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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Symmetric Matrix with Positive Eigenvalues

Not all matrix with positive eigenvalues is positive definite, i.e. $\mathbf{x}^\mathsf{T}A\mathbf{x}>0$ for all non zero vector $\mathbf{x}$. For example consider matrix $$A = \begin{bmatrix} 1 ...
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14 views

Physical meaning of cofactor and adjugate matrix

I like the way there a physical meaning tied to the determinant as being related to the geometric volume. Since the determinant can be calculated through Laplace's formula where the cofactor matrix is ...
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1answer
19 views

Finding an orthonormal basis for $\mathbb{C}^2$ with respect to the Hermitian form $\bar{x}^TAy$

Problem Statement: Let $A=\begin{bmatrix} 2 && 1 \\ 1 && 2 \end{bmatrix}$. Find an orthonormal basis for $\mathbb{C}^2$ with respect to the Hermitian form $\bar{x}^TAy$. I am ...
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9 views

satellites attitude determination TRIAD - how are orbital reference frame vectors constructed?

I posted this same question on space.stackexchange but never received any answer. So I am posting here hoping to get an answer as this is a quite mathematical topic. I am trying to fully understand ...
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1answer
15 views

How can we know if the minimal polynomial of a matrix has no multiple products?

If the characteristic polynomial $f_A(x)$ has multiples of the same product, for example $f_A(x)= (x+2)^2(x-1)$ so $(x+2)$ has a multiple of $2$, then is there a condition on $A$ such that we know ...
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24 views

If the characteristic polynomial of $A,B$ is the same does it mean that $A,B$ are similar?

If the characteristic polynomial of $A,B$ is the same does it mean that $A,B$ are similar? So I read that it's true only if $A,B$ are diagonalizable, but why? if the characteristic polynomial is ...
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20 views

What is the maximum value of coefficient $f_v$ with the constraInt that the matrix is positive semi-definite?

I am trying to solve this equation my self with my knowledge about characteristic polynomials, etc but I have placed it here earlier because I'm not a mathematician and maybe you give me ideas to ...
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1answer
25 views

If $x^{T}By = 1$, should $\operatorname{Tr}(Byx^{T}) = 1$?

would appreciate any hints with this question: Assume $x$, $y$ are both $n \times 1$ vectors, and that $B$ is $n\times n$. Given that $x^{T}By = 1$, should $\operatorname{Tr}(Byx^{T}) = 1$ ? Thank ...
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1answer
18 views

If $A^4=4A^2$ then $m_A(x)=x^2-4$ and if it isn't diagonalaziable over $\mathbb R$ then $0$ is an eigenvalue

Given $A_{n\times n} \in \mathbb R$ such that $A^4=4A^2$ then if $A$ is invertible and isn't of the form $cI, c\in \mathbb R$ then $m_A(x)=x^2-4$. if $A$ isn't diagonalaziable over ...
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1answer
31 views

Calculate A^8 using Cayley Hamilton Therorem

Find $A^8$ using Cayley Hamilton Therorem, when $$A = \begin{pmatrix} 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 0 ...
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3answers
34 views

Representing a linear operator on $V$ with an element of $V \otimes V^*$

I got interested by the first sentence of this wikipedia subsection. It claims that any linear operator $f:V\to V$ can be represented by an element of $V\otimes V^*$ in a very concrete way: the ...
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33 views

how to compare trace($B^4$) and trace[$(B-D)^4$] with $D$ the diagonals? [on hold]

If $B=(b_{ij})_{n\times n}$ is a real symmetric $n$ by $n$ matrix, $D = (d_{ij})_{n\times n}$, defined as $d_{ij}=b_{ij}$ if $i=j$ and $0$ otherwise. then how to compare $\text{Trace}(B^4)$ and ...
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35 views

Given $A$, $A^{-1}$ can be expressed with: $A^{-1}=bA+dI$

Given the matrix $A=\begin{pmatrix} -1 &3 &3 \\ 3& -1 & 3\\ 3& 3 & -1 \end{pmatrix}$ then $A$ is invertible and $A^{-1}$ can be expressed with: $A^{-1}=bA+dI, ...
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2answers
28 views

Let $A$ be a real symmetric matrix with rank $1$ , then can all the diagonal entries of $A$ be $0$ ?

