Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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How to calculate the value of vectors after multiplication with matrix

I have the following equation: $Y = Gh \ \circ \left[ D x \right] $ where $Y$ is a vector of size $L \times 1$, $G$ is a unitary matrix with size $L \times L$, $h$ is vector $L \times 1$ with first $...
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Calculating the rank of a particular singular matrix

Question. Let $A=(a_{ij})\in\mathbb R^{(2n+1)\times (2n+1)}$, while $a_{ii}=0$, for all $i=1,\ldots,2n+1$, and $a_{ij}\in\{-1,1\}$, if $i\ne j$. If $$ \sum_{j=1}^{2n+1}a_{ij}=0, \quad \text{for all $i=...
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What should be the value of $p,q,r,s,$ and $t$ so that the following matrix is orthogonal?

What should be the value of $p,q,r,s,$ and $t$ so that the following matrix is orthogonal? $$\left[ {\begin{array}{cc} p & q & r \\ \frac{1}{\sqrt{3}} & q & s \\ \frac{1}{\sqrt{3}} &...
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How to augment a matrix to a tall orthonormal matrix to make it complete orthonormal

We are given a tall orthonormal matrix $Q_1 \in \mathbb{R}^{m \times n}, m > n$. Describe a procedure to find a matrix $Q_2$ so that $Q = [Q_1 Q_2]$ is a complete $m \times m$ orthonormal matrix.
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Is $\lVert A^{-1}_n \rVert$ bounded if $\lVert A_n \rVert$ is bounded?

Let $n \in \mathbb{N}^{\ast}$ and $A_n$ be a nonsingular $n \times n$-matrix. I know that $\lVert A_n \rVert$ is bounded for some matrix norm $\lVert \cdot \rVert$. I would like to know if $\left\...
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A potential criterion for product of nilpotent matrices to be $0$

Given a set $X := \{A_1,\dots,A_n\}$ of $n \times n$ matrices such that any product of $k$ of these matrices is nilpotent (where $1 \leq k \leq n$), does it follow that $\prod_{i=1}^n A_i = 0$. To be ...
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Nilpotent product of nilpotent $2\times 2$ matrices is $0$

Let $A,B$ be $2 \times 2$ matrices such that $A^2 = B^2 = (AB)^2 = 0$ does it then follow that $AB =0$?
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How to calculate the solution set for an equation with vectors?

I am given the following task: Calculate the the real solution set $L$ for $$(\begin{array}{cc} -10 & 17 & 70 \\ \end{array})\begin{bmatrix}x \\ y \\ z \end{bmatrix} = 0$$ I really don'...
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Rank$(AB) <$ Rank$(B)$ when $A,B,AB$ are nilpotent matrices

Let $A,B$ be $n\times n$ matrices, then we have inequality $\operatorname{rank}(B) \geq \operatorname{rank}(AB)$. I was wondering whether it is possible to make this inequality strict if $A,B,AB$ are ...
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Determine an orthonormal basis for the subspace $W$

The subspace $W$ in $\mathbb{R}^5$ is given as the solution space to the linear system of equations (you can see it in the picture). enter image description here Determine an orthonormal basis for $W$ ...
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What does the transition matrix for this diagram look like?

The problem states that: I did it the wrong way, but I fully understand their method. I tried to do it by finding that $P(X_1 =1)$ or $P(X_2 = 0)$ or $P(X_2=1)$ and so on, it got me to: $P(X_2=0) = P(...
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Does adding a zero column change the matrix behavior? Can we add a zero column to a matrix so that it can be multiplied to another matrix?

Suppose I have two matrices \begin{gather} A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, B = \begin{bmatrix} e \\ f \\g \end{bmatrix} \end{gather} I understand that A X B is NOT defined....
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Inverse Matrix or Cramer's Rule?

We consider the system $$\begin{pmatrix}-5 & 2 & 0.5\\ 0 & -20 & 1 \\ 4 & -1 & 4\end{pmatrix}\vec{x}=\begin{pmatrix}4 \\ -30\\ 42\end{pmatrix}$$ I have applied the Jacobi ...
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Define a matrix that properly defines an inner product and induces the greater distance between 2 given vectors:

I need some help. Given 2 vectors how can I choose the matrix that properly defines the inner product and induces the greater distance between x and y?
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Solve matrix equation: XA= B + 3X

I have this matrix equation: $XA = B + 3X$ $$A =\begin{pmatrix} 4 & 2 & 1\\0 & 8 & -4\\0 & 2 & 1 \end{pmatrix}$$ $$B = \begin{pmatrix}2 & 3 & 2\\1 & -6 & 7\\ -3 ...
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Existence of a power of a binary matrix with non-ziro entires

Let $A$ an $n \times n$ matrix with $0$ and $1$ entries. Is there any condition/s on $A$ that make us sure that there is a power $m$ such that $A^m$ has all entries non-zero?
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Prove that A^k = 0 implies A^2 = 0 for A_2x2 [duplicate]

