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
votes
2answers
33 views

For which values of t does a matrix not have eigenvalues

I need help solving this problem "For which values of real parameter t does the matrix: \begin{bmatrix} π^2t^2 & 36\\ -36 & 0 \\ \end{bmatrix} NOT have real eigenvalues. Thank you.
0
votes
0answers
18 views

Matrix Calculation Significance and Multivariate Bayesian Methods

Suppose I have the matrix given by: $$X = \begin{bmatrix}1 & 0 & 0\\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{bmatrix}$$ This matrix actually represents whether a user interacted with a ...
2
votes
2answers
25 views

Non-strict column diagonally dominant matrix inner product

Let $A \in \mathbb{R}^{n \times n}$ be a normalized non-strict column diagonally dominant matrix, that is: $$a_{j,j} = \sum_{i \ne j} \left|a_{i,j}\right|$$ where $$0 \le a_{j,j} \le 1$$ and $$-...
1
vote
1answer
28 views

Operatornorm of $(\mathbb{R}^d, \|\cdot\|_1) \to (\mathbb{R}^d, \|\cdot\|_{\infty})$

Determine the operatornorm of the mapping $I:(\mathbb{R}^d, \|\cdot\|_1) \to (\mathbb{R}^d, \|\cdot\|_{\infty})$! Unfortunately I haven't many ideas for this task. I know that the definition of the ...
0
votes
0answers
22 views

commutativity of log(I + A) and log( A−1) (matrix function)

I'm self-(re)learning linear algebra since the beginning of the summer, and i have a problem with the following exercice entitled additive logarithmic. If i'm right, we need to prove the ...
0
votes
1answer
10 views

Convex set or not?

This question is related to my previous question Set of all positive definite matrices with off diagonal elements negative I know that the set of all positive definite matrices form a convex set. ...
1
vote
0answers
14 views

nth product of sequential matrices

$\forall n \in \mathbb{N}$, let: $$P_n = \left( \begin{matrix} a & 1-a \\ b_n & 1-b_n \end{matrix} \right). $$ Whereby $\{b_n\}_{n \in \mathbb{N}}$ is a monotonically increasing sequence of ...
1
vote
2answers
52 views

Order of $\mathrm{SL}(n,\mathbb{F}_p)$ (Constructive proof)

Most proofs of $$ \vert ~\mathrm{GL}(n,\mathbb{F}_p) ~\vert = \prod_{k=0}^{n-1} (p^n-p^k) $$ I have seen so far, are done by counting the possibilities to build up invertible matrices i.e. counting ...
0
votes
1answer
52 views

Find $B(B^{T}B)^{-1}B^{T}$.

To find: $$B(B^{T}B)^{-1}B^{T}$$ for $B=[0,1,-1]^T$ I have $$\begin{bmatrix} 0\\ 1\\ -1 \end{bmatrix} \left ([0,1,-1]\begin{bmatrix} 0\\ 1\\ -1 \end{bmatrix} \right )^{-1}[0,1,-1]$$ but ...
2
votes
0answers
44 views

Taylor series of a square matrix

Let $A$ be a constant square matrix, $\Delta t$ is a scalar. I would imagine a taylor series of its exponential like this: $$e ^{A\,\Delta t} = I + \sum_{n=1}^\infty \frac{(A\,\Delta t)^n}{n!} $$ ...
0
votes
1answer
18 views

Write summation of vector outer products into matrix form

My question is as follows: Given the weighted summation of vector outer products $\sum_i\sum_jh_{ij}{\bf v_i}{\bf u_j}^T$, where $h_{ij}$ is the weight, and ${\bf v_i,u_j}$ are column vectors, I was ...
0
votes
2answers
34 views

Mathematical calculation

I encountered during my reading to ridge regression that $$(X^TX+\lambda I)^{-1}X^TX = I-\lambda(X^TX+\lambda I)^{-1}$$ What mathematical manipulation has been done here? Thanks in advance
1
vote
2answers
39 views