Let $A$ be a square real symmetric matrix with rank $1$ , then can all the diagonal entries of $A$ be $0$ ? I know that real symmetric matrices are diagonalizable . Also if all the diagonal entries be ...
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0answers
35 views

How to find a onto homomorphism between two groups?

Consider the following subgroups of $\text{SL}(2,\mathbb{Z})$ : $A$ the subgroup of matrices with determinant $1$ : ...
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1answer
16 views

Frobenius norm and submultiplicativity

I read (page 8 here) that if $A$ and $B$ are rectangular matrices so that the product $AB$ is defined, then $$(1)\quad||AB||_F^2\leq ||A||_F^2||B||_F^2$$ Does that mean that the inequality above ...
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27 views

how to find out generators of the following free group?

following is the subgroup of SL($2,\mathbb{Z}$) \begin{bmatrix}2\mathbb{Z}+1&4\mathbb{Z}\\2\mathbb{Z}&2\mathbb{Z}+1\end{bmatrix} how to find out its generators? i know it is free group of ...
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0answers
13 views

how to find the index of following subgroup?

if I denotes the principal congurence group of level 2 i.e. $I=\{ M \in SL(2,Z) ; \:M \:\:\text{congruent to I} \mod(2)\}$. or I= ...
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1answer
32 views

Given a normal $A_{n\times n}$ matrix, then $\lVert A^*v \rVert = \lVert Av\rVert$ and $\langle Av,v\rangle = \langle A^*v,v\rangle$

Let a normal $A_{n\times n}\in \mathbb C^n $ matrix, then: $\forall v \in \mathbb C^n:\lVert A^*v \rVert = \lVert Av\rVert $ $\forall v \in \mathbb C^n : \langle Av,v\rangle = \langle ...
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1answer
43 views

Finding the inverse of A where A is of the form $A = D (I − N)$, where $D$ is diagonal with nonzero entries and $N$ is nilpotent

If a matrix can be written as $A = D (I − N)$, where $D$ is diagonal with nonzero entries and $N$ is nilpotent, then $A^{−1} = (I − N)^{−1}D^{−1}$. Use this to find inverse of: $\begin{bmatrix} 2 ...
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22 views

Why does $\bar A = \left\{ {{P_\Delta }(\lambda ):\left\| {{\Delta _j}} \right\| \le \varepsilon ,j = 0,1,2…m} \right\}$? [on hold]

Suppose ${P_\Delta }(\lambda ) = ({A_m} + {\Delta _m}){\lambda ^m} + ....... + ({A_1} + {\Delta _1}){\lambda ^1} + ({A_0} + {\Delta _0})$ is a matrix polynomial, and $\lambda $ is a complex ...
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32 views

Show $SO_2(\mathbb{R}) \cong\{A \in GL_2(\mathbb{R}) : A^tA = I, \det A = 1\}$.

Show $SO_2(\mathbb{R}) \cong\{ A \in GL_2(\mathbb{R}) : A^tA = I, \det A = 1\}$, where $SO_2(\mathbb{R})$ is the group of rotations of the circle under the operation of composition. Attempt: ...
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0answers
20 views

Finding transform matrix from resulting multiplypoint function

Two matrix transformation functions exist within the Unity3D API: 1) MultiplyPoint 2)MultiplyPoint3X4 3X4 matrix (2) preforms a standard transform against a vector (And ofc is easily replicated ...
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1answer
30 views

