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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1
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1answer
203 views

Smoothness of trace of a matrix

Let $X$ be a square matrix. Is $\operatorname{trace}(X)$ a smooth function of $X$? Why? What if $X$ positive-semi-definite?
0
votes
1answer
92 views

Convert Arrays of Reals into an Equivalent Array of Complex

Given a collection of $n$ real number arrays of length $m$, for example: $$[r_{11},\ \dots, r_{1m}]$$ $$\vdots$$ $$[r_{n1},\ \dots, r_{nm}]$$ is it possible to transform the entire collection into ...
2
votes
2answers
2k views

Can QR decomposition be used for matrix inversion?

Is there any simple algorithm for matrix inversion (that can be implemented using C/C++)? Can QR decomposition be used for matrix inversion? How?
2
votes
1answer
106 views

Determinant of a series of Hadamard matrix

Given: $$H=\ \left[ \begin{array}{cc|r} 1 & 1 \\ 1 & -1 \end{array} \right]$$ a Hadamard $H_2$ matrix. and the series: $$S=\sum_{k=0}^{N}{\frac{H^k}{k!}}$$ Is it possible to calculate ...
2
votes
2answers
728 views

Positive semi-definite matrix

Suppose a square symmetric matrix $V$ is given $V=\left(\begin{array}{ccccc} \sum w_{1s} & & & & \\ & \ddots & & -w_{ij} \\ & & \ddots & ...
0
votes
1answer
68 views

Some smaller piece of information that can be derived from a specific type of matrixes?

Given a matrix: $$ A_{m,n} = \begin{pmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots ...
6
votes
3answers
1k views

Is it true that any matrix can be decomposed into product of rotation,reflection,shear,scaling and projection matrices?

It seems to me that any linear transformation in $R^{n\times m}$ is just a series of applications of rotation(actually i think any rotation can be achieved by applying two reflections, but not sure), ...
6
votes
3answers
458 views

An application of Vandermonde determinant

Let $\lambda_1,\ldots,\lambda_n$ be complex numbers such that for each positive integer $k\geq 0$, $$\sum_{i=1}^n \lambda_i^k=0.$$ Here I am supposed to show that $\lambda_i=0$ for each $i\in ...
6
votes
2answers
3k views

Determining the left inverse of a non-square matrix

I understand that non-square matrices do not have an inverse, that is, both a left inverse and a right inverse. Yet, I am fairly certain that it is possible for a non-square matrix to have either a ...
6
votes
4answers
816 views

How to prove that the normalizer of diagonal matrices in $GL_n$ is the subgroup of generalized permutation matrices?

I'm trying to prove that de normalizer $N(T)$ of the subgroup $T\subset GL_n$ of diagonal matrices is the subgroup $P\in GL_n$ of generalized permutation matrices. I guess my biggest problem is that I ...
1
vote
0answers
55 views

Nonsingularity of a matrix

I have a smooth function $g(x) \colon \mathbb{R}^n \to \mathbb{R}$ that is homogeneous of order one: $g(\lambda x) = \lambda g(x)$. Let $(x_1,...,x_n) \circ (y_1,...,y_n) = (x_1 y_1,...,x_n y_n)$. Are ...
4
votes
2answers
1k views

Power of a matrix

$A$ is a $n\times n$ matrix, $ A^m = 0 $ for some positive integer $m$. Show that $A^n = 0$. My attempt: For $n > m$, it's obvious since matrix multiplication is associative. For $n < ...
0
votes
1answer
2k views

Coordinate matrix of a vector in terms of a basis

I am taking a linear algebra class currently and working through Hoffman's textbook. One of the exercises I am unsure about is, Find the coordinate matrix of the vector $\alpha=(1,0,1)$ in the ...
0
votes
2answers
129 views

Solving a 3 equation system using elimination

Solve the system using elimination: X=? Y=? Z=? I'm trying to solve this problem by putting the system into the form of an augmented matrix and using gaussian elimination but I can't find a way to ...
2
votes
1answer
141 views

question in linear algebra, matrices

Given $A$ and $B$, $2\times 2$ matrices, which of the following is necessarily true? If $A$ and $B$ are both Unitary matrices over $R$ and $\det(A)=\det(B)=1$ then $A$ is similar to $B$. If $A$ and ...
2
votes
2answers
201 views

Matrix multiplication, equivalent to numeric multiplication, or just shares the name?