I shall prove the statement that if A is a 2x2 matrix and $A^2 \neq 0$, then it follows that $A^k \neq 0$ for $k > 2$ as well. I intended to prove the contraposition, but I’m not quite sure how to ...
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Cardinality of set of the rank of consecutive matrix powers

Let $A$ be a $n$ x $n$ matrix with complex entries. Define $m=|\text{rank}(A^k) $-$ \text{rank}(A^{k+1})$:$k$ $\in$ $N^{*}$ }|. Prove that $n+1$ $\ge$ $\frac{m(m+1)}{2}$. I have tried writing $A$ in ...
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Matrix pivots with unknowns

I'm reading this series of articles on matrix determinants. It is said: Cofactor expansions are also very useful when computing the determinant of a matrix with unknown entries. Indeed, it is ...
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Prove that $I + AB$ invertible

Let $A$ and $B$ be square complex matrices such that $A^3 = A^2$ and $A + B = I$. Prove that $I + AB$ invertible. I am a beginner so I do not know how to solve it. Please help with this problem! ...
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Game theory problem with one player has more advantage

I am having trouble with this game theory exercise. 2 students are trying to be class president, so they have to compete by choosing to do a campaign or not. They can try to campaign or not in three ...
1 vote
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The equation of matrix only containing A and I(Identity Matrix) could be solved as the scalar equation?

When I try to solve the equation $A^2-5A-4I=o$ for any square matrix A and its compatible identity matrix, I factored the euqation's left into $(A-\frac{5}{2}I)^2=\frac{41}{4}I^2$ so that the answer ...
6 votes
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Prove that $A^3\equiv I\mod p$.

Let $p$ be a prime. Let $A$ be a $p\times p$ matrix whose $(i,j)$th-coordinate is ${i+j-2\choose i-1}$. Prove that $A^3\equiv I\mod p$. Source: problem 10 from this problem set. We need to show that ...
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Is $||A^\top A + B^\top B||_2 \ge || A^\top A ||_2$ true?

I'm trying to prove the following statement, which feels very natural, but I am having very little success. For matrices $A,B$, we have $||A^\top A + B^\top B||_2 \ge || A^\top A ||_2$. I feel like ...
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3 votes
4 answers
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Find the eigenvalues of a 5x5 (symmetric) matrix containing a null 4x4 matrix

Find the eigenvalues of $$A=\begin{bmatrix} 0 & 1 & 1 & 1 &1 \\ 1 & 0& 0 & 0& 0\\ 1 & 0& 0 & 0& 0\\ 1 & 0& 0 & 0& 0\\ ...
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Nilpotent matrix $n=2$

A nilpotent matrix has all eigenvalues zero, so a suggestion is: $\begin{pmatrix}0&1\\ 1&0\end{pmatrix}$ However, when I square this, I don't get the zero matrix. Is there a rule for how to ...
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Do equivalent matrices have the same image

Let say we have R-module homomorphisms $T_1 ,T_2 : R^m \rightarrow R^n $ with matrices $A$ and $B$ respectively. If $B$ can be obtained from $A$ by just performing elementary column and row operations,...
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Convergence of an integral involving matrices

Let $A$ be any positive definite square matrix of size $n$ and $B$ be any symmetric matrix of size $n$. I guess the integral $$\int_0^\infty e^{-tA}Be^{-tA}~\mathrm{d}t$$ is convergent. I am not sure ...
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Convergence of matrix iteration

Let say $I$ is the $n \times n$ identity matrix, $B$ is an $n \times n$ symmetric positive definite matrix, and $\{W^{(k)}\}_{k=0}^{\infty}$ is a sequence of diagonal matrices, i. e. $W^{(k)} = \...
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1 answer
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Derivative of a vec operator with respect to another vec operator

Let A be a symmetric matrix and $A^{-1}$ its inverse. Denote $ a = \textrm{vec}[A] $. Find the derivative of the following expression wrt $a$. $$ \textrm{vec}^T[A^{-1}] - \textrm{vec}^T[A^{-1}BA^{-1}]...
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Why must the determinant of a matrix be nonzero for it to be invertible and is there a work around for physical systems?

Leading on from a question on the practicality of inverse matrices used in physical engineering systems which I've asked previously, I have come across an issue with my system where I want to take the ...
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How to prove Concavity using Hessian matrix

I have a function say f(x,y) in two variables x and y, defined as $f(x,y) = x \log_2(1+\frac{y*c}{x})$, where c is constant. Help me to prove this as concave (is it a concave function?) using Hessian ...
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Matrix where each element is an inner product with the same vector. What is a compact notation of such a matrix?