Matrix multiplication to make all numbers in a 3x3 matrix negative

Let's say I have the matrix called Delta, $$ \begin{matrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{matrix} $$ What would I have ...
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2answers
26 views

Operations on 3x3 matrix through matrix products

What would I have to multiply the following matrix ... $$ \begin{matrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{matrix} $$ by so ...
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1answer
19 views

Determinant of block matrix with null row vector

I'm a bit confused on a problem. I've been given an $(n+1)\times(n+1)$ square matrix, which is written in the form of a block matrix with the following dimensions $ \begin{bmatrix} (1x1) ...
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24 views

Let $X : S_3 → GL_2(\mathbb{R})$ . Compute the six matrices {$X(\pi) : \pi \in S_3$} and show they faithfully represent $S_3$.

Consider an equilateral triangle $V_1V_2V_3$ with center at the origin, and vertex $V_1 = (0,1)$ and vertices $V_1, V_2, V_3$ in counterclockwise order. Consider the action of the symmetric group ...
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6answers
179 views

Can we say that $\det(A+B) = \det(A) + \det(B) +\operatorname{tr}(A) \operatorname{tr}(B) - \operatorname{tr}(AB)$.

Let $A,B \in M_n$. Is this formula true? $$\det(A+B) = \det(A) + \det(B) + \operatorname{tr}(A) \operatorname{tr}(B) - \operatorname{tr}(AB).$$
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12 views

How to compute homography matrix H from four corresponding points [duplicate]

I am using 4 point correspondence to compute elements in Homography matrix $H$. \begin{align*} [x']={}& [h_1 h_2 h_3] [x] \\ [y']={}& [h_4 h_5 h_6] [y] \\ [(1)]={}&[h_7 h_8 h_9] [(1)] ...
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1answer
11 views

calculating the orientation of an object

If you have a rotation matrix (or an attitude/direct cosine matrix, which are all synonyms). This matrix actually transforms vectors from one reference frame to another. But if your goal is to ...
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1answer
54 views

The eigenvalues of $AB$ and of $BA$ are identical for all square $A$ and $B$ …a different approach

The eigenvalues of $AB$ and of $BA$ are identical for all square $A$ and $B$. I have done the proof in a easy way… If $ABv = λv$, then $B Aw = λw$, where $w = B v$. Thus, as long as $w \neq 0$, it is ...
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1answer
32 views

linear transformation proof problem

So question is : For any $m\times n$ matrix $A$,let $T_A$ be the linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ defined by $T_A(x) = Ax$ for all $x \in \mathbb{R}^n$.Let $A$ and $B$ be ...
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2answers
33 views

Finding Eigenvectors for $3 \times 3$ matrix with rows of zeros.

For a $3 \times 3$ matrix: $ $[A]$ = \begin{bmatrix} 6 & -2 & 2 \\ -2 & 3 & -1 \\ 2 & -1 & 3 \end{bmatrix} $ I have the eigenvalues: ...
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1answer
19 views

A question in limit matrix polynomial

Suppose ${A_j},\,{\Delta _j} \in {\mathbb C^{n \times n}},\quad\big(\,j = 0,\,1,\,2,\,\ldots,\,m\,\big)$ ${P_\Delta }\left(\lambda\right) = \left({A_m} + {\Delta _m}\right){\lambda ^m} + \, \cdots ...
0
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0answers
7 views

Uniqueness and invariance of the LDLT decomposition

A real symmetric positive semi-definite matrix $A$ can be decomposed in the form $A = P^TLDL^TP$, where $P$ is a permutation matrix, $L$ is a lower unit triangular matrix and $D$ is a diagonal ...
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0answers
24 views

Matrix Inequality for the identity and a traceless matrix

Given a traceless matrix $C$ $\in M_n(\mathbb{F})$, i.e., tr$(C)=0$, what is the relationship between tr$|\mathbb{I}+C|$ and tr$|C|$? The two matrices are of dimension $n$. This was cross-posted to ...
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3answers
58 views

Is the following set a group?

