# Tagged Questions

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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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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\\ & \...
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### 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 ...
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### 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}\...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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, ...
Problem I need to find a simpler formula for the following series: S = $\sum_{a=1}^{\infty} \frac{1}{a} \sum_{b=1}^{a} X^{b-1}MX^{a-b} = \sum_{a=1}^{\infty} \frac{1}{a} X^{a-1} \sum_{b=0}^{a-1} X^... 1answer 23 views ### Doubt on Prasolov's notation:$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: ... 1answer 45 views ### Diagonalizability of a given matrix I must find out under which conditions the matrix $$A= \left[\begin{array}{ccc|cc}& & & c_0 &\\ & & &c_1&\ddots\\ & & &c_2 &\ddots& c_0\\ &... 1answer 22 views ### Mutually orthogonal vectors in a complex vector space? Consider a Matrix A \in \mathbb C^{m \times n}, m<n which is build by vectors like$$ A = \begin{pmatrix} | & | & & | \\ \vec a_1 & \vec a_2 & \cdots & \vec a_n \\ | &... 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 ... 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., $$V = \left[\begin{array}{cccc} V_{11}&V_{12}&\cdots&V_{1m}\\ V_{21}&V_{22}&\cdots&V_{2m}\\ \vdots&\... 0answers 14 views ### Generate a class of matrices via optimization I want to generate a matrix (using Matlab) with the following properties: (1) A = (a_{ij}) \in \mathbb{R}^{n \times n}; (2) a_{ij} \in \{0,1\} and a_{ii} = 0 for all i\in\{1,2,\cdots, n\}; (... 0answers 18 views ### How do I calculate similar matrix with arbitrary change of base matrix P=[\begin{matrix}\overrightarrow v_1 & \overrightarrow v_2 & \overrightarrow v_3\end{matrix}] with \overrightarrow v_1, \overrightarrow v_2, \overrightarrow v_3 \in \Bbb R^3 and A= \... 0answers 20 views ### Is there an algorithm for finding the largest possible linear subspace of a given vector space having this specific property? Let G_1,G_2,\dots,G_k be n\times n real matrices, and let \mathcal{G} = \operatorname{span}\left\{ G_k\right\}. Let \mathcal{V} be a linear subspace of \mathcal{G}, i.e. \mathcal{V} \... 2answers 58 views ### What are the constraints on \alpha so that AX=B has a solution? I found the following problem and I'm a little confused. Consider$$A= \left( \begin{array}{ccc} 3 & 2 & -1 & 5 \\ 1 & -1 & 2 & 2\\ 0 & 5 & 7 & \alpha \end{... 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 ... 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 ... 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 ... 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. ... 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 ... 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}(...
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 ...