthogonal matrices and $SO(n)$ the set of special orthogonal matrices. Find the cardinal ...
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1answer
18 views

why is $ \{(0,x,z)|x,z\in R\}$ is a two dimensional subspace space of $R^{3}$ over $R$ but $\{(0,0,z)|z\in R\} $ $\cup$ $\{(0,x,0)|x\in R\}$ is not?

why is $ A=\{(0,x,z)|x,z\in R\}$ is a two dimensional subspace space of $R^{3}$ over $R$ but $B=\{(0,0,z)|z\in R\} $ $\cup$ $\{(0,x,0)|x\in R\}$ is not? i Think both are two dimensional as A has ...
1
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1answer
51 views

Properties of upper triangular matrix

let $A$ be the set of all invertible upper triangular matrics in $\mathbb{M}_n (\mathbb{R})$ where $n \ge 2$ then $A $ is choose the correct option $1.$ dense $2.$Nowheredense $3.$open ...
1
vote
1answer
336 views

Reconciling the 'column' and the 'row' pictures for matrices

I understand the column picture (namely considering a matrix equation $Ax=b$ as a vector equation with each column of the matrix $A$ scalarly multiplied by each coefficient of the vector $x$ to give ...
0
votes
1answer
21 views

Revisit “Inequality between Frobenius and nuclear norm”

I am reading the following question: Inequality between Frobenius and nuclear norm I know $$\|X\|_* = \sum_{i=1}^r \sigma_i(X)$$and $$\|X\|_F = \bigg(\sum_{i=1}^r \sigma^2_i(X)\bigg)^{1/2}$$ I try ...
1
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0answers
57 views

Householder transformations to upper triangular form

Let $A=\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{pmatrix}$. How to transform this matrix with Householder transformations to an upper ...
5
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1answer
42 views

Prove that $\langle\mathbf{A}, \mathbf{B}\rangle \leq \delta$ equals with $\|\mathbf{A}\|_*\leq\delta$ given $\mathbf{B}^T\mathbf{B}=\mathbf{I}_n$

Given an arbitrary matrix $\mathbf{A}\in R^{n\times n}$ and the basis matrix set $\mathbb{S}=\{\mathbf{B}\in R^{n\times n}: \mathbf{B}^T\mathbf{B}=\mathbf{I}_n\}$. Then how to prove: 1:If we have $\...
0
votes
2answers
22 views

If A is a nxn singular matrix, then it has a singular value = 0

This is a question on a testexam. But am I correct in assuming that a singular matrix has det = 0, which gives it an eigenvalue of 0 and that gives it a singular value of 0?
1
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1answer
852 views

Regression with a Vandermonde matrix.

From what I understand, when doing a least squares regression with a Vandermonde matrix, you're essentially solving the equation $y=Xa$ Where y is a vector of y-values, X is the Vandermonde matrix, ...
0
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0answers
15 views

Quadratic eigenvalue problem (QEP)

$Q(\lambda)x=0$ and $Q(\lambda) = \lambda^2 M+\lambda C +K$ are defined in enter link description here matrix M,C,K are $n\times n $-matrices. The thesis said that when M,C,K are real or hermitian. ...
0
votes
1answer
837 views

Prove the orthogonal matrix with determinant 1 is a rotation

Let's define "preserve orientation" in the following way (I am not sure it is right, pls point out if there is something wrong): For a linear transformation, we only need to check non-parallel ...
0
votes
2answers
49 views

determinant of the sum of two matrices det(A+B)

I have met with a formula, that $$\det(\lambda^2A + \lambda B +C) = \lambda^{2n} \det(A) + \text{ lower order terms}.$$ Here, $\lambda$ is a scalar and $A,B,C$ are $n \times n$-matrices. Can you help ...
6
votes
2answers
64 views

Showing $ 2 $ matrices are similar [duplicate]

