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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Prove of disprove.There exists 3x3 real valued matrix such that … [closed]

Prove of disprove this statement. There exists 3x3 real valued matrix B such that $B^2=A$. $$ A=\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\1 & 1& 0 \\ \end{bmatrix}$$
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1answer
43 views

Gradient of $X^{T}AX$ wrt matrix X

I need to calculate the gradient of $X^{T}AX$ with respect to $X$, where $X$ and $A$ are $nxn$ matrixes. Using the matrix cookboox here (pag 9), I see that the gradient is obtained for each element $...
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2answers
44 views

How can I find the rank of this linear transformation

Suppose $Q \in M_{3 \times 3}\mathbb(R)$ is a matrix of rank $2$. Let $T : M_{3 \times 3}\mathbb(R) \to M_{3 \times 3}\mathbb(R)$ be the linear transformation defined by $T(P) = PQ$. Then rank of T ...
2
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2answers
84 views

Prove that this determinant equals zero

I recently came up with a problem in which I need to use the fact that the determinant below is equal to zero : \begin{vmatrix} 0 & a_{21} & 0 & a_{41} & 0 & \dots&0 &a_{...
1
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1answer
37 views

To prove a property of a positive semidefinite matrix with the zero first entry.

I want to prove the following: If the matrix $$ M=\begin{pmatrix} 0&\vec{q}^T \\ \vec{q}&N \end{pmatrix} $$ is PSD, then $\vec{q}=\vec{0}$. The only three properties of a positive ...
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0answers
38 views

Comparing condition numbers

I have two matrices, $A$ and $B$ where $A,B \in R_{mxn}$ ($m>>n$). I need to answer the question: Which one is better-conditioned among $A$ and $B$ ? However, I will be making these comparisons ...
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1answer
11 views

start T in matrix equation

In many start equations I see T (transpose) or -1 (inverse). Why is there using T, but not original matrix? Example https://en.wikipedia.org/wiki/Sherman%E2%80%93Morrison_formula ($\mathbf{A}$ + $\...
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2answers
41 views

Similarity transformation into symmetric matrix

I have a matrix of the form: $$ \begin{bmatrix} 0 & q & 0 & 0 & 0 & 0 & \cdots \\ p & 0 & q & 0 & 0 & 0 &...
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0answers
26 views

The solution space of $A_{12}X=0$ and $B_{12}X=0$ is isomorphic?

Let $A$ be a $n\times n$ invertible matrix. Suppose $$A=\begin{pmatrix}A_{11}&A_{12}\\ A_{21}&A_{22} \end{pmatrix}$$ $$A^{-1}=\begin{pmatrix}B_{11}&B_{12}\\ B_{21}&B_{22} \end{pmatrix}...
2
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1answer
81 views

Quadratic matrix equation $XAX=B$

Let $A$ and $B$ be two positive semidefinite $n \times n$ matrices. Does the following quadratic matrix equation have a solution in the set of real symmetric matrices? $$XAX=B$$ It's a special case ...
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1answer
55 views

Convexity of $t\mapsto b'e^{-A-Bt}b$

Let $b$ be some given vector and $A,B$ arbitrary symmetric matrices, and $A$ is positive definite. Is the function $$t\mapsto b'e^{-A-Bt}b$$ convex? This is related to another question: Convexity of $...
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2answers
72 views

Do real eigenvalues $\implies$ symmetric matrix? And why is a positive definite matrix symmetric?

Proof: $Av$ = $\lambda v$ $\implies \bar{v}^{T}Av = \lambda \bar{v}^{T} v$ ------(1) And, $Av$ = $\lambda v \implies \bar{A}\bar{v}$=$\bar{\lambda}\bar{v} \implies \bar{v}^{T}\bar{A}^{T}=\bar{\...
3
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1answer
51 views

Showing that there is unique matrix $B$ such that $B^k=A$ for some $A$

Let $A$ be a $n$ by $n$ real matrix with distinct positive eigenvalues $\lambda_1$,...,$\lambda_n$. And let $k$ be an odd integer. Then, I was able to show that there exists a real matrix $B$ such ...
0
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1answer
44 views

