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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2answers
51 views

Find the determinant, inverse of a matrix and under what condition it is positive

The matrix is $B=[(1-\rho)I_n+\rho\textbf{1}\textbf{1}']$ where $\textbf{1}=[1\;\cdots\;1]'$, an $n\times 1$ vector with every entry $1$. So what's the determinant, inverse of this matrix and under ...
0
votes
1answer
276 views

Step in Euler's rotation theorem

I have been examining the matrix proof for Euler's rotation theorem on Wikipedia. I have deduced every step up to proving that $\det (R - I) = 0$ for any rotation matrix R. However, I'm having ...
1
vote
2answers
232 views

Rotate rectangles in a rectangle

Is there a general and easy way to calculate the new x and y coords of the rectangles in the big rectangle if I rotate the parent rectangle about 90deg. I know the x and y coords of each rectangle in ...
0
votes
3answers
1k views

2-norm vs operator norm

I have read that we define the "2-norm" of a matrix as $$\max_i \,{|\sigma_i|},$$ which I have also heard called the "operator norm" (here $\sigma_i$ are the singular values). Also we have the ...
1
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1answer
74 views

Are there any mathematical/physical concepts or theories for dealing with a matrix in which the values are changing in a certain way?

As a matter of fact, my application scenario is a recommender system in which the interests/preferences of the users change. I have such a global user-interest matrix: the rows are the records of many ...
3
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4answers
320 views

If $X$ is an orthogonal matrix, why does $X^TX = I$?

It's not immediately clear to me why this is true. My notes say that putting $n$ orthonormal vectors $ v_1, ..., v_n$ in the columns of $X$ gives $X^TX = I$, and it follows from this that the rows of ...
1
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1answer
144 views

How can you tell how many invariant factors a matrix has?

In my notes, I have a $4\times 4$ complex matrix $A$ with the following properties. The characteristic polynomial of $A$ is $(x-1)(x+1)^3$, and the geometric multiplicity of $-1$ is $2$. That is all ...
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0answers
85 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, ...
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2answers
250 views

Looking for a closed form to determine whether a symbol is part of the ith combination nCr

Hi I'm new to this, feel free to correct or edit anything if I haven't done something properly. This is a programming problem I'm having and finding a closed form instead of looping would help a lot. ...
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1answer
118 views

Can This Matrix Proof Be Done Without the Definition? [duplicate]

The following was taken from Stephen Friedberg's Linear Algebra, 2nd Edition. Let $A$ be a $m \times n$ matrix, $B$ and $C$ be $n \times p$ matrices. Show that $A(B+C) = AB + BC$ and, for any $k \in ...
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2answers
137 views

Matrix (correct) notation

Say I have a real $m \times n$ matrix $\mathbf{M}$. Shall I write $\mathbf{M} \in \mathbb{R}^{m \times n}$ or $\mathbf{M} \in \mathbb{R}^{m,n}$? What is commonly accepted and most beautiful and ...
2
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3answers
641 views

Matrix representation of a linear transformation between vector spaces

Let $v$ be an $n$-dimensional vector space over a field $F$ and $\psi: V \to V$ and isomorphism. Show that there exist bases $B_1$, $B_2$ (possibly different) such that the matrix representation of ...
1
vote
4answers
751 views

Reduced row-echelon form of a matrix with variables

I have been staring at this for an hour. How would you reduce such a matrix? \begin{bmatrix} p & 0 & a \\ b & 0 & 0 \\ q & c & r \end{bmatrix} $abc\neq0$
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1answer
39 views

Unicity of identity

I've two doubts: 1) If $A,B$ are square matrices and $AB=I_n$, but not necessary $BA=I_n$, is true that $A$ is invertible and $A^{-1}=B$? 2) If $AB=B$, then $A=I_n$. Well, I know that the ...
0
votes
2answers
75 views

Multiplying matrices

I'm solving a exercise and the following doubt has arisen: Let $A,B,C$ be non-zero matrices, if $AB=CB$ then $A=C?$ I already have a partial solution when $A,B,C$ are square matrices and $B$ is ...
1
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1answer
317 views

Calculating the product of tridiagonal matrix times its transpose

Denote by tridiag($a$,$b$,$c$) the tridiagonal matrix of size $n \times n$ with diagonal elements $b = (b_1, \ldots,b_n)$. Let $a = (a_1, \ldots, a_{n-1})$ and $c = (c_1,\ldots,c_{n-1})$ be the ...
3
votes
2answers
141 views

Formula for Nth Derivative of Matrix Inverse

I was looking for an equation for the nth derivative of a matrix inverse, ie $\frac{d^n \bf{A}^{-1}}{dx^n}$ I know that the first derivative $\frac{\text{d} \bf{A}^{-1}}{\text{d}x} = -\bf{A}^{-1} ...
2
votes
1answer
97 views

Am I right about this exercise of Matrices?