I gotta show if or if not those $ 2 $ matrices are similar: $$ \left(\begin{matrix} 3 & 2 & -2 \\ 1 & 4 & 0 \\ -2 & 1 & -1 \\ \end{matrix}\right) $$ $$ ...
0
votes
0answers
30 views

Limit of Matrix power equals $\pi$ [on hold]

Denote matrix \begin{equation} A= \begin{pmatrix} 1-t&2-t\\-t&1-t \end{pmatrix} \qquad (0<t<1) \end{equation} and vector $u_n\equiv(u_{1n},u_{2n})^\top=A^n v$, where $v=(1,1)^\top$. $N(t)...
0
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1answer
14 views

Eigenvalues of a block off-diagonal matrix

Let $A_1,A_2 \in \mathbb{R}^{n \times n}$. Construct the block matrix $A$ as follows: $$A: = \left[ {\begin{array}{*{20}{c}} 0&{{A_1}}\\ {{A_2}}&0 \end{array}} \right]$$ My observation is that ...
3
votes
1answer
22 views

Find rank of AB, given that A has linearly independent columns and B has rank 2

I'm trying to prove to myself that given... Matrix A, which has linearly independent columns, and at least 2 columns... Matrix B, which has rank of 2 Their product, AB, will have rank of 2. I ...
0
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2answers
24 views

transition from one base to another

Given the polynomials $p_1,p_2,p_3,v_1,v_2,v_3 \in P_2$: \begin{gather*} p_1(t) := t^2 − 2t + 5, \qquad p_2(t) := 2t^2 − 3t, \qquad p_3(t) := t + 1, \\ v_1(t) := t^2 + 4t − 3, \qquad v_2(t) := t − 1,...
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2answers
15 views

Procedure to find a pair of matrices whose product has rank less than each matrix in the pair

I am trying to figure out how to find two rank-deficient matrices (not necessarily square) which when multiplied will have rank less than either of the original matrices. In other words I am looking ...
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votes
1answer
32 views

How to write relationship of b and c [on hold]

Hi I am doing a research which involves matrices, and I wish to find the relation between $b$ and $c$. The nxn matrix is given as follows: $$ \begin{bmatrix} 1/2 & -c & 1/2 & 0 & 0 ...
0
votes
3answers
39 views

If $A, B$ and $C$ are all non-zero matrices, and $AB = BC$, does it imply $A = C$?

I don't see how $A = C$ but I can't find an example that proves otherwise.
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1answer
26 views
1
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1answer
32 views

A matrix equation equivalence

Let $\Omega_{\, m\times m}$ be a real square positive definite symmetric matrix, $u_{m\times 1}$ is a vector, $I_{m\times m}$ is the identity matrix. Let $x$ be a solution of a matrix equation $$ u^...
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votes
2answers
27 views

If there are two identical rows or columns identical in a determinant, the value is 0. Is the converse true?

I know that $$A= \begin {vmatrix} q&a&a \\ z&b&b \\ x&c&c \\ \end {vmatrix} =0 $$ Because in determinants, if we have to two identical rows or columns of $A$ that are identical ...
0
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0answers
13 views

Is working out the smith normal form trivial?

What's the point of using elementary column operations to work out the smith normal form of a matrix of reals when it is easy to work out exactly how many redundant equations there are from the row ...
0
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0answers
33 views

How to choose $B$ to have $ \operatorname{trace}(H) \ge \operatorname{trace}(B^{\top}HB)$?

Considering $H \in \mathbb{R}^{N \times N}$ a real symmetric matrix which has both positive and negative eigenvalues, and $B \in \mathbb{R}^{N \times M}$ a real matrix with positive entries. Can I ...
0
votes
1answer
23 views

A matrix equation involving eigendecomposition

Let $p<n$ and -$H\in\mathbb{R}^{n\times n}$ be symmetric with eigendecompoistion being $H=U\Lambda U^{\text{T}}$, -$A\in\mathbb{R}^{n\times p}$, -$D\in\mathbb{R}^{p\times p}$ be a diagonal ...
1
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1answer
77 views

Let $A$ be complex matrices and $x$ be eigenvalue of $A\overline{A}$.

