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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System of linear equations with a free variable.

I am new to linear algebra. The following matrix $A_n$ is an $(n-1) \times n$ matrix. I need to solve $A_nX = 0$, where $X = (x_1, \cdots, x_n)$. \begin{equation} A_n = \begin{bmatrix} b_2 & a_2 &...
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Number of invertible matrices in $GL(2, 4)$

The formula for the order of $GL(2, 4)$ suggests there are 180 invertible matrices with entries in the field with 4 elements. However, when running a Python script to find the invertible matrices, the ...
0 votes
4 answers
32 views

Eigenvalues of an inverse matrix

Given the 2x2 matrix A has a negative eigenvalue -2 and a positive eigenvalue 6, then find the eigenvalues of the inverse matrix A From my studies, Inverse of a matrix can be found by: $A^{-1} = \...
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the determinant on the identity matrix. proof1

Show that the determinant of a matrix is a differentiable function at all points (matrices), calculate the differential of the determinant on the identity matrix evaluated at B
-2 votes
1 answer
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Linear algebra. Finding three matrices in Vectorspace $V$

I have this vector space $$V = \left\{\begin{bmatrix}a & b \\ c & -a\end{bmatrix}\, \middle|\ a, b, c \in \mathbb{R}\right\}$$ and am supposed to find three matrices ($A$, $B$ and $C$) in it ...
0 votes
0 answers
19 views

Solving the probabilities of a combinatorial optimization

The optimization is to find $X$ to maximize $$\max_X\|(I-\alpha X)^{-1}(X\circ A E)\|_F$$ where $X,A$ are $n$ by $n$ square matrices, $E$ is a vector $E=(1,\cdots,1)^\top$. More specifically, consider ...
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0 answers
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Matrix set generation interpretation problem

I don't know if I understand the problem statement correctly. This means generating infinitely many matrices in the interval $$[0, tk]$$. Because I don't know how to obtain the maximum and the minimum ...
4 votes
1 answer
27 views

Why does the semigroup of matrices form an epigroup?

An epigroup is a semigroup $S$ in which every element has a (positive) power that lies in a subgroup of $S$. (The subgroup may depend on the element). Note that if $x^n\in G$, where $G$ is a subgroup ...
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0 answers
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Are all Hadamard matrices equivalent to their transposes?

Hadamard matrices are matrices such $H * H^T = nE$, and all columns are pairwise orthogonal(and all raws are pairwise orthogonal) The Hadamard matrices are equivalent(~) if it is possible to obtain ...
0 votes
3 answers
68 views

Let $A$ be a square matrix such that $A(I-A)=I$, prove $A^2-I$ is invertible

Let $A$ be a square matrix such that $A(I-A)=I$, prove $A^2-I$ is invertible. $A^2-I=(A-I)(A+I)$, clearly $A-I$ is invertible, but how do I prove that $A+I$ is also invertible?
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How are a matrix, a result matrix of a couple of matrix multiplies, the inverse of that matrix, and doing the matrix multiplies backwards connected?

The matrices I use are for the canvas in webbrowsers. https://developer.mozilla.org/en-US/docs/Web/API/CanvasRenderingContext2D/transform where [a,b,c,d,e,f] is ...
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1 answer
20 views

How to show that a map is $K$-multilinear

Hey I have this exercise where I have some questions For a field $K$, let $x =\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}, y = \begin{pmatrix}y_1\\y_2\\y_3\end{pmatrix}, z = \begin{pmatrix}z_1\\z_2\\z_3\...
0 votes
0 answers
32 views

Fréchet derivative of a similarity transformation

Suppose $h(Q) = Q^{T} A Q$, then the Fréchet derivative is given by $D_{h} [Q] (H) = H^{T} A Q + Q^{T} A H$. I am bit unsure about this so-called Fréchet derivative is obtained. I would have just said:...
0 votes
1 answer
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Show that if $X$ is a solution of $X'=A(t)X$ with $A$ a matrix and $X(0)$ with positive coefficients, $X(t)$ has positive positive coefficients

Let $A : \mathbb R^+ \to M_n(\mathbb R^+) \in C(\mathbb R^+,M_n(\mathbb R^+) )$ and $X$ a solution of $X'=AX$ such that $X(0)$ with positive coefficients $(\ge 0)$. I need to show that : $\forall t\in ...
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How to solve this post? [closed]

How long does it take to download a two-hour HD movie from the iTunes store? According to Apple’s technical support site, support.apple.com/en-us/HT201587, downloading such a movie using a 15 Mbit/s ...
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The difference between the sum of the squares of the diagonal elements of WAW (for Wishart matrix W) and matrix A (for any A)

For any symmetric matrix ${ A}\in R^{K\times K}$, we can compute ${ B} = { W}{ A}{ W}$ where $W\in R^{K\times K}$ is a Wishart matrix with $N$ degrees of freedom (N>K). I want to bound term $\sum_{...
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1 vote
1 answer
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Partial isometry, Isometry and Gram matrix

Suppose I have a set of matrices $\mathcal{H}$ s.t. $\forall H\in\mathcal{H}, \ H = CU$ where $C$ is a fixed matrix and $U$ is an orthonormal matrix. We know that this set can be characterized ...
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-1 votes
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Up to similarity, there is a unique 3 × 3 matrix with minimal polynomial (x−1)^2 (x−2) [closed]

Is this true. Because I can't find any invertible P, such that M = P D P^-1, so that the similarity satisfies. Can any one guide me regarding my approach to the problem? Thanks @Gerry Myerson, my ...
0 votes
1 answer
28 views

Find $v>0$ so that $Av>0$ [closed]

Let \begin{align} A=\frac{1}{h^2}\begin{pmatrix} 1 & 0 & 0 & &\cdots & 0\\ -1 & 2 & -1