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
33 views

solve a system of equations by matrices

Is there any way to solve the system of linear equations by matrices ?? for example: $$\left\{\begin{aligned} 2x+3y-z&=p \\ px+(p+1)y-pz&=1 \\ x+y+pz&=2 \end{aligned} \right. $$ Thank ...
1
vote
2answers
53 views

Does it always hold, that the product of the eigenvalues of a matrix is it's determinant?

I know that this relation holds in several cases but I'm not sure about the full scope of it. So are there any cases when it doesn't hold?
0
votes
1answer
29 views

Projection and trace of the matrix

For a given vector: $$ x= \begin{pmatrix} 3e^{j^{45^\circ}}\\ 2-3j \end{pmatrix} $$ I need to find projection matrix $P$ of $\hat x$ then trace of the matrix $tr P$ Could anyone advice?
5
votes
1answer
61 views

Does PSD imply on average diagonal dominant?

Suppose $A$ is a $N \times N$ positive semidefinite matrix. This does not necessarily imply that $A$ is diagonally dominant. But does it imply the following "average diagonal dominance" i.e. ...
7
votes
1answer
145 views

nilpotent elements of $M_2(\mathbb{R})$, $M_2(\mathbb{Z}/4\mathbb{Z})$

Let $R$ be a commutative ring. We define $\mathfrak{N}(R)$ to be the set of nilpotent elements in $R$. Find $\mathfrak{N}(R)$ for: $R = M_2(\mathbb{R})$ $R = M_2(\mathbb{Z}/4\mathbb{Z})$
3
votes
4answers
570 views

Why is positive (semi-)definite only defined for symmetric matrices?

When we are defining positive (semi-)definite matrices, we do so for symmetric matrices only. Why do we need symmetry in the definition?
1
vote
1answer
56 views

Expressing $C(x) = \tilde{x} = (\langle x,a_i \rangle )$ as a product of matrices in the form $Cx = \tilde{x}$

Le that $(a_i)^{n}_{i=1}$ be an orthonormal basis and $C(x)$ be a transformation defined as follows: $$C(x) = \tilde{x} = \left( \begin{array}{c} \langle x, a_1 \rangle\\ \vdots \\ \langle x, a_k ...
2
votes
2answers
148 views

All matrices which commute with all $2\times 2$ matrices

I would like to find all matrices which commute with all $2\times2$ matrices. I started solving problem in this way: 1) I have this matrix $A$ with real numbers: $$A=\left[\begin{array}{cc}a ...
1
vote
0answers
58 views

Is there an interesting interpretation of the ROWS of an affine transform matrix?

Context: I have a question about affine transform matrices in 3-space. Matrices are 4x4, with the right-most column being the translation, and the bottom row being [0,0,0,1]. In discussions I read ...
0
votes
1answer
89 views

Geometric interpretation of 2D-Translation's Matrix representation

I just learned the trick of writing a translation of a 2-dimensional real vector as a matrix multiplication in a 3-dimensional space - wikipedia explains it here. Basically it shows: $$ ...
1
vote
1answer
92 views

Matrix Inversion acceptable Condition Numbers

When considering matrix inversion it is worth while worrying about the condition number of the matrix you wish to invert. Matrices that are poorly conditioned can often create inaccurate results. This ...
0
votes
1answer
49 views

Can the Kernel of the commutator of two matrices with empty Kernel sets be non-empty?

The motivation for this question arises from the following: Is it possible, given two Quantum Mechanical observables $A$ and $B$ with associated operators $\hat{\mathbf{A}}$ and $\hat{\mathbf{B}}$ ...
2
votes
3answers
72 views

If $f_A(x) \ne m_A(x)$ and $A^3=I$ then $A=I$?

Suppose $A \in M_{3\times3}(\mathbb R)$ and $f_A(x) \ne m_A(x)$ where $f_A(x)$ is the characteristic polynomial of $A$ and $m_A(x)$ is the minimal polynomial of $A$. If we were to assume that ...
1
vote
0answers
66 views

The maximum notation as regards the absolute value?

