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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8 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 ...
0
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1answer
22 views

Matrix function to express pair-wise distances of rows in $X, Y$

There are two real matrices: $X, Y$ with $X$ being of dimension $n_1$ x $p$, $Y$ of dimension $n_2$ x $p$. The goal is to form the matrix $D$ of dimension $n_1$ x $n_2$ where each element $d_{ij}$ ...
4
votes
3answers
27 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 ...
1
vote
4answers
87 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
13 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 ...
1
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0answers
9 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 (i.e. HxH=H). I am trying to find a proof to this but everyone claims it's obvious. It is ...
2
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0answers
13 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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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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1answer
29 views

Solution for set of matrix equations involving an inverse

I am encountering the following set of three matrix equations for which I search a solution in terms of ${\bf M}\in\mathbb{R}^{N\times N}$ and ${\bf D}\in\mathbb{R}^{Q\times N}$, $${\bf M}{\bf W} = ...
1
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0answers
28 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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0answers
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 ...
2
votes
2answers
177 views

Determinants of symmetric tridiagonal matrix and of Toeplitz matrix

Is there any fast way to compute the determinant of this matrix: $$ \begin{vmatrix} a & b & 0 &0 &0 \\ b & a & b &0 &0 \\ 0 & b & a &b &0 \\ 0 & 0 ...
1
vote
1answer
12 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.
3
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0answers
15 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 ...
0
votes
1answer
16 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 $A_{ij} B_{ik}$, ...
1
vote
0answers
59 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& ...
5
votes
1answer
36 views

How do I show that $\inf\limits_{\det(X)\neq0}\|X^{-1}AX\|^{2}_{F}=\sum\limits_{\lambda\in{\Lambda}}|\lambda|^{2}$?

Show that $$\inf\limits_{\det(X)\neq 0}\|X^{-1}AX\|^2_F=\sum_{\lambda\in\Lambda}|\lambda|^{2}$$ holds, where $\Lambda(A)$ is the set containing all eigenvalues of A, and $\|\cdot\|_{F}$ is the ...
8
votes
1answer
94 views

Is a normal matrix satisfying $A^TA=…$ circulant?

Let $A=\{a_{ij}\}$ be a normal matrix such that $a_{ij}\geq 0$ with equality iff $i=j$. Suppose that $$ A^TA=\begin{pmatrix} a & b & \cdots & b\\ b & a & \ddots & \vdots\\ ...
2
votes
0answers
36 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 ...
5
votes
2answers
93 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 ...
0
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0answers
25 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 ...
0
votes
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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0answers
15 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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2answers
5k views

How to check if a matrix is positive definite

i want to know how to check if a matrix M is positive definite ,assume that M is 3x3 real numbers matrix i think one way is to put the matrix in a quadratic form $X^TMX$ , where X is a vector ...
2
votes
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 ...
5
votes
1answer
147 views
+100

Why a tensor product of $2\times 2$ unitaries cannot implement a $3\times 3$ unitary?

Let $\{v_1, \dotsc, v_m\} \in \mathbb{C}^{2^n}$ be a set of orthonormal vectors. Define a map $R_m$ from $2^n \times 2^n$ to $m \times m$ matrices as follows: $$R_m(M) := \sum_{i,j=1}^m (v_i^*M v_j) ...
0
votes
1answer
11 views

Issues with a particular bilinear form and determining rank, signature, etc. of its restriction

Let $b: M_2(\mathbb{R}) \times M_2(\mathbb{R}) \to \mathbb{R}$ such that $b(X,Y)=trace(X^tAY)$, where $X^t$ is the transpose of $X$ and $A=\begin{pmatrix} 2 & 1\\1 & 0\end{pmatrix}$. In my ...
0
votes
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 ...
10
votes
5answers
190 views

Invertibility of a Kronecker Product

Prove that $A\otimes B$ is invertible if and only if $B\otimes A$ is invertible. I don't have a clue where to start to be honest. I am not very familiar yet to the Kronecker Product so could you ...
0
votes
0answers
14 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 ...
1
vote
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 ...
0
votes
2answers
153 views

How to show $U(n)$ is a group?

The first part of the question is the following: Let $A$ and $B$ be complex $n \times n$ matrices. If $A = (a_{ij})$ then we define its complex conjugate as $\overline{A} = (\overline{a_{ij}})$ . ...
1
vote
2answers
36 views

Define a $2 \times 2$ matrix that is the lower $2 \times 2$ block in $A$ (Matlab)

First of all, on this Matlab exercise sheet that I am currently working through what does the term 'the lower $2 \times 2$ block' mean in the question below? $A = \left[\begin{array}\ 1 & 2 ...
3
votes
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 \\ ...
0
votes
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 ...
2
votes
1answer
24 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, ...
0
votes
1answer
16 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 ...
0
votes
1answer
89 views

Prove that a matrix is the permutation matrix of a permutation

Prove that a matrix is the permutation matrix of some permutation just when ...
1
vote
1answer
20 views

Prove that the LDU factoriztion is unique [on hold]

How would one prove that the LDU factorization of a matrix is unique?
3
votes
1answer
259 views

Express Hadamard product as a normal matrix product

I have $N^2$ equations which I can write as the following Hadamard product. Is there a way I can get rid of the Hadamard product and express this using usual matrix operations? $\left[ ...
2
votes
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 ...
2
votes
1answer
21 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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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 $X$, ...
0
votes
2answers
56 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 $$A=\begin{pmatrix} 2&-1\\ -1&2 \end{pmatrix}$$
0
votes
1answer
17 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 ...
0
votes
2answers
21 views

Inverse of matrix sum

I found on the Wikipedia page "Determinant" the following property: For any invertible $m \times m$ matrix $X$, $\det(X + AB) = \det(X) \det(I_m + BX^{-1}A)$. Is this true? If so, how is this ...
0
votes
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 ...
4
votes
2answers
262 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 ...
0
votes
2answers
340 views

Given a $4\times 4$ symmetric matrix, is there an efficient way to find its eigenvalues and diagonalize it?

I have a $4\times 4$ matrix $$A=\left(\begin{array}{cccc}8 & 11 & 4 & 3\\11 & 12 & 4 & 7\\4 & 4 & 7 & 12\\3 & 7 & 12 & 17\end{array}\right).$$ I want to ...
3
votes
2answers
57 views

Skew symmetric $4\times 4$ matrix of full-rank

I have come across the fact that a $4\times 4$ skew-symmmetric matrix of full-rank is equivalent to \begin{pmatrix} 0 &\theta_1& 0 &0 \\ -\theta_1& 0 &0 &0 \\ 0& 0&0 ...