A matrix is defined as: $$ \begin{bmatrix} {\bf a}^T{\bf b}_{1,1}, \ldots, {\bf a}^T{\bf b}_{1,N} \\ \cdots, \cdots, \cdots, \\ {\bf a}^T{\bf b}_{N,1}, \ldots, {\bf a}^T{\bf b}_{N,N} \end{bmatrix}...
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Relating a dispersion equation to an eigenvalue equation in a Fourier transformed system of PDEs

I am reading through the paper "Dynamics of Membranes Driven by Actin Polymerization" by Nir S. Gov and Ajay Gopinathan. In it a set of coupled differential equations for a mathematical ...
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3 answers
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Check that $A^2=A$

I want to check something rather basic, but I can't get it to add up. Let $A$ be the $2\times2$ matrix. \begin{equation} \begin{pmatrix} |\alpha|^2&\bar{\alpha}\beta\\ \alpha\...
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Least Square and Power Methods MATLAB [closed]

I'm working on a numerical analysis exercise and need an explanation and answer to help me learn. Use these x & y numbers: ...
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2 answers
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Writing $e^A$ as an expression of matrix A

Let $A\in\mathscr{L}(x)$ and satisfies the condition $A^2=A$. Write the function $e^A$ as expression of A. My attempt: Let's start with $e^x$, which we re-write as a series: $$f(x) = \sum\limits_{j=0}...
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What properties do the matrix $A^HA$ have?

The matrix $\mathbf{A}$, written in the form of column vectors, is $\mathbf{A}=(\mathbf{a}_1, \mathbf{a}_2,...\mathbf{a}_P)$ where $\mathbf{a}_k$ is an $M\times 1$ complex column vector that is given ...
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Coefficients of polynomial with at least $r$ non-zero roots.

You have a polynomial $f$ of degree up to $k$ over a finite field with $q$ elements where $k|q-1$. If polynomial $f$ has $r<k$ distinct non-zero roots, what we can say about the non-zero ...
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Matrix linear mapping [closed]

Not sure how to start on the following problem: Determine the matrix A for the linear mapping $T:\Bbb R^3 \to \Bbb R^3$ defined by first reflecting in the plane $x-4y+3z=0$ and then projecting down ...
1 vote
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Find the $4 \times 4$ Vandermonde determinant

I'm currently doing an exercise question from the textbook. The question is: The goal of this problem is to find the $4 \times 4$ Vandermonde determinant. $V_4 = \begin{bmatrix}1&a&a^2&a^...
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1 vote
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Intuitive reason as to why the spectral norm is the largest singular value

Why is the spectral norm, the largest singular value? I understand the proof behind this, but I can not intuitively explain why the spectral norm is the largest singular value. It makes sense that it ...
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If A and B idempotents and tr A = tr B, prove that A and B are similar [closed]

Given A,B $\in {M_{nxn}(\mathbb{R})}$, A,B idempotents and tr A = tr B. Prove that A and B are similar.
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1 answer
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Is it true that for a matrix $A$ with non-zero entries, $\det{A}=0 \iff$ its cofactors are $0$?

We know that through cofactor expansion, the determinant of an $n \times n$ matrix $A$ may be written as a linear combination of the determinants of it's cofactors. Let $A_{i,j}\neq 0$ denote the $i,j$...
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How to interpret take derivative a matrix Y with respect to another matrix X?

Let $X, A$ be lower triangular $pxp$ matrices where $A=(a_{ij})$ is a constant matrix with $a_{jj}>0,j=1,...,p$ and $X$ is a matrix of functionally independent real variables. Then $Y=X+X' \...
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Operator norm of a singular matrix and implications for law of large numbers

I came across the following in a book: where the step before this showed that $\textbf{K}_n(l)=\frac{l}{n}\pmb{\Lambda}^{1/2}\textbf{Y}_1\left(l\textbf{I}_n-\frac{1}{n}\textbf{X}_2^*\textbf{X}_2\...
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Calculate the inverse of a matrix that is a sum of outer products

I am trying to find the inverse of matrix in the following form $\mathbf{R}=\sigma_0\mathbf{I} + \Sigma_{k=1}^{P}\sigma_k\mathbf{d}_k\mathbf{d}_k^H$ where $\mathbf{I}$ is an $M\times M$ identity ...
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The transformation matrix of an dimetric projection onto the plane Z=0

I need to make a dimetric projection of the point onto the Z=0 plane. I found the transformation matrix of an isometric projection onto the Z=0 plane and it looks like this: \begin{align*} M &= \...
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Row echelon form using exclusive-or operation in MATLab

By using MATLab, How to reduce a large matrix of "binary bits and symbols" to row echelon form. For instance consider: ...
1 vote
2 answers
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Derivative of a matrix divided by its Frobenius norm

I have a function $f$ that takes $\textbf{X}\in \mathbb{R}^{m \times n}$ as input and return $\textbf{Y} \in \mathbb{R}^{m \times n}$ matrix as output. $$ \textbf{Y}=f(\textbf{X}) = \textbf{X} \frac{1}...

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