In my book of Linear Algebra asks me the following exercise: For any square matrix $T$, over $\mathbb{R}$, it defines $p(T)$ to be the matrix $c_nT^n+\cdots + c_1T+I$ with $c_i \in \mathbb{R}$ for all ...
6
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2answers
878 views

Does anyone know any resources for Quaternions for truly understanding them?

I've been studying Quaternions for a week, on my own. I've learned various facts about them but I still don't understand them. My goal is to understand rotation quaternions specifically. I don't want ...
4
votes
1answer
1k views

Eigenvalues of matrix with entries that are continuous functions

For each $t \in [0,b]$, let $M(t)$ be an $n \times n$ matrix with entries $m_{ij}(t).$ The matrix $M(t)$ is invertible and positive-definite, so the eigenvalues of $M(t)$ exist and are positive for ...
1
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1answer
157 views

Invariant subspaces and minimal polynomial

I wanted to know whether every invariant subspace $U$ of an endomorphism $A$ with minimal polynomial $m_A= \Pi_{i=1}^n p_i$, where the $p_i$ are mutually coprime polynomials, can be written in the ...
15
votes
1answer
154 views

How to calculate the number of factorizations of a square matrix?

I need to write a function, that, given a square matrix M of non-negative integers, calculates the number of representations of M as a product of two square matrices of non-negative integers. Could ...
2
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1answer
135 views

Does same characteristic polynomial and same rank imply similar?

Are two matrices with the same characteristic polynomial and the same rank necessarily similar? Where can I find the proof for such a thing?
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2answers
58 views

How do I see that for every self-adjoint and positive-definite $A$ there is an $S$ such that $A=S^*S$?

How do I see that for every self-adjoint and positive-definite $A \in \mathbb{C}^{n \times n}$ there is an $S \in \mathbb{C}^{n \times n}$ such that $A=S^*S$? Is something that I am trying to ...
1
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1answer
45 views

About a matrix, its powers, and a particular value…

Is it possible to find two matrices, $A$ an $B$ such that: $$A B^0 x = \begin{pmatrix} c_0 \\ ? \\ ? \\ \end{pmatrix}, $$ $$A B^1 x = \begin{pmatrix} c_1 \\ ...
3
votes
1answer
157 views

$A^TA+I$ is always invertible?

How to prove a general matrix invertible given by as below? How to prove that $A^TA+I$ is always invertible for $\forall A \in \mathbb{R}^{n\times n}$?
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0answers
45 views

How to show that entries of this matrix are in $L^\infty(0,T)$?

I have a problem. Let $A(t)$ be a $n \times n$ matrix for each $t \in [0,b]$ with the property for all vectors $x$ that $$x^TA(t)x \geq C|x|^2$$ where $C$ doesn't depend on $t$. Can I use this fact ...
1
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1answer
95 views

Why is it true that there exists an $n$ such that $V = \operatorname{im}(A-\lambda \operatorname{Id})^n \oplus \ker(A-\lambda\operatorname{Id})^n$

Let $A : V \rightarrow V$ be an endomorphism. Then we have that $$V = \operatorname{im}(A-\lambda \operatorname{Id})^n \oplus \ker(A-\lambda\operatorname{Id})^n$$ for some $n \in \mathbb{N}$. I ...
2
votes
1answer
109 views

Show the following matrix recursion is symmetric

I need to show the following matrix equation is symmetric and I'm not sure where to start: $A_i=\sum_{j_1=1}^{i-2}2(i-j_1-1){i-2 \choose j_1-1}A_{j_1}\Big(\sum_{j_2=1}^{i-j_1-1}{i-j_1-2 \choose ...
2
votes
0answers
131 views

Lattice Reduction Problem: Minimizing the “Longest” Basis Vector

Suppose we have a basis for an integer lattice formed by the vectors $\vec v_1, \vec v_2, \ldots,\vec v_n$. Then let $A$ be the augmented matrix $( \vec v_1| \space \vec v_2| \cdots |\space \vec ...
3
votes
1answer
112 views

Polynomial roots or discriminant

I was wondering if it is possible to find the roots of the following polynomial $$ P(x)=x^n+ax^m+b $$ or at least can I get the discriminant of it, which is the determinant of the Sylvester matrix ...
0
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0answers
77 views

Eigenvalues of time-dependent matrix bounded?

For each $t \in [0,b]$, let $A(t)$ be a $n$ by $n$ matrix with entries $a_{ij}(t) \in L^\infty(0,b).$ Suppose I know that the eigenvalues of $A(t)$ for each $t$ are positive. Can I conclude that the ...
3
votes
1answer
169 views

How to simplify the characteristic polynomial of a given matrix?

Reading through this paper I've come across a statement that I don't follow, could someone give some pointers/hints? Let $A$ be the $2n\times 2n$ matrix given by $$A=(I_n\otimes F)+(G\otimes ...
8
votes
3answers
977 views

Proof for the funky trace derivative : $d (\operatorname{trace} (ABA'C))$?

