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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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Inequality between Frobenius norm and L2 Norm

$u , w \in S ^ { n - 1 }$ and $v , z \in S ^ { m - 1 }$ which means u,w are unit vector in $R^n$, v,z are unit vector in $R^m$ Prove $\left\| u v ^ { \mathrm { T } } - w z ^ { \top } \right\| _ { F } ...
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41 views

Minimal polynomial?

Let $K$ a field, $p,n\in \mathbb{N}$, $B\in \mathcal{M}_p({K})$ and let denote $S_B=\{X\in \mathcal{M}_p(K) \ \mid \ X^n=B\}$. If $X\in S_B$, I have to prove that $\mu_X$ (the minimal polynomial of $...
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1answer
57 views

Unitary Matrices Proof

Problem: Let $A, B \in \mathbb{C}^{n\times n}$ be selfadjoint ,such that $[A,B] := AB − BA = 0$ Show that there is a unitary matrix $U \in \mathbb{C}^{n\times n}$ such that $U^*$A$U$ and $U^*$$...
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1answer
28 views

Connection between selfadjoint and normal matices

Let $A, B \in \mathbb{C}^{n\times n}$ be selfadjoint, such that $[A,B] := AB − BA = 0$ . Show that $C := A + iB$ is normal matrix. Could someone give me a hint on this problem ? I think that as ...
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2answers
37 views

A positive integer matrix with no integer eigenvalues

Let $n \in \mathbb{N}$ and $A\in M_n(\mathbb{N})$ with $Tr(A)=0$ and $A^3+A-2I_n=O_n$.Prove that $n$ is a multiple of $3$ and $\det(A^2)=\det(A^2+I_n)$. I tried to find $A$'s eigenvalues, but the ...
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1answer
48 views

How to extract a positive definite submatrix from a PSD matrix?

Let $M$ be a real symmetric positive semi-definite matrix s.t. there is only one zero eigenvalue. Question: Is it true that there is a unique principal submatrix that is positive definite? If so, ...
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1answer
32 views

Mean vector and covariance matrix

I am given a home work for one subject, but my probability theory course is just started, so I dont have enough information. Could someone help me with that? Given: $$\begin{equation} p_\underline x(...
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1answer
43 views

Inverting a power series matrix

Let's assume I want to invert a matrix function $M=M(x)$, which is expressed as a power series of the small parameter $\epsilon$ $$M = M_0(x) + M_1(x) \epsilon + M_2(x) \epsilon^2 + \mathcal{O}(\...
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15 views

Perturbing a matrix with asymptotic constraints

I want to expand the following function in series of $\epsilon = a/l$: $$f=\frac{h \cos\alpha + l\sin\alpha}{l \cos \alpha + c \sin \alpha}$$ I know from the physics of the problem that also $$\frac{...
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19 views

Convert velocity vector to yaw roll pitch Tait Bryan

I have a cartesian position and velocity vector describing the flight path of an object in the format "time posX posY posZ velX velY velZ" and want to convert it to a "time posX posY posZ ang1 ang2 ...
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1answer
37 views

On the supremum norm of matrices

Let $D=diag (d_{ii}) \in M_n(\mathbb R)$ be a diagonal matrix and $E\in M_n(\mathbb R)$ be such that $||E||_\infty < \min _{i\ne j} \Bigg|\dfrac{d_{ii}-d_{jj}}{2}\Bigg|$. Then how to show that ...
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2answers
68 views

Two $m \times n$ matrices$\ A$ and$\ B$ satisfy that$\ A^TB=B^TA$. How to find a matrix $\ Q$ s.t.$\ A=QB$? [closed]

There has two $m \times n$ matrices$\ A$ and$\ B$ satisfy that$\ A^TB=B^TA$. Find a matric$\ Q$ s.t.$\ A=QB$
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38 views

Derivative of gradient of vector wrt vector

I am trying to solve for the derivative of the following equation: $t(\pmb {m_1} , \pmb {m_2} )= \frac{1}{2} \: ||\nabla \pmb {m_1} ||_2^2 \: ||\nabla \pmb {m_2}||_2^2 \: - \frac{1}{2} \:|\nabla \pmb ...
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15 views

Square matrix geometric meanings

What is the geometric meaning of 1) Determinant of square matrix 2) Inverse of square matrix 3) Trace of square matrix
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3answers
44 views

Orthogonal matrices with eigenvalues equally spaced over unit circle?