Is matrix multiplication equivalent to numeric multiplication, or do they just share the same name? While there are similarities between how they work, and one can be thought of being derived from ...
0
votes
1answer
100 views

Translation on Matrix

Suppose I have a 2D matrix as such: $$\begin{bmatrix} 1 &0 &0\\ 0 &1 &0\\ 0 &0 &1\\ \end{bmatrix} $$ If I apply a rotation of 180 degrees and a scale of 2, what will my ...
0
votes
2answers
201 views

Plugging a matrix multiplied by an imaginary number in the exponential function

I want to prove $\exp(iAx) = I\cos x + iA\sin x$, where $I$ is the identity matrix $\in M_n(\mathbb{C})$, $A\in M_n(\mathbb{C})$ s.t. $A^2 = I$ and $A$ is normal, $x \in \mathbb{R}$, and $\exp(iAx)$ ...
1
vote
0answers
196 views

Matrix Calculus example

How do I use chain rule to calculate the derivative of $ h(\textbf{x}) $, where $ h(\textbf{x}) = f(\textbf{Sx}), f:R^n \to R$ and $ \textbf{S}$ is a matrix. I know how to use chain rule to ...
2
votes
1answer
120 views

Which topic(s) does this matrix come from? What is the name of this matrix?

Which topic(s) does this matrix come from? What is the name of this matrix? $$\pmatrix{1&1&1\\1&\omega&\omega^2\\1&\omega^2&\omega}$$
3
votes
1answer
467 views

Can axis/angle notation match all possible orientations of a rotation matrix?

The rotation group is isomorphic to the orthogonal group $SO(3)$. So a rotation matrix can represent all the possible rotation transformations on the euclidean space $R3$ obtainable by the operation ...
2
votes
2answers
167 views

finding the determinant

Is there any fast way to compute the determinant of this matrix $$ \left( \begin{array}{ccccc} a & b & 0 &0 &0 \\ b & a & b &0 &0 \\ 0 & b & a &b ...
3
votes
1answer
110 views

Let $ \rho(P)$ be the spectral radius of $P$. Show $ \rho( \dfrac{P}{ \rho(P) + \epsilon } ) < 1 \text{ for all } \epsilon >0. $

Let $P$ be a square matrix and $ \rho(P)$ the spectral radius of $P$. How to show \begin{align} \rho\left( \dfrac{P}{ \rho(P) + \epsilon } \right) < 1 \text{ for all } \epsilon >0. \end{align} ...
1
vote
1answer
46 views

Simultaneous equation, is my reasoning correct here?

Determine the value of $k$ such that the matrix is the augmented matrix of a linear system with infinitely many solutions. $$\left(\begin{array}{cc|r} 8 & -4 & 5\\ 16 & k & ...
2
votes
0answers
177 views

Simultaneous equation matrices, how do my answers look here?

How many solutions does each system have A. Unique solution B. No solutions C. Infintely many solutions D. None of the above. $$\left(\begin{array}{cc|r} 1 & 0 & 6\\ 0 & 1 ...
0
votes
1answer
411 views

Create transformation matrix from scale and angle.

Suppose I have an angle e.g. 180 degrees, and I know that an image scales to 1.5 of its size, how do I map these to a 2D transformation matrix? Thanks!
3
votes
6answers
1k views

Matrix-Square Root

I was wondering about matrix square roots. What is the procedure to evaluate $(X^{T}X)^{-1/2}$? Is it by a spectral decomposition of $(X^{T}X)^{-1}$ as $U\lambda U^{T}$ followed by the square root $S$ ...
7
votes
1answer
144 views

matrix equation $(A-B)CA=B$

let $A,B,C$ be $n\times n$ matrices with real entries such that $A$ is invertible. if $(A-B)CA=B$ show that $AC(A-B)=B$. any Ideas??
3
votes
2answers
8k views

How to show a matrix is full rank?

I have some discussion with my friend about matrix rank. But we find that even we know how to compute rank, we don't know how to show the matrix is full rank. How to show this?
0
votes
0answers
103 views

matrix norm in terms of norm of the row

Let A be $n\times m$ matrix with independent identical distributed rows $X_i$valued \in $R^m$. Let $Q \in R^m$ be orthogonal projection. Suppose, Euclidean norm $P(\lVert ...
7
votes
1answer
548 views

On matrix norm equivalence

For finite dimensional spaces, all norms are equivalent, i.e. there exist constants say $A,B$ such that for all matrices from the $\mathbf M \in R^{d\times d}$ (let $d$ be a fixed positive integer) ...
2
votes
1answer
292 views

What is the expected root mean square determinant of an $n\times n$ matrix?

The expected mean determinant of random nxn matrices of 0s and 1s is 0. What is the expected root mean square determinant? e.g. $\frac{\sqrt{3}}{2\sqrt{2}}$ for a $2\times 2$
1
vote
1answer
622 views

Determinant of the “bordered” Hessian of a composition

Write $H_{f}$ for the Hessian of a real function $f:\mathbb{R}^n\mapsto \mathbb{R}$, and define the bordered Hessian as $$ H_{f} = \left(\begin{matrix}0 & \nabla f' \\ \nabla f & H ...
5
votes
3answers
257 views

computing determinant of a matrix

let $A$ be an $n\times n$ matrix with entries $a_{ij}$ such that $a_{ij}=2$ if $i=j$. $a_{ij}=1$ if $|i-j|=2$ and $a_{ij}=0$ otherwise. compute the determinant of $A$. using the famous formula ...
1
vote
1answer
302 views

Characteristic and minimal polynomial - leading coefficient and norming

When calculating the characteristic polynomial as $$\det \; (A−t E_n)$$ I get the same polynomial as when I calculate the characteristic polynomial as $$\det\;(t E_n−A).$$ Only the signs are changed. ...
0
votes
2answers
232 views

Finding the dimensions of matrices?