Getting characteristic polynomial from a small matrix

Sorry I don't know how to format matrices, but if I have this matrix $\pmatrix{1& 1& 0\\ 0& 0& 1\\ 1 &0& 1\\}$ How is the characteristic polynomial $λ^3 − 2λ^2 + λ − ...
0
votes
1answer
15 views

Matrix for a recurrence

The matrix for a recurrence of the form $a_{k+2} = ka_{k+1}+a_{k}$ where $a_0 = 0$ and $a_1 = 1$ is given by $$\begin{bmatrix}k & 1\\ 1 & 0 \end{bmatrix}^n = \begin{bmatrix} a_{k+1} & a_k \...
1
vote
2answers
46 views

Find non-diagonal matrices $A$ and $B$ such that $B^TAB$ is diagonal

Here $B^T$ denotes the transpose of $B$. $A$ and $B$ are invertible $3\times 3$ matrices with integer entries. $A$ is symmetric positive definite with at most two zero entries. We want the ...
0
votes
0answers
22 views

sending basissen

Lets say we have this $3\times3$ matrix: $$ \begin{bmatrix} 4 &−4 &12\\ 1& -1& 3\\ −1& 1 &−1 \end{bmatrix} $$ What is the algorithm to find a basis of $\Bbb R^3$ for which ...
2
votes
1answer
33 views

Row sum of inverse of a matrix

Let's say I have a matrix A, $$A= \begin{bmatrix} a_{11}& a_{12} & a_{13} \\ a_{21}& a_{22} & a_{23} \\ a_{31}& a_{32} & a_{33} \end{bmatrix} $$ All the elements of A are ...
1
vote
2answers
63 views

Compute $R^{2016}$ of a given counterclockwise rotation.

Write out the matrix $R$ of counterclockwise rotation by 30$^{\circ}$ in $\mathbb{R}^2$. Compute ${R}^{2016}$. Now this is an easy question to answer overall; 30 goes into 360 12 times and one twelfth ...
1
vote
1answer
37 views

Element-wise derivative of matrix logarithm

$E = ln(C) = -\sum_{a=1}^{\infty}\frac{1}{a}(I-C)^a$ I want to find a simple formula for $\frac{\partial E_{ij}}{\partial C_{pq}}$ $\frac{\partial C_{ij}}{\partial C_{pq}} = \delta_{ip}\delta_{...
0
votes
1answer
49 views

$A^2$ is bounded $\implies$ $A$ is bounded?

Let $A_n$ be a sequence of $k \times k$ real matrices. Assume $A_n^2$ is bounded w.r.t some norm. Is $A_n$ also bounded? I was able to show this is true if $A_n$ are symmetric matrices (using SVD). ...
0
votes
1answer
53 views

Is there an effect for the eigenvalues on vectors other than the Eigenvectors?

Does having an eigenvalue greater than one mean that the magnitude of any vector multiplied by the matrix will be increased?
1
vote
1answer
61 views

Linear independence of standard basis vectors from Vandermonde style vectors

Is it true a statement that all $n$ dimensional vectors of the standard basis (e.g. $[1 \ 0 \ 0 \ ...]^T$, $[0 \ 1 \ 0 \ ...]^T$ etc ..) are linearly independent from the set of the $n-1$ vectors $...
0
votes
1answer
25 views

QR decomposition subcases

Is the full QR decomposition the most general, which includes the reduced QR, i.e, is it alright to always compute the full QR Decomposition for a given matrix blindly? What's the point of having two ...
0
votes
0answers
29 views

Linearize Matrix Equation

I want to find a linearized formula for G in terms of A. $G = B^TC^{-1}T(I+BA)$ $G$ is 4x2 $B$ is a constant matrix 2x4 $A$ is a variable matrix 4x2 $C = I + A^TB^T + BA + BAA^TB^T$, so $C$ is ...
2
votes
1answer
28 views

Characterization of a square matrix.