Let $A$ be non-singular $n\times n$ complex matrices and $x$ be negative eigenvalue of $A\overline{A}$. Show that algebraic multiplicity of $x$ is even number. I show that the $\det(A\overline{A})>...
0
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1answer
28 views

Finding values of x for linear dependence

Find all real numbers $x$ for which the matrices are linearly dependent $$M_1=\begin{bmatrix} 1&1\\x&0 \end{bmatrix}\quad M_2= \begin{bmatrix} 1&-x\\x-1&3 \end{bmatrix}$$ $$M_3= \...
30
votes
2answers
4k views

necessary and sufficient condition for trivial kernel of a matrix over a commutative ring

In answering Do these matrix rings have non-zero elements that are neither units nor zero divisors? I was surprised how hard it was to find anything on the Web about the generalization of the ...
0
votes
1answer
25 views

inverse matrix search algorithm

For ex, $$A^{-1}\cdot A=\left[\begin{matrix}x_1 & x_2\\x_3 & x_4\end{matrix}\right]\cdot\left[\begin{matrix}1 & 2\\1 & 3\end{matrix}\right]=\left[\begin{matrix}1 & 0\\0 & 1\...
0
votes
0answers
22 views

On $\bigg[\begin{matrix}E&O\\O&F\end{matrix}\bigg],B_b=\left[\begin{matrix}2b&-1&0&-1\\0&b&0&0\\0&-1&0&-1\\0&1&0&b\end{matrix}\right]\in M_4(\Bbb R)$

Consider the $4\times4$ matrices $A=\bigg[\begin{matrix} E&O_2\\O_2&F \end{matrix}\bigg]$, where $E,F$ are any nilpotent $2\times2$ matrices, and $B_b=\left[\begin{matrix}2b&-1&0&-...
2
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0answers
44 views

Is the differential of left multiplication still left multiplication?

Let $A$ be a matrix in the real n-dimensional general linear group $G$. Let $l: G \to G$, $l(B)=AB$ be left multiplication by $A$. Consider the differential of $l$ at the identity matrix $I$ of $G$, $$...
1
vote
0answers
25 views

eigenspace corresponding to eigenvalue 1?

Let $A$ be in $GL(2,n)$ means $n$ by $n$ invertible matrix with entries in $Z_2$, If $A$ has eigenvalue $1$, Is there something we can say about structure of matrix $A$ or about $\dim(\ker(I-A))$ or ...
0
votes
0answers
14 views

Multiplication of cyclic diagonals.

We denote by $D_k$ the set of all matrices of the form $A=(a_{i,j})_{i,j=0}^{n-1}$ , such that $a_{i,j}\neq 0$ for $i-j=k$ or $k-n$, and $k=1,2,\cdots n-1$. Then we have to show that if $A\in D_k$ and ...
3
votes
2answers
51 views

How to show $\text{Tr}(M\log N)=\sum_{i,j}^n\lambda_i\log(\tilde{\lambda_j})(u_i^{\top}\tilde{u}_j)^2$?

The above question is the equation $(2.4)$ of the following paper: MATRIX EXPONENTIATED GRADIENT UPDATES. Let $M$ and $N$ be two $n \times n$ positive definite matrices where $M=U\Lambda U^{\top}$, $...
0
votes
0answers
20 views

Finding inverse of Elementary matrices

I wish to prove that for an identity matrix I of order n , n is a + ve integer, inverse of a matrix A = [a_ij] obtained by interchanging any two rows / columns of identity matrix I is again the same ...
0
votes
1answer
49 views

Show that $\exp(C^{-1} AC ) = C^{-1} \exp(A C)$

Show that $\exp(C^{-1} AC) = C^{-1} \exp(A C)$ for any matrices $A \in L_{n}(\mathbb{R})$ and $C \in GL_{n}(\mathbb{R})$. The hint of the question is given below: Consider the linear operator $\...