We know that the eigenvalues of orthogonal matrices have norm 1, and thus they are all on the unit circle. However, I wonder if there is a way to construct a orthogonal matrix (with real entries) ...
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3answers
67 views

Another proof for Sherman Morrison Formula?

The proof of Sherman Morrison Formula is on wikipedia as well as this question Proof of the Sherman-Morrison Formula. Isn't there a proof which does not uses multiplication of the inverse and the ...
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2answers
49 views

Proof of Jordan-Chevally-Decomposition

Let A be a square matrix over $\mathbb{C}$, prove there are matrices $D$ and $N$ such that $A = D + N$ such that $D$ is diagonalizable, $N$ is Nilpotent and $DN = ND$. I can see that any ...
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1answer
24 views

Orbits of the conjugation action of $GL_3 (\mathbb{R})$ on the nonsingular symmetric $3\times3$-matrices

Let $S$ be the space of all symmetric $3 \times 3$ matrices of full rank and with real entries. $GL_3 (\mathbb{R})$ acts on this space by conjugation, \begin{align*} g.A = (g^{-1})^T A g^{-1}, \quad ...
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2answers
33 views

For a square matrix A over $\mathbb{C}$, Proofs that matrices D and N exist with A=D+N under different conditions

(i) D is Diagonalizable This one i believe to be fairly straightforward, if D is diagonalizable then we can allow $D^t = I$ (where I is the identity) and therefore D id diagonalizable and therefore A=...
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13 views

Should I use pseudo-inverses to prove this?

Let $A=A(x)$ be a square matrix and let $x^*$ be such that $A(x^* )z=0$ and $z≥0$. Let $A(x)y≫0$ for $x≠x^*$. Then $z≯0$. How to prove this? Should I use pseudo-inverses?
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1answer
35 views

Check a word given a generator matrix G

Take the binary code C with generator matrix: $$ \left( \begin{array}{ccccc|cccc} 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 ...
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2answers
23 views

Solve linear equation system with Gauss

I have the following matrix and have to see if it has solutions depending on $a$. My solution: $M= \left[ {\begin{array}{cc} a & a^2 &| &1 \\ -1 & -1& | & -a \\ 1 &...
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1answer
32 views

If A is a symmetric square matrix. I need to show that it is positive definite only if all eigenvalues are positive.

I understand that a positive definite matrix by the definition is a symmetric matrix where all eigenvalues are positive. I also know that if $ (x,y) = {x^T}{\cdotp}M{\cdotp}y$ then it is positive ...
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1answer
10 views

Is the transformation matrix of an upper triangular matrix to its Jordan normal form always triangular?

Assume that $A$ is an upper triangular matrix. In the case where $A$ is 2x2, I've checked that a transformation matrix $P$ such that $J = P^{-1}AP$, with $J$ Jordan normal form of $A$, is always upper ...
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1answer
41 views

A linear map $T: \mathbb{R^3 \to \mathbb{R^3}}$ has a two dimensional invariant subspace.