I would like to see a proof to this fact. For a square matrix the following are equivalent: $A$ has a right inverse. $A$ has rank $n$, where $A$ is $n \times n$. $A$ is invertible.
5
votes
1answer
57 views

Rank of a lower triangular block matrix

For $$A= \begin{bmatrix}B&0\\C&D\end{bmatrix}$$ where $B, C, D$ are matrices that may be rectangular, is it true or false that $$\text{rank}(A)=\text{rank}(B)+\text{rank}(D)$$ I think that if ...
1
vote
1answer
31 views

Right inverse matrix

I know that if $A, B$ and $C$ are square matrices such that $$ AC=I \quad \mbox{and} \quad BA=I, $$ then \begin{eqnarray*} AC=I & \Rightarrow & BAC=B\\ & \Rightarrow &IC=B\\ & \...
1
vote
2answers
47 views

A proof of the Continuity of the inverse matrix function

I would like to see a proof to this fact. If $A$ is an invertible matrix and $B \in \mathcal{L}(\mathbb{R}^n,\mathbb{R}^n)$, that is an bounded linear opertor in $\mathbb{R}^n$. Then, if there ...
1
vote
1answer
59 views

How to prove that the vectors of the Krylov space of A are linearly independent if A is nonsingular.

$\mathbf{K}$ is a Krylov matrix. \begin{align} \mathbf{K}&= \left[ \begin{array}{ccccc} \mathbf{b} & \mathbf{A}\mathbf{b} & \mathbf{A}^2\mathbf{b} & \cdots & \mathbf{A}^{N-1}\...
4
votes
4answers
127 views

Not understanding derivative of a matrix-matrix product.

I am trying to figure out a the derivative of a matrix-matrix multiplication, but to no avail. This document seems to show me the answer, but I am having a hard time parsing it and understanding it. ...
2
votes
2answers
52 views

Given a symmetric matrix $A$, find $P$ such that $P^T A P$ is a diagonal matrix

Given $$A =\begin{pmatrix} 0 & 3 & 0 \\ 3 & 0 & 4 \\ 0 & 4 & 0\end{pmatrix}$$ find a matrix $P$ such that $P^T A P$ orthogonally diagonalizes $A$. Verify that $P^TAP$ is ...
3
votes
1answer
95 views

Is it possible to endow $\text{GL}_2(\Bbb R)$ with a ring structure?

My question is the following: Is it possible to find a binary operation $*$, seen as an addition, such that $(\text{GL}_2(\Bbb R),*,\cdot)$ has a ring structure (not necessarily with a unit)? [We ...
0
votes
0answers
20 views

Change eigenvalues of correlation matrix and transform into original basis

I use the Random Matrix Theory to filter out the information from the correlation matrix that is associated with noise - Marcenko Pastur band. That is straight forward. Then I follow Rosenow, Bernd, ...
0
votes
0answers
49 views

Simplify Series composed by Noncommutative Matrices

Problem I need to find a simpler formula for the following series: S = $\sum_{a=1}^{\infty} \frac{1}{a} \sum_{b=1}^{a} X^{b-1}MX^{a-b} = \sum_{a=1}^{\infty} \frac{1}{a} X^{a-1} \sum_{b=0}^{a-1} X^...
0
votes
1answer
23 views

Doubt on Prasolov's notation: $A=||a_{ij}||_1^n$.

I'm reading Prasolov's: Problems and Theorems in Linear Algebra. He defines the following notation: $||a_{ij}||_p^n$ as a notation for a matrix, where $p\leq i, \;j\leq n$. And there is a problem: ...
2
votes
1answer
45 views

Diagonalizability of a given matrix

I must find out under which conditions the matrix $$A= \left[\begin{array}{ccc|cc}& & & c_0 &\\ & & &c_1&\ddots\\ & & &c_2 &\ddots& c_0\\ &...
2
votes
1answer
22 views

Mutually orthogonal vectors in a complex vector space?