We know that $\max(\textbf{A})$ gives the maximum element of the array $\textbf{A}$. What is the notation, or a short formula, if we seek the element having the largest absolute value? e.g., ...
0
votes
0answers
18 views

Deducing linear independence from row operations

If I have a set of vectors and I use the transpose of these vectors as rows of a matrix, in the reduced row echelon form, does a leading one in every column imply linear independence or is it a ...
3
votes
1answer
161 views

Induction proof about entries of powers of strictly upper triangular matrix

Let $A$ be a $n \times n$ strictly upper triangular matrix. Prove that, for $k \ge1$, the matrix $A^k$ has the property that $(A^k)_{i,j} = 0$ for all $(i,j)$ with $j-i < k$. Also, show that $A^n ...
0
votes
1answer
34 views

2- norm 2 and A- norm

Let $x\in \mathbb{R}^n$ and let $A\in \mathbb{R}^{n\times n}$ be a symmetric positive definite matrix. We know that $\Vert x \Vert_2=\sqrt{x^Tx}$ and $\Vert x ...
1
vote
1answer
45 views

A problem about lub and glb of matrix

For any matrix $A\in \mathbb{C}^{n\times n}$, define $$lub_K(A):= \inf\{\alpha\geq 0: AK\subset \alpha K\},$$ and $$glb_K(A):= \sup\{\alpha\geq 0: \alpha K\subset AK\},$$ where $K$ is a equilibrated ...
4
votes
1answer
117 views

If $\operatorname{rank}\left( \begin{bmatrix} A &B \\ C &D \end{bmatrix}\right)=n$ Prove that $\det(AD)=\det(BC)$ [closed]

Let $A,\ B,\ C,\ D \in \mathcal{M}_n(\mathbb{C})$. If $\operatorname{rank}\left( \begin{bmatrix} A &B \\ C &D \end{bmatrix}\right)=n$, prove that $\det(AD)=\det(BC)$
0
votes
1answer
26 views

Let $A_{3x3},B_{3x3}$, and $AB=0, BA \neq 0$ the span of the homogeneous for $Ax=0$ is $Sp\{(1,1,1)\}$ find $\ rank(B)$

I have this problem : Let $A_{3x3},B_{3x3}$, and $AB=0, BA \neq 0$ the span of the homogeneous for $Ax=0$ is $Sp\{(1,1,1)\}$ find $rank(B)$. Since $BA \neq 0$ and $B_{3x3} \implies 0 < rank(B) ...
0
votes
1answer
39 views

Proof/Disproof - Exist $A_{nxn}$ so all $b \in R^n$ the linear system $Ax=b$ has infinite solutions

I have this problem : Proof/Disproof - Exist $A_{nxn}$ so all $b \in R^n$ the linear system $Ax=b$ has infinite solutions. I think its wrong, but I don't know how to proof it. Any ideas? Thanks!
1
vote
1answer
61 views

A and P*A*Q have same singular values being P and Q orthonormal matrixes:

Let P and Q be two orthogonal matrices such that it makes sense to calculate PAQ. Show that A and PAQ have the same singular values. So far, I've come to the fact that the SVD of an orthogonal matrix ...
-1
votes
1answer
81 views

Change of basis problem

Let $$v = \left( \begin{array}{c} 1\\ 7\\ \end{array} \right)$$ $$w = \left( \begin{array}{c} 2\\ -1\\ \end{array} \right)$$ Let $B = (v,w)$ and $$A = \left( \begin{array}{c} 1&2\\ -1&0\\ ...
0
votes
1answer
37 views

Re-arranging matrices/vectors

If I have an equation $$r_1(r_1a+r_2b+r_3c)+r_2(r_1b+r_2d+r_3e)+r_3(r_1c+r_2e+r_3f)=1$$ where all $r_1,r_2,r_3$ are variables and $a,b,c,d,e,f$ are constants. I want to re-arrange them such that ...
0
votes
1answer
70 views

lower bound on matrix norm inequality of sum

The question is simple: can we say this? $\|A\|-\|B\|<\|A+B\|$ for any norm you like.
4
votes
1answer
36 views