Let $T: \mathbb{R^3 \to \mathbb{R^3}}$ be an $\mathbb{R}$-linear map. Then I want to show that $T$ has a $2$ dimensional invariant subspace of $\mathbb{R^3}.$ I considered all possible minimal ...
3
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2answers
118 views

Let $A\in M_n(\mathbb{Q})$ with $A^k=I_n$. If $j$ is a positive integer with $\gcd(j,k)=1$, show that $ \operatorname{tr}(A)= \operatorname{tr}(A^j)$. [closed]

Let $A\in M_n(\mathbb{Q})$ with $A^k=I_n$. If $j$ is a positive integer with $\gcd(j,k)=1$, show that $ \operatorname{tr}(A)= \operatorname{tr}(A^j)$. I don't know how to start to prove that. ...
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16 views

Differentiate quaternion differential equation

I'm reading at the moment this paper, but I'm not clear with some steps the do: On page 5: When I differentiate $\dot{\mu}$ by $mu$ there should be also $ -M_q w_q$ in the A matrix at position 0,0? ...
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2answers
27 views

How to rewrite matrix formula for Diagonalizable matrix $A=PDP^{-1}$

I am working on an old exam containing a question about Diagonalizable matrix, I am quite confident about the subject overall but there is one simple thing that bothers me, a lot! We are given the ...
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0answers
30 views

$A$ and $B$ are $3 X 3$ real matrices such that $rank (AB) = 1$ [duplicate]

If $A$ and $B$ are $3 X 3$ real matrices such that $rank (AB) = 1$ ,then rank (BA) _______? since $rank (AB) =1 |AB|=|A||B|=0$ It implies, atleast |A|=0 or |B|=0 Hence, $rank (BA)$ cannot be 3. ...
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0answers
40 views

Are eigenvectors a basis set for any invariant subspace?

Let $A\in \mathbb{R}^{n\times n}$ (i.e., a real square matrix). Assume that $v_1,\dots, v_m$ ($m\le n$) are eigenvectors of $A$. Can any $d-$dimensional invariant subspace of $A$ be constructed using ...
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1answer
20 views

Made an error solving a linear system

Can anybody explain to me what I did wrong here? I am working on practice problems for my linear algebra course. The problem in question is as follows: Suppose the system below is consistent for all ...
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2answers
40 views

Determine whether a linear system may have no solution

Say we have a matrix $A\in\mathbb{R}^{n\times n}$ that can be written as $A=(M+\lambda I_n)^\top (M-\lambda I_n)$, where $M\in\mathbb{R}^{n\times n}$ and $\lambda\in\mathbb{R}\backslash\{0\}$. I am ...
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0answers
46 views

what is the equivalent of $xy=a$, then $y→∞$ as $x→0$ for matrices

For scalars, given the equation $xy=a$, then $y→∞$ as $x→0$, i.e. as $x$ tends to being non-invertible. I wanted to find an equivalent theorem for matrices. I came up with: For matrices and vectors, ...
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1answer
49 views

What would be a good mathematical model to measure the degree of homogeneity of a mixture?

At my current workplace, we are looking to quantify a batch to say how "similar/dissimilar" the items are. The problem can be stated like so (transformed for public posting): We have parts that can ...
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1answer
55 views

Find the determinant $Δ_n$

Find the determinant $\Delta_n$ $A_n = \begin{bmatrix} 0 & 1 & 0 &\dots &\dots&0\\ -1 & 0 &1 & 0&&\vdots\\ 0&-1 & 0 &1 &\ddots&\vdots\\ \...
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1answer
64 views

Is the set $\{AB-BA\colon A,B\text{ are } n\times n\text{ matrices}\}$ a subspace of all square matrices of order $n$?

[${AB-BA }$ , where $A_{n\times n}$ and $B_{n\times n}$ are square matrices] is this set a subspace of the vector space of all square matrices of order $n$. If yes then can you please make me ...
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1answer
134 views

If $\,\operatorname{trace}(C) = 0\; $ then there exist matrices $A$ and $B$ such that $AB - BA =C$

If $c_{11} + c_{22}=0$ then there exist matrices $A$ and $B$ such that $AB - BA =C$, where $$C = \begin{pmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{pmatrix}.$$ I cannot understand where ...
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24 views

Is there a difference between the definition of the group $L_{n}(R)$ and that of $GL_{n}(R)$?