I'm trying to figure out if this matrix operation is possible: $$\begin{bmatrix}1&2\\3&7\end{bmatrix}\times\begin{bmatrix}1\\5\end{bmatrix}$$ I know that in order to do that I need to find ...
1
vote
2answers
508 views

Cayley Transform, Exponential Mapping and more…

Assume a self-adjoint operator, represented as hermitian matrix $H=H^\dagger$. To my knowledge there are at least 2 mappings of $H$ onto unitary matrices: Cayley's Transformation with ...
1
vote
2answers
612 views

proof that if $AB=BA$ matrix $A$ must be $\lambda E$ [duplicate]

Let $A \in Mat(2\times 2, \mathbb{Q})$ be a matrix with $AB = BA$ for all matrices $B \in Mat(2\times 2, \mathbb{Q})$. Show that there exists a $\lambda \in \mathbb{Q}$ so that $A = \lambda E_2$. ...
3
votes
2answers
227 views

Diagonalize a matrix

We have a matrix $A = \left( \begin{array}{ccc} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\ \end{array} \right)$. How do you find a diagonal matrix $D$ and an ...
2
votes
1answer
104 views

elegant way to identify the order of a group's element - $A \in (GL(2,\mathbb{R}), \; \cdot \;)$

I'm looking for an elegant way to identify and justify in writing the order of a group's element. $$ A \in (GL(2,\mathbb{R}), \; \cdot \;) \quad , \quad A= \left( \begin{array}{cc} 0&1\\ ...
3
votes
1answer
215 views

Orthogonal matrix, translations, and fixed points

Is it true that $Ax+b$ where $A^\dagger A=I=AA^\dagger$ and $A$ is an $n\times n$ real matrix, $x,b\in \mathbb R^n$ must have either a fixed point or a fixed $n-1$ hyperplane? If not, is it true for a ...
4
votes
1answer
877 views

Quick ways to _verify_ determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors …

What are easy and quick ways to verify determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors after calculating them? So if I calculated determinant, minimal ...
1
vote
1answer
103 views

Help with the mathematical representation of operations on a matrix

I need to know how to represent the following as a mathematical formula using proper math notation: I have a $1\times n$-matrix of $3$-tuples $[a, r, x]$. I need to represent the following logic ...
2
votes
4answers
182 views

Is there a classic Matrix Algebra reference?

I'm looking for a classic matrix algebra reference, either introductory or advanced. In fact, I'm looking for ways to factorize elements of a matrix, and its appropriate determinant implications. ...
21
votes
2answers
399 views

Showing $\prod\limits_{i<j} \frac{x_i-x_j}{i-j}$ is an integer

Let $x_1,...,x_n$ be distinct integers. Prove that $$\prod_{i<j} \frac{x_i-x_j}{i-j}\in \mathbb Z$$ I know there is a solution using determinant of a matrix, but I can't remember it now. Any ...
5
votes
1answer
741 views

3D Rotation Matrix Uniqueness

Given a 3D rotation matrix R in a basis B. Can we consider R as being unique in B? Is there any other 3d rotation matrix R' representing the same 3D rotation in B? How could I prove that? Note: I do ...
5
votes
4answers
464 views

$A^{100} = 0$ implies $A^2 = 0$ when $A$ is $2\times 2$

How to show the following claim $A^{100} = 0 \implies A^2 = 0$ with $A \in Mat(2 \times 2, K)$ If A is the matrix of a linear map $\phi$ then for all $v \in K^2$ the following identity should be ...
0
votes
0answers
138 views

Singular value decomposition, possible property

Suppose a singular value decomposition on matrix $P\in\mathbb{R}^{n\times m}$ is given, $P=U\Sigma V^T$ with $U=[u_1,\dots, u_n]\in\mathbb{R}^{n\times n}$, $u\in\mathbb{R}^{n}$, containing the ...
6
votes
3answers
521 views

Characterization of the trace function

We know that the trace of a matrix is a linear map for all square matrices and that $\operatorname{tr}(AB)=\operatorname{tr}(BA)$ when the multiplication makes sense. On the Wikipedia page for ...
1
vote
0answers
104 views

Need help to understand one line in the proof

Lemma. If the matrices $\mathbf{X}$ ($n\times p$ design matrix of full column rank ) and $\mathbf{L}_{2}$ satisfy $\mathbf{L}_{2}^{\prime}\mathbf{X}=\mathbf{0}$ and $\Omega$ is positive definite ...