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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7 views

Reduce Matrix to echelon Form

Suppose that to reduce a matrix A to row echelon form are necessary n elementary operations E1,...,En. Suppose that En1,...,Enk are the permutation operations that are needed. How to prove that to ...
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1answer
13 views

Invertible Matrices and Linear independence

If a matrix is invertible, what does this tell us application wise? I am familiar with what it implies in regards to the properties of the matrix, i.e: the determinant is non-zero, and for a matrix ...
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2answers
28 views

Knowing the eigenvalues for A find the matrix A

I know the eigenvalues for a matrix. Let's say they are 2 and 1. How can I find the matrix A for them (all members of A are not null) ?
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1answer
22 views

How can I find values that make a matrix Linearly Dependent?

Let $$A=\left(\begin{smallmatrix}2&-1&0&1\\0&0&a&3\\0&0&0&b\end{smallmatrix}\right)$$ For what choices (if any) of real numbers a and b are the rows of A linearly ...
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1answer
29 views

Matrix norm relation between $A^*A$ and $AA^*$

Let $A^*$ denote the complex conjugate transpose of a matrix $A$. In the Euclidean norm, if $$||A^*A+AA^*||=||A^*A||$$ does this imply that $$||A^*A+AA^*||=||A^*A-AA^*||$$ Related question: Matrix ...
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1answer
10 views

Infinity norm greater $1$ implies that spectral radius greater $1$?

Suppose I have an arbitrary real matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$ such that the sum of absolute values in each row is greater than $1$: $$\sum_{j=1}^n |A_{ij}|>1,\quad \forall ...
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3answers
58 views

Rank of $ A^2 +A + I$

Let $A$ be $6 \times 6$ real symmetric matrix of rank $5$. Find the rank of $A^2 +A +I$. Well i dont know any tool that can solve this question. The book says answer is $5$ but it could be wrong ...
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1answer
22 views

Matrix norm question

Let $A^*$ denote the complex conjugate transpose of a matrix $A$. In the Euclidean norm, if $$||A^*A+AA^*||=||A^*A||$$ does it imply that $AA^*=0$. If not, could you give a counter-example?
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1answer
26 views

Product of symmetric matrices

Let $A \in \mathbb{R}^{n \times n}$ be symmetric. I am trying to understand under which conditions on $B \in \mathbb{R}^{n \times n}$ the product $AB$ is also symmetric. It is clear that if $B$ is ...
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0answers
9 views

What Boolean matrices are reachable from the NXN identity only by adding columns mod 2?

This problem arose from work on a Boolean software problem. Starting from an $NxN$ identity matrix, the only operation allowed is to add some column i to another column j (mod 2) - i.e. for all $k$, ...
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3answers
32 views

Alternative matrix representation for translation

The ''usual" way to write translation for $\textbf{v}\in \mathbb{R}^2$ is with the following $3\times3$ matrix $$ \left( {\begin{array}{ccc} 1 & 0 & x_{0} \\ 0 & 1 & y_{0} \\ ...
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1answer
20 views

Transformation Matrix for cube in 2D

My task is to transform the cube from the left corner to the big cube in the middle: What I did was: First i scale the cube: $$ \begin{pmatrix} 4 & 0 & 0 \\ 0 & ...
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1answer
10 views

Row Vector (sum 1), Matrix (rows sum 1), Matrix product rows sum to 1

Am trying to prove that the product of a row vector $$\kappa$$ size (1 x n) and $$Matrix B$$ where the sum of the rows equals 1 produces a matrix C (1 x n) where the sum of the row is also 1. So ...
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1answer
13 views

Form of Matrix for Reflection about a Line

I've seen a bunch of variations on the wonderful properties of this specific matrix. My textbook gives one algebraic form in particular that I'm having a bit of trouble verifying: Any help here? I ...
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0answers
17 views

SVD of a parametrized matrix.

Suppose we have a parametrized matrix $Z(λ)\in R^{m\times n}$ where $λ\in(a,b)$ and $Ζ(λ)$ is an analytic function of $λ$, e.g. $Z(λ)=λA+(1-λ)B$ where $A,B \in R^{m\times n}$. In general, the ...
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0answers
17 views

How to prove matrix in hermite canonical form is idempotent

It says in Rao that it's "easy to prove" that a matrix H in hermite canonical form is necessarily idempotent under matrix multiplication (i.e. HxH=H). I am trying to find a proof to this but everyone ...
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0answers
18 views

Maximizing inner product

Suppose we have two row vectors $a$ and $b$ of nonnegative real numbers such that, for $j<k$ $a_j\leq a_k$ and $b_j\leq b_k$. Let P be a permutation matrix. Can we prove (or disprove) that $$ ...
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3answers
37 views

Calculation of determinant

Is there any easier way to make sure the determinant of the following matrix is n (the dimension of square matrix)? $ \begin{vmatrix} 1 & -1 & -1 & -1 & \cdots & -1 \\ 1 ...
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1answer
21 views

What values of $a$ and $b$ does this system have infinitely many solutions?