Is there a difference between the definition of the group $L_{n}(R)$ and that of $GL_{n}(R)$? I already know the definition of $GL_{n}(R)$.
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2answers
35 views

Is the row space of a matrix A a subspace A? If so, what are the objects of A?

I have problems with an assertion that I read in a definition of the row space. I hope somebody can help me :) This part is clear: Let A be a mxn-matrix, with rows $ r_{1},...,r_{m} \in K^{n} $ ...
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1answer
22 views

The congruence of two partitioned matrices

Two $(m+n)\times (m+n)$ real symmetric matrices $X$ and $Y$ can be partitioned as follows: $X=\left( \begin{matrix} A_{n\times n} & O_{n\times m} \\ O_{m\times n} & C_{m\times m} \end{matrix} ...
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1answer
22 views

Does the matrix norm inequality or the Cauchy-Schwarz inequality hold for L2,1 norms

I read here https://statweb.stanford.edu/~souravc/Lecture32.pdf that Cauchy-Schwarz inequality holds for the Hilbert-Schmit or Frobenius norms. I wanted to know if the same holds for other norms too ...
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3answers
50 views

Minimal polynomial of a matrix having only 1s on the counter diagonal

Consider the matrix $A=a_{ij}$ where $$a_{ij}=\begin{cases}1\ \ \text{if}\ \ i+j=n+1\\0\ \ \text{otherwise}\end{cases}$$. Then, what can be said about the minimal polynomial of the matrix $A$. Note ...
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1answer
40 views

The exponential of a skew-symmetric matrix in any dimension.

The Rodrigues rotation formula gives the exponential of a skew-symmetric matrix in three dimensions, and the exponential of a skew-symmetric matrix in two dimensions is given by Euler's formula. Is ...
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1answer
60 views

Notation inconsistencies in matrix differentiation?

1. Suppose $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$. Let $g(z) = f(Xz)$ where $x \in \mathbb{R}^{n}$ and $X \in \mathbb{R}^{n,n}.$ I am interested in $\nabla_{z} g(z).$ $$\nabla_{z} g(z) = \nabla ...
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1answer
16 views

Get from a transformation matrix to the resultant span of the solution set

Guess that's a very basic question, but anyway: I have the following transformation matrix: $$\begin{bmatrix}1 & 0 & 0 & -\tfrac{1}{3}\\ 0 & -6 & -3 & -1\end{bmatrix}$$ And I ...
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0answers
19 views

Alternatives in Farka's Lemma as boudaries

I am attempting to solve a problem in the field of Economics, and for that purpose I have devised the following lemmas. Lemma 1: Let $A(x)$ be a square matrix and let $x^*$ be such that $A(x^* )z=0$ ...
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1answer
24 views

If $A$ and $Q$ are unitary, then $U = Q^{-1}AQ$ is unitary…

Here is what I have so far... Since $A$ and $Q$ are unitary, by definition, we have that $AA^* = A^*A = I$ and $QQ^* = Q^*Q = I$, in other words, $A^* = A^{-1}$ and $Q^* = Q^{-1}$. We can then define ...
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1answer
24 views

Amount of integer multiplications performed with matrices - how much faster?

I am trying to determine how much faster a multiprocessing system would be at multiplying matrices than a single processor system. Here is my thought process/example: Assume Matrix A is k x l and ...
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1answer
16 views

Recurrence relations in Matrices raised to natural powers

Let some matrix be $A$, what methods are there for attaining a recurrence relation which allows us to calculate - for example - the leading term in the matrix $A^n$ when given the leading terms of a ...
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0answers
23 views

Permutation matrix and reducible matrix .

Consider the matrix: $\begin{bmatrix} 1&0&1&0&0\\0&1&1&1&1\\1&1&1&1&1\\1&1&0&0&0\\1&0&1&0&0\end{bmatrix}$. I checked ...