Convexity of $x\mapsto \mathrm{tr}(e^{-E\langle a,x\rangle}bb')$

Let $E$ be a matrix with entries equal to one minus the identity matrix and $a,b$ known vectors. Is the function $$x\mapsto \mathrm{tr}(e^{-E\langle a,x\rangle}bb')$$ convex?
1
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1answer
21 views

Find inverse of specific matrix

So, I have to find an inverse of this matrix: $$\begin{pmatrix} A & B\\ 0 & C \end{pmatrix}$$ where, $A\in M_m(\mathbb{R})$,$B\in M_{mn}(\mathbb{R})$, $C\in M_n(\mathbb{R})$ and $A$ and $C$ ...
1
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1answer
36 views

For $4\times 3$ matrix $M$, for any $3\times4$ matrix $N$, $\exists 0 \neq v \in \mathbb{C}^4$ such that $MNv = 0$.

Define $M := \left(\begin{matrix}1&-1&2\\2&-1&1\\-4&1&0\\3&-2&3\end{matrix}\right)$. Prove or disprove: For every $3 \times 4$ complex matrix $N$, there is a non-zero ...
2
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3answers
61 views

$V=\{A\in M_n(\mathbb Q): \operatorname{tr}A=0\}$, Prove that $V\oplus \operatorname{Span}\{I_n\}=M_n(\mathbb Q)$

Q: Let $V=\{A\in M_n(\mathbb Q): \operatorname{tr}A=0\}$. Prove that $V\oplus \operatorname{Span}\{I_n\}=M_n(\mathbb Q)$ Since $\dim(V\cap \operatorname{Span}\{I_n\})=0$, $\dim(\operatorname{Span}\{...
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2answers
57 views

Inductively simplify specific Vandermonde determinant

From Serge Lang's Linear Algebra: Let $x_1$, $x_2$, $x_3$ be numbers. Show that: $$\begin{vmatrix} 1 & x_1 & x_1^2\\ 1 &x_2 & x_2^2\\ 1 & x_3 & x_3^2 \end{vmatrix}=(...
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0answers
24 views

Euclidean distance of adjacent pairs of matrix

I have a matrix defined as; $A$= $$\begin{pmatrix}12 & 17\\\ 30 & 52 \\\ 50 & 73 \\\ 25 & 54 \\\ 67 & 21\end{pmatrix}$$ and i want to calculate the euclidean distance, $eucli(A_i,...
1
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1answer
21 views

Linearly dependent matrix columns implies zeroes in the solution of the system of equation given by this matrix?

Let's assume that we have a system of equations: $a_{11} x_1 + ... + a_{1n} x_n = 0$ $a_{21} x_1 + ... + a_{2n} x_n = 0$ $......................$ $a_{m1} x_1 + ... + a_{mn} x_n = 0$ Let us denote by ...
1
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1answer
32 views

Product of two matrices in block form: step in proof

Let $R$ be a commutative ring with $1$ and $M_{k,l}(R)$ denote set of matrices of size $k\times l$ over $R$. $A\in M_{m,n}(R)$ and $B\in M_{n,p}(R)$. Partition $m$, $n$ and $p$ as $$ m=m_1+\cdots + ...
2
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0answers
42 views

When is $\left\{\begin{pmatrix} x & yb \\ y & x \\ \end{pmatrix} : x,y\in \mathbb{Q}\right\}$ an integral domain?

Any hint about this exercise I am struggling with? Let $b \in \mathbb{Z}$ and $$A=\left\{ \begin{pmatrix} x & yb \\ y & x \\ \end{pmatrix} : x,y\in \mathbb{Q}\right\}.$$ Now, $...
2
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1answer
94 views

Quadratic matrix equation $X^TBX=A$

Let $A$ and $B$ be two positive semidefinite $n \times n$ matrices. Does the following matrix quadratic equation have a solution? $$X^TBX=A$$ When $B$ is positive definite the solution is $$X=B^{-...
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0answers
13 views

Can a dominance matrix be used for ranking?

I have only seen competition scenarios for an application for dominance matrix so far, as shown in the screenshot below. Can a dominance matrix be possibly used for a ranking of exam marks perhaps? I ...
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0answers
82 views

Is it possible to find the explicit inverse of $I+AB$, where $A$ is diagonal and $B$ symmetric matrix?