As a disclosure, this question is more for me to confirm that I did my work correctly. More specifically, the "solution" provided to me claims there are two values of $a$ and $b$ that yield infinite ...
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4answers
90 views

Show that a given matrix always has an eigenvector in $\mathbb{R}$ Can somebody give a hint?

The given exercise is, for all $\theta$ in $\mathbb{R}$, show that the matrix always has an eigenvector in $\mathbb{R^2}$ $$ A = \begin{pmatrix} \cos(\theta) & \sin(\theta) \\ \sin(\theta) & ...
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0answers
32 views

Compute the Frobenius norm

I'm trying to compute the Frobenius norm of $L^{-1}$ where $$L^{-1}= I + N + N^2 + ... + N^{n-1}$$ and $N$ is strictly lower triangular. Can anyone suggest some way to do this?
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17 views

What's the name of this kind of matrix in which entries in each column and row are distinct?

I want to generate an n by n matrix, in which each column and each row contains each of n letters (or integers if you prefer) exactly once. This is like a sudoku board but without the 3x3 boxes ...
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0answers
24 views

Understanding averaging of symplectic matrices via Haar measure

In McDuff and Salamon's Intro. to Symplectic Topology (2nd edition), there's a proof that $U(n)$ is a maximal compact subgroup of $Sp(2n)$ which I'm trying to understand. The proof uses the Haar ...
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1answer
13 views

unitary matrix decomposition using orthogonal matrices

Is it possible to decompose an n by n unitary matrix U, such that $U=O_1DO_2 $ with D being diagonal(obviously just has complex phase factors) and $O_1,O_2$ being real orthogonal matrices.
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1answer
18 views

Index notation with non-commuting matrix entries

Just a contradiction I came across working with matrix multiplication in index notation: I'm probably using some rule wrong, but I can't figure out which one. Consider the expression $A_{ij} B_{ik}$, ...
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0answers
39 views

Jordan normal form book

I am currently reading the book Basic Algebra [modern] Anthony W. Knapp about Jordan canonical form Is there any detailed oriented book about Jordan Normal Form which explain : An Algorithm to put ...
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0answers
27 views

Jordan normal form Upper or lower

I am reading a jordan form book at the moment, Basic Algebra [modern] Anthony W. Knapp page $231$, but I feel the lack of understanding : should we have to start with the Bigger Jordan blocks of ...
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2answers
95 views

Probability of building an Invertible Matrix

If we build a 10X10 matrix,randomly filling with 1's and 0's, is it more likely to be invertible or singular? First of all until we have 10 1's its not going to be invertible. With 10 1's on the ...
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0answers
16 views

What is the mathematically operation referred to where you put a matrix of lower dimension into a matrix of higher dimension

Suppose I have a vector $A = [a_1, a_2, a_3]$; now let's construct another vector $B =[A, 1]$ What is this process of putting a smaller matrix into a larger one generally referred to? Embedding?
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0answers
63 views

Determine all $t\in\mathbb{R}$ for which $A_t$ is diagonalizable.

I have this matrix: $$A_t=\begin{pmatrix}\phantom{-}2+t&\phantom{-}4&\phantom{-}2+t&\phantom{-}2+t\\\phantom{-}t-2&\phantom{-}0& -6+t& ...
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1answer
33 views

Eigenvector Problem

Given a matrix $X$, let $eigvec(X)$ be its eigenvector associated with the largest eigenvalue. Is there a relationship among $eigvec(X+X^T)$, $eigvec(X)$ and $eigvec(X^T)$? In other words, can I use ...
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1answer
29 views

Finding the corresponding Perron eigenvalue

Find the Perron root and the corresponding Perron eigenvector of A. $\begin{bmatrix} 0 &1 &1 \\ 1&0&1 \\ 1&1&0 \end{bmatrix}$ I figured out the Perron root which happens to ...
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0answers
36 views

Transpose of the adjacency matrix

As homework I had to do an adjacency matrix for the following graph: My solution was the following: $$ \begin{bmatrix} 0&0&1&0&0 \\ 1&0&0&1&0 \\ ...
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0answers
15 views

$A \times B^{-1}$ has irreducible characteristic polynomial when $A,B$ are random integer matrices — simple proof?