Let $ G= \begin{pmatrix} a & a\\ a & a\\ \end{pmatrix} $ where $a\in \Bbb R, a \neq0$. I need to show that $G$ is a group under matrix multiplication. The ...
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1answer
29 views

Matrix equation implies invertibility

Let $D = \operatorname{diag}(\lambda_1, \ldots, \lambda_n)$ be a diagonal matrix with positive entries $\lambda_i > 0$ (some of them might coincide). If we have the matrix equation $A D A^t = ...
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Singular matrices over prime fields

Show that if a given matrix A with integer coefficients over $\mathbb{F}_{p}$ is singular for infinitely many primes $p$, then it is for all primes.
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2answers
18 views

The relationship between matrix rank and its characteristic polynomial coefficients

Given the matrix characteristic polynomial coefficients. Is there a quick way to determine the rank of the matrix?
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218 views

Factorial of a matrix: what could be the use of it?

Recently on this site, the question was raised how we might define the factorial operation $\mathsf{A}!$ on a square matrix $\mathsf{A}$. The answer, perhaps unsurprisingly, involves the Gamma ...
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5answers
4k views

How could we define the factorial of a matrix?

Suppose I have a square matrix $\mathsf{A}$ with $\det \mathsf{A}\neq 0$. How could we define the following operation? $$\mathsf{A}!$$ Maybe we could make some simple example, admitted it makes any ...
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1answer
154 views

Proving positive definiteness of matrix $a_{ij}=\frac{2x_ix_j}{x_i + x_j}$

I'm trying to prove that the matrix with entries $\left\{\frac{2x_ix_j}{x_i + x_j}\right\}_{ij}$ is positive definite for all n, where n is the number of rows/columns. I was able to prove it for the ...
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2answers
42 views

Proving matrix properties: [closed]

Prove: (i) $A(I+BA)^{-1}=(I+AB)^{-1}A$ (ii) $(I+AB)^{-1}=I-A(I+BA)^{-1}B$ (i) Consider $A(I+BA)=(A+ABA)=(I+AB)A$ Taking inverse on both sides (invert) ...
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0answers
24 views

Writing a Rotation Matrix About an Angle

I am asked to find a rotation matrix $R_O$ of an angle $O$ about axis $u\in R^3$, with $u$ having length of 1. I've looked up this concept on the web but I have no idea where to get started...could ...
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1answer
100 views

Is $\det (A + B)=\det (A) + \det (B) + \operatorname{tr}(A \operatorname{adj}(B))$?

Let $A,B \in {M_n}$. Is this true that $\det (A + B) = \det (A) + \det (B) + \operatorname{tr}(A\operatorname{adj}(B))$?
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1answer
44 views

Generate a random neutrally stable matrix

I need to generate random real matrices such that all eigenvalues have real part equal to 0 -- i.e. random neutrally stable matrices. What's the simplest way to do this? Note that I don't care about ...
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11answers
480 views

Why represent a complex number $a+ib$ as $[\begin{smallmatrix}a & -b\\ b & \hphantom{-}a\end{smallmatrix}]$? [duplicate]

I am reading through John Stillwell's Naive Lie Algebra and it is claimed that all complex numbers can be represented by a $2\times 2$ matrix $\begin{bmatrix}a & -b\\ b & ...
0
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1answer
380 views

Do entries in augmented columns count as pivot?

I am in a basic linear algebra course, and we are learning to solve linear equations with augmented matrices. We learned that when an augmented matrix is in row echelon form or reduced echelon form, ...
1
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1answer
113 views

Linear map invertible if and only if associated matrix invertible

Theorem: Let $V$ and $W$ be finite dimensional vectorspaces with ordered bases $\beta$ and $\gamma$ resp., and let $T: V \rightarrow W$ be linear. Then $T$ is invertible if and only if the associated ...