Question. Is it possible to find the explicit inverse of $I+AB$, where $A$ is diagonal and $B$ symmetric matrix? $A$, $B$ are non-singular, and $I+AB$ is invertible. For $B=S-I$, where $S=[s_{ij}]=[c*...
0
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1answer
68 views

Why does $(A^TA)^{-1} = I$ imply that $T_A$ is injective?

Let $T_A : \mathbb R^n \to \mathbb R^m$ be given by the matrix $\mathbf{A}$. I have been told A has a left inverse if $T_A$ is injective. Also, I have been told that $\mathbf{A}$ has a left inverse if ...
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1answer
23 views

Mathematical notation for orthogonal projection of a set of points on a line

How could i mathematically denote the following: Assume a matrix P which represents the coordinates of a set of points (each row = a single point). Each point in the matrix P is projected on a line ...
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1answer
22 views

Counting combinations in a table/matrice with fixed row sums and column sums

I have a problem which I can't seem to google my way to. I have a table where all the sums for the rows and columns are fixed. I would like to know how to find how many possible combinations/...
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2answers
31 views

How to find one-sided inverse of a non-invertible linear transformation?

Suppose I am working with the linear transformation from $\mathbb R^3$ to $\mathbb R^2$ given by a $2\times3$ matrix say $$ \begin{bmatrix} 1 & 2 & 3 \\ 4 & 0 & 5 \\ ...
1
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1answer
37 views

Representing a Bilinear Form in a Matrix

Let $b : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be a bilinear form, and $\langle,\rangle$ be the standard inner product of the Euclidean space and $e_1,...e_n$ be the standard basis for the ...
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2answers
31 views

Negative powers of matrix

For any matrix $A$ is it true that, $A^{-n} = (A^{-1})^n = (A^n)^{-1}$? Does this also then apply to powers of diagonalizable matrices? That is, if $A = PDP^{-1}$ then $A^{-n} = PD^{-n}P^{-1}$.
2
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1answer
62 views

Similarity class of $3 \times 3$ matrices with entries in $\mathbb{F}_3$

I've been trying to solve the following problem. Find a representative for each similarity class of $3 \times 3$ matrices $A$ with entries in $\mathbb{F}_3 = \mathbb{Z}/3\mathbb{Z}$ such that $A^4 = ...
2
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1answer
35 views

Show that polynomial with non-zero constant term of nilpotent matrix is invertible

So I got a polynomial: $f(x) = a_nx^n+...+a_1x^1+a_0$ and $a_0 \neq 0$ And a matrix $A$ such that $A^k = 0$ for some $k$. I have to prove that $f(A)$ is invertible. So, I know there is a fact ...
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1answer
30 views

Show that $\mathrm{rank}(A)=n$ iff $\mathrm{rref} A$ has no zero columns.

Let $A$ be an $m\times n$ matrix. Show that $\mathrm{rank}(A)=n$ iff $\mathrm{rref}A$ has no zero columns, using only the definition of rank and reduced row echelon form ($\mathrm{rref}$). My attempt:...
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2answers
49 views

Diagonalizable matrices over the complex numbers

I'm having a hard time proving the following question: Let $A\in M_{n\times n}(\mathbb C)$ be a diagonalizable matrix, show that there exists $B\in M_{n\times n}(\mathbb C)$ such that $B^{2018}=A$. ...
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1answer
38 views

Question regarding a theorem about the number of eigenvalues in a matrix

I'm having a problem with this theorem: Matrix $A_{n\times n}$ has no more than $n$ different eigenvalues. Now I have the matrix $A = \begin{pmatrix} 0&2&0\\ 0&0&1\\ 4&0&0 \...
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0answers
21 views

Chain rule for differential calculus of 3D rotations

In the paper "A Primer on the Differential Calculus of 3D orientations" by Bloesch et al. they give an example on how to apply the identities derived for the derivatives of various rotation ...
1
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1answer
25 views

Get translational component of particular matrix multiplication

Say I have a $4\times4$ rotation matrix $R$ and $4\times4$ translation matrix $T.$ If I multiply the matrices in this order $T * R$, the translation component of $T$ will not get affected. But if ...
1
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1answer
19 views

Solving a matrix equation using vectorization [closed]

Does anyone know how to solve the following matrix equation for X: $C = X+AXA^T$, where $C$, $X$ are positive semi-definite and $A$ is the adjoint of an SE3 transformation.
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2answers
48 views

Show that $g(A)$ is the transformation matrix of the endomorphism $(g(f))$

Let $V$ be a finite-dimensional vectorspace. Given an endomorphism $f:V \to V$. Let $A$ be the transformation matrix of $f$ with respect to a basis $v$ of $V$ and let $g(x)=g_0x^m+ \cdots g_{m-1}x+g_m$...
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1answer
53 views

How does matrices magically solve a simultaneous equation?

I know how to solve a simultaneous equation using matrices. But I don't understand how the answer suddenly come out. The stuff I don't understand: 1. What is the logic behind matrix multiplication? ...
3
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1answer
74 views

Solution of coupled non-linear matrix difference equations encountered in calculating the determinat of a block partitioned matrix.

The determinant of the following $n N \times n N$ block partitioned complex symmetric matrix ($N \times N$ blocks) $$\begin{bmatrix}\mathbb{A} & \mathbb{B} &\cdots &\mathbb{B} \\ \mathbb{B}...
1
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1answer
28 views

I need help understanding a simple proof about determinants of matrices.

What I don't understand: I don't understand why in the proof det(Q) = $λa_{i1}C_{i1}$(A) + $λa_{i2}C_{i2}$(A) + ... + $λa_{in}C_{in}$(A). Shouldn't the det(Q) = $λa_{i1}C_{i1}$(Q) + $λa_{i2}C_{i2}$...
3
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2answers
58 views

Is the set of all invertible 2x2 matrices a subspace of all 2x2 matrices?

Is the set of all invertible 2 x 2 matrices a subspace of all 2 x 2 matrices? If not, can someone give me a counterexample to disprove this statement.
2
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2answers
56 views

The differential of $\lVert D^{1-\lambda} U^* D^{2\lambda}\ UD^{1-\lambda} \rVert_F^2$ with respect to $\lambda$

Let a square matrix $A=WDV^*$ by the SVD where $D$ is diagonal with positive entries, $U=V^*W$ is unitary, and $0<\lambda<1$. let $$ f_{(\lambda)} = \lVert D^{1-\lambda} U^* D^{2\lambda}\ UD^{...
1
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2answers
20 views

Theorem about operators on complex vector spaces always having an upper triangular matrix

I'm reading the following proof of a theorem in Linear Algebra done right enter image description here why is it obvious that $(T-\lambda I)u \in U$ from the definition of $U$? Surely if $u \in U$ ...
4
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0answers
71 views

Find B such that $\det(B)>0$ but there is no real $A$ with $\exp{(A)}=B$

I have proved that $\exp(\text{tr}(A)) = \det (\exp (A))$. An easy corollary is that $\exp (A)$ has positive determinant whenever $A$ is real. Now the question asks us to find a B such that $\det(B)&...
1
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1answer
44 views

Prove that all of the solutions of a 2 x 2 dynamical system approaches the origin iff these three conditions are true

Prove that if $X_{n+1} = AX_n$, where $X_n \in \Bbb R^2$ and $A$ is a 2 x 2 matrix all the solutions of the planar system approaches the origin when $n \to \infty$ iff: $\DeclareMathOperator{\Tr}{...
3
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0answers
26 views

Is there literature on the study of “eigenmatrices”?

Starting with a disclaimer: I will not be able to describe this with the correct terminology because I am trying to find the literature which I am unsure of it's existence. I will try to explain what ...
1
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2answers
61 views

Characterize all real-valued $2\times 2$ matrices with eigenvalues $\pm c$, for $c > 0$.

Characterize all real-valued $2\times 2$ matrices that have as eigenvalues $\lambda_1 = c$ and $\lambda_2 = −c$, for $c > 0$. Use your result to generate a matrix that has its eigenvalues $-1$ and $...