Let $A,B$ be $d\times d$ integer matrices with each entry drawn uniformly from $[0,2^n)$, and define the rational matrix $C = A \times B^{-1}$. Is there a simple way to prove that $C$'s characteristic ...
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0answers
26 views

Linear maps and subspaces

The set-up for my question is this, let $k \le n$, let $E \subseteq \mathbf{R}^n$ be a $k$-dimensional subspace. Let $I \subseteq \{1,\ldots, n\}$ such that $|I| = k$, then we can define coordinate ...
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1answer
17 views

Inversing badly-conditioned square matrix: methodology

I have a badly-conditioned square matrix. I need to inverse it. For inversing, currently I'm doing the following steps: I take the badly-conditioned matrix with size of $n$ by $n$ By reduced row ...
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1answer
38 views

matrix vs vector span {} linear algebra

I am in a University Linear Algebra course and am confused by the term span and its relation to both matrices and vectors. Can someone help clarify what they mean? =Span= Can it only be made of ...
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1answer
21 views

Prove that the LDU factoriztion is unique [on hold]

How would one prove that the LDU factorization of a matrix is unique?
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1answer
24 views

If $R$ is a commutative simple ring with identity , then is any matrix ring $M_n(R)$ also simple? [duplicate]

If $R$ is a commutative simple ring with identity , then is any matrix ring $M_n(R)$ over $R$ of matrices of size $n$ also simple ?
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1answer
16 views

Generators of $Sp(2n)$

Let $\sigma =\begin{pmatrix} 0 & 1 \\ -1 & 0\end{pmatrix}$. Define $J_{2n} = \underbrace{\sigma \oplus \cdots \oplus \sigma}_{\text{$n$ copy}}$. We define a $2n \times 2n$ real matrix matrix ...
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0answers
22 views

Is there a formula for the sum of absolute eigenvalues in terms of matrix elements?

Given a symmetric matrix $X \in \mathbb{R}^{n \times n}$. We know the following: trace$(X) = \sum_{i=1}^n x_{ii} = \sum_{i=1}^n \lambda_i$ where $x_{ii}$ is the $i$th element on the diagonal of $X$, ...
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2answers
57 views

Find a positive definite matrix B such that $B^2=A$. [on hold]

Find a positive definite matrix B such that $B^2=A$, where $$A=\begin{pmatrix} 2&-1\\ -1&2 \end{pmatrix}$$
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1answer
22 views

How to prove that $B$ is positive definite when $\|A-B\|\leq\lambda_\min(A)$ for some positive definite $A$?

Denote by $\mathbb R^{n \times n}$ the vector space of $n \times n$ matrices with real entries. For $A \in \mathbb R^{n \times n}$, the notation $A\succ 0$ means that $A$ is symmetric and positive ...
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0answers
8 views

Need to find the coefficient matrix [on hold]

I have $f([1 0 0]^t) = [ 3 -2 -1]^t f([0 1 1]^t) = [ 1 1 1]^t f([1 1 1]^t) = [ 1 2 3]^t$ if $f(x) = AX$ for any vector where $x$ belongs to $\mathbb R^3$. Find Coefficient matrix A $^t$ stands for ...
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2answers
263 views

Why does Gaussian elimination not preserve similarity of a matrix?

I am trying to understand reduction of an unsymmetric real square matrix to Hessenberg form from Numerical Recipes Vol. 3. In it, the author states that one does not use Gaussian elimination for ...
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0answers
16 views

If through row reduction of a square matrix you can produce a row of zeros does that automatically make that matrix singular?

If through simple row operations such as adding one row to another I can produce a row of entirely zero does this mean that the matrix will be singular? For example: $$\left\{ \begin{matrix} 1 & ...
2
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1answer
26 views

extended PCA (tangled matrices)

Given an $m$ by $n$ matrix $A$ and the constant $r$, the principal component analysis allows us to find matrices $W$ and $H$ so that the $WH$ gives a lower rank approximation of $A$. In other words, ...
3
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1answer
48 views

Real matrix with the property that every nonzero vector in $\mathbb{R}^n$ is an eigenvector of $A$. [duplicate]

so I'm supposed to let $A$ be a square real matrix with the property that every nonzero vector in $\mathbb{R}^n$ is an eigenvector of $A$. And I'm supposed to show that $A=\lambda I$ for a constant ...
2
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1answer
29 views

Generators of $\Gamma_0(N)$

Let $\textbf{T}:=\bigl(\begin{smallmatrix} 1&1\\ 0&1 \end{smallmatrix} \bigr)$, $\textbf{S}:=\bigl(\begin{smallmatrix} 0&1/\sqrt{N}\\ -\sqrt{N}&0 \end{smallmatrix} \bigr)$ and $H$ the ...
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1answer
40 views

What is the derivative of this?

I have a function of the following form: $J = \|W^TW-I\|_F^2$ Where, $W$ is a matrix and $F$ is the Frobenius Norm. How can I find the derivative of $\frac{\partial J}{\partial W}$ ?