Given three matrices $A$, $B$ and $C$ such that $ABA^T C$ is a square matrix, the derivative of the trace with respect to $A$ is: $$ \nabla_A \operatorname{trace}( ABA^{T}C ) = CAB + C^T AB^T $$ ...
2
votes
1answer
49 views

Constant in the positive-definite condition of matrices

Suppose I have a positive definite matrix $A$, so $$x^TAx \geq C_A|x|^2$$ holds for all $x$. $A^{-1}$ is positive definite too: $$y^TA^{-1}y \geq C_B|y|^2,$$ is there any way I find the constant ...
1
vote
2answers
85 views

Basis means determinant of matrix of inner products is non-zero

Let $x_i$ be a basis of Hilbert space $X$ (NOT necessarily orthogonal) How do I show that $\text{det}((x_i,x_j)_H)_{ij} \neq 0$ for $i,j=1,...,n$? I see this fact used in Galerkin approximation ...
1
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0answers
45 views

Matrix with elements in $L^\infty(0,T)$ means inverse matrix also has elements in $L^\infty(0,T)$?

Let $A(t)$ be a matrix with entries $a_{ij} \in L^\infty(0,T)$. Suppose $A(t)^{-1}$ exists, and is positive-definite. Does it follow that $A(t)^{-1}$ has elements in $L^\infty(0,T)$ too?
1
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0answers
80 views

Lower bound on Gram determinant

Is there a lower bound for the determinant of the Gram matrix when the elements are linearly independent? I am talking about Hilbert space setting not just vectors. Thanks.
2
votes
1answer
92 views

How to represent a Matrix Space?

How can we represent the space of matrices? E.g. A vector $z\in{}R^m$ in the column-space of matrix $A\in{}R^{m\times{}n}$ can be represented as $$ z=Ax $$ for some $x$. Context: In the following ...
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0answers
76 views

Matrix differentiation.

Given $A$ is square matrix depends on a scalar $x$, is there rules to find $df(a)/dx$? For example, indices rule or chain rule? I found this kind of information is not much on the internet. Thanks.
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3answers
82 views

Name for this matrix operation?

Suppose two matrices have the same number of rows. I want to perform an operation of element-wise product between all possible column pairs between the two matrices. For example, if $A = ...
1
vote
1answer
65 views

Linear Algebra :: Multiplying by Matrices on both sides of equation

I was wondering, for scalar equations, it is true that $A=B$ implies $PA=PB$ where P is also any scalar. Are these true for matrices as well? Is $PAx=Pb$ true when $Ax=b$? Assume that dimensions ...
4
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0answers
144 views

General Solution to $\operatorname{Tr} \ln ( I + A)$ where A is complex and symmetric and zero on the diagonal

Are there any useful identities which would help me to find a general formula for $ \operatorname{Tr} \ln ( I + A ) $ Where I is the identity matrix and A is some N by N complex and symmetric ...
0
votes
1answer
76 views

Mapping a plane in $\Bbb R^3$ to $\Bbb R^2$

I have three points that represent a rigid body. The rigid body undergoes a planar transformation in $\Bbb R^3$ due to rotation and translation. I am working with angular velocity with nonzero $\vec ...
1
vote
2answers
179 views

Quadratic forms of two matrices are equal then the matrices are equal

$A,B\in M_n$, then prove that if $x^HAx=x^HBx$ for all $x\in C^n$, then $A=B$
2
votes
0answers
68 views

Inverses of the sums of all possible subsets of a set of symmetric and positive definite matrices

I have a set of $c$ matrices $A_1 ... A_c$ which are all symmetric and positive definite. I would like to calculate the inverses of all the possible sums, i.e. ...
1
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4answers
203 views

Calculate the Determinant?

$$D=\begin{bmatrix} 246 & 427 & 327 \\ 1014 & 543 & 443 \\ -342 & 721 & 621 \\ \end{bmatrix}$$ What's the trick? Hints? Of course I know calculate by definition... Please ...
3
votes
2answers
162 views

Is there meaning for $\bf uv^T$?

Let $\bf{u,v}$ be two column vector in $\mathbb R^n$, which can be represented by $n\times1$ matrix. $\bf u^T v$ is the inner product of $\bf u,v$, then is there meaning for $\bf uv^T$, which is a ...
0
votes
1answer
169 views

What does it mean by “the origin is moved by the transformation” in linear transformations?

Linear transformations have the special property that the origin is not moved by the transformation. I don't really understand what this means. The example I'm given is that the following ...
5
votes
1answer
134 views

Proving a well-known formula regarding adj(A)

The adjugate of a matrix $A$ is defined as $$ (\mathrm{adj}(A))_{ij} = (-1)^{i+j}M_{ji}(A) $$ where $M_{ji}(A)$ is the determinant of the matrix $A$ after row $j$ and column $i$ have been removed. It ...