Consider a Matrix $A \in \mathbb C^{m \times n}$, $m<n$ which is build by vectors like $$ A = \begin{pmatrix} | & | & & | \\ \vec a_1 & \vec a_2 & \cdots & \vec a_n \\ | &...
0
votes
1answer
51 views

What is the determinant of exp(matrix)? [duplicate]

Given a square matrix $A$, form the Lie series of it, which is defined by: $$ e^A = I + A + \frac{1}{2} A^2 + \frac{1}{3!} A^3 + \cdots + \frac{1}{n!} A^n = \sum_{k=0}^\infty \frac{1}{k!} A^k $$ Is ...
1
vote
1answer
33 views

How to find the derivative of the following matrix?

Let $V$ be $n$ by $m$ matrix and let $x$ be $m$ by $1$ vector, i.e., $$V = \left[\begin{array}{cccc} V_{11}&V_{12}&\cdots&V_{1m}\\ V_{21}&V_{22}&\cdots&V_{2m}\\ \vdots&\...
0
votes
0answers
14 views

Generate a class of matrices via optimization

I want to generate a matrix (using Matlab) with the following properties: (1) $A = (a_{ij}) \in \mathbb{R}^{n \times n}$; (2) $a_{ij} \in \{0,1\}$ and $a_{ii} = 0$ for all $i\in\{1,2,\cdots, n\}$; (...
0
votes
0answers
18 views

How do I calculate similar matrix with arbitrary change of base matrix

$P=[\begin{matrix}\overrightarrow v_1 & \overrightarrow v_2 & \overrightarrow v_3\end{matrix}]$ with $\overrightarrow v_1$, $\overrightarrow v_2$, $\overrightarrow v_3 \in \Bbb R^3$ and $A= \...
0
votes
0answers
20 views

Is there an algorithm for finding the largest possible linear subspace of a given vector space having this specific property?

Let $G_1,G_2,\dots,G_k$ be $n\times n$ real matrices, and let $\mathcal{G} = \operatorname{span}\left\{ G_k\right\}$. Let $\mathcal{V}$ be a linear subspace of $\mathcal{G}$, i.e. $\mathcal{V} \...
2
votes
2answers
58 views

What are the constraints on $\alpha$ so that $AX=B$ has a solution?

I found the following problem and I'm a little confused. Consider $$A= \left( \begin{array}{ccc} 3 & 2 & -1 & 5 \\ 1 & -1 & 2 & 2\\ 0 & 5 & 7 & \alpha \end{...
0
votes
0answers
34 views

Matrix that changes basis

Does change of basis matrix we use in linear transformations change both the domain and range of the transformation matrix? By the way, I have a hard time calculating the change of basis matrix. I've ...
2
votes
4answers
46 views

Show that the diagonal elements are not all $0$

If the rank of a real symmetric matrix be $1$, show that the diagonal elements of the matrix can not be all zero. Since the rank is $1$, the determinant of the entire matrix is $0$, so it is ...
0
votes
1answer
16 views

Embedding A Matrix

Okay, I have a matrix $A \in M_k(\mathbb{C})$ that I want to view it as embedded in some larger matrix in $M_n(\mathbb{C})$, which means $k < n$, with zeros filling in the rest of the entries so as ...
0
votes
0answers
43 views

represent an image in linear algebra

Can we represent a grayscale image as a matrix of values, and then apply all our linear algebra techniques to that? Like finding the column space and null space, reason about the matrix structure. ...
2
votes
1answer
58 views

How can I divide a vector by a matrix?

I am trying to go backwards through a neural network. I have an output and I want to see what input would lead to that output. To go forwards I start with a vector and multiply by a matrix and then ...
1
vote
0answers
26 views

Is there a non-trivial special orthogonal transform which preserves the diagonal elements of a symmetric matrix with positive entries?

This problem is at the interface of matrix algebra and spectral graph theory. Let $\mathbf{S}$ be a symmetric $n\times n$ matrix, with positive entries $S_{ij}\geq 0$, and $\mathbf{D} = \mathrm{diag}(...
2
votes
2answers
32 views

Matrix decomposition in unipotent matrices

Consider the positive definite and symmetric matrix $$A = \begin{pmatrix} 1 & 2 & 0 \\ 2 & 6 & -1 \\ 0 & -1 & 1 \end{pmatrix}$$ Find a decomposition with unipotent $U ...