Algorithm concerning orthogonal matrices

Say I have a n-dimensional orthogonal matrix, with some of its elements given and these others unknown. Does there exist an effective algorithm to find out the unknown elements and restore the whole ...
2
votes
0answers
34 views

Why do entries of Maurer-Cartan 1-form span space of left-invariant 1-forms

Suppose $G$ is a $k$-dimensional Lie group of $n\times n$ matrices of the form $[g_{ij}(x_1,...,x_k)]$ where $g_{ij}$ are smooth functions. As a follow-up to the question I posted recently, I now ...
0
votes
1answer
79 views

Sum of M matrices

For a symmetric $M$-matrix $M$ and a positive diagonal matrix $D$, I'm trying to evaluate whether $$ x^TDMx = \frac{1}{2}x^T(DM+MD)x > 0 $$ for all $x \in \mathbb{R}^n$. Since $M$-matrices are ...
1
vote
1answer
49 views

Understanding an eigen decomposition notation

So I think I understand eigen decomposition where a matrix M is equal to Q D inverse(Q), where Q is a matrix formed by the eigenvectors of M, and D is a diagonal matrix of the eigenvalues. But I came ...
4
votes
2answers
87 views

Finding determinant of $n \times n$ matrix

I need to find a determinant of the matrix: $$ A = \begin{pmatrix} 1 & 2 & 3 & \cdot & \cdot & \cdot & n \\ x & 1 & 2 & 3 & \cdot & \cdot & n-1 \\ x ...
0
votes
2answers
59 views

Polynomials over finite field

I tried to calculate the characteristic polynomial of a 4x4 matrix over the finite field with two elements. I got two results: $x^4+x^3+x+1$ and $(x+1)^3$. First I thought that this must be an error, ...
1
vote
1answer
165 views

How to prove linearity, find matrix relative to canonical basis and determine kernel and the image with given map?

I've got $v \in \mathbb{R}^n$ and considering the map $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with given $f(x) = v x^T v$. How can I proove, that $f$ is linear? How can I find matrix $f$ relative ...
1
vote
2answers
48 views

A is singular. Show that A is diagonalizable over $\Bbb{R}$

Let $A\in M_{10}(\Bbb{R})$ be singular matrix such that $\mathrm{rank}(I-A)=4 \ \mathrm{and} \ \mathrm{rank}(3I-A)=7$. Show that $A$ is diagonalizable over $\Bbb{R}$. Which diagonal matrix $A$ is ...
3
votes
0answers
46 views

The relation between $\inf_{R\in \mathsf{U}_n} \left\Vert A - BR\right\Vert^2_F$ and $\left\Vert AA^*-BB^*\right\Vert$

Suppose that $A$ and $B$ are two arbitrary $m\times n$ matrices with $m>n$. Let $\mathsf{U}_n$ denote the set of $n\times n$ unitary matrices. I'd like find positive constants $c_1$ and $c_2$ such ...
1
vote
1answer
42 views

Finding base $B'$

I am given $$ B = \begin{pmatrix} 0 & -1 & 2 \\ -2 & 2 & -1 \\ -1 & 2 & 1 \\ \end{pmatrix} $$ How can I find a base $B'$, so that for ...
3
votes
2answers
109 views

How to prove, that solution of system $Ax = b$ exists only if there is no solution of $A^T y = 0$ and $b^T y = 1$?

I have a little linear algebra problem here: How can I prove, that there is a solution of system $Ax = b$ only if there is no solution of $A^T y = 0$ and $b^T y = 1$?
0
votes
0answers
144 views

How to calculate the square root of matrix $A+B$ perturbatively?

$A=diag\{\lambda_1,...,\lambda_n\}$ and $\lambda_i>0$, $B$ is a positive definite symmetric matrix and $max\{B_{ij} \}\ll min\{\lambda_i\}$ Note that the perturbative calculation of square root ...
1
vote
0answers
36 views

Trouble finding the values of a matrix using rref

I'm working on a school project in which I have to get all the values of a missing matrix. To make a small test in my program, I used a simplified example just to see if the math was right, but for ...
1
vote
1answer
56 views

How did the author find the vector v prime perpendicular to n

I'm reading the $3D$ Math Primer for Graphics and Game Development by Fletcher Dunn and Ian Parberry, but I've gotten stuck. If you look at the attached image where it says, "Now we can see the ...
0
votes
2answers
111 views

Trace identity of product of matrices

I have to express the trace of a product of matrices A and B in terms of traces of individual matrices. I know that $\operatorname{tr}(AB) \neq \operatorname{tr}(A)\operatorname{tr}(B)$, and for the ...
0
votes
1answer
386 views

Converting quaternion or $4\times 4$ matrix to $3\times 3$ matrix representation.

I'm working on some code that manipulates an Axis-Aligned Bounding Box (AABB), so it always encompasses the object it borders. I use a $3\times 3$ matrix to re-size the box when it rotates. The ...
0
votes
1answer
153 views

subtraction between sum of all elements of two symmetric matrices

Let assume that I have an $n\times n$ symmetric matrix $A$ and I know $A^{-1}$. Now, I have a new matrix $$M = \begin{pmatrix} A & b \\ b^T & c \end{pmatrix},$$ where $b$ is a vector and ...
2
votes
1answer
22 views

Multiplying matrices to get a specific result

Are there matrices $A,B$ (of dimension $n$), that give \begin{equation} AB-BA=I \end{equation} I have tried getting a result in small scale by using $2\times 2$ matrices and got a false equation ...
2
votes
1answer
54 views

how to determine $A^{15}$

let $A$ be the $3\times 3$ matrix such that ,for any $v \in \Bbb R^3,Av$ gives the projection of $v$ onto the plane $x+y+z=0$. Determine $A^{15}$. I think I know $A^{15}=PD^{15} P^{-1}$. But I don't ...
2
votes
5answers
146 views

Why is the left inverse of a matrix equal to the right inverse? [duplicate]

Given a square matrix $A$ that has full row rank we know that the matrix is invertible. So there is a matrix $B$ such that $$ AB=1 $$ writing this in component notation, $$ A_{ij}B_{jk}=\delta_{ik} ...
1
vote
1answer
39 views

How to prove or disprove the matrices formula

Could some one give me some hints about the following prove of disprove: (a) If $PXX^TP^T=QXX^TQ^T$, then $PX=QX$; (b) If $PXX^T=QXX^T$, then $PX=QX$. In the above formulas, $P$, $Q$ and $X$ are ...
2
votes
1answer
124 views

Number of self-inverse matrices over prime field

Regarding the cryptosystem known as the Hill Cipher, my textbook by Douglas R. Stinson has an exercise asking you to find the number of involutory keys for $m=2$ over $\mathbb Z_{26}$. This means that ...
3
votes
1answer
217 views

How does Maurer-Cartan form work

I have seen similar post asking for interpretation of the Maurer-Cartan form, but I am still struggling to understand it, so let me try to work a specific example and pose a specific question. Let ...
0
votes
1answer
32 views

Find the commutator $[I+aE_{i,j},I+bE_{j,r}]$.

Let $G=GL_n(K)$ where $K$ is a field, $E_{i,j}$ is the matrix with "1" in the $i,j$ spot and zeros in any other spot. What is the commutator $[I+aE_{i,j},I+bE_{j,r}]$? For those of you who don't know ...
5
votes
2answers
171 views

Given a finite metric space, are the matrices of triangle inequality errors invertible?

I have been working on some problems regarding finite metric spaces and have already proven/positively answered the following statement/question if the underlying metric has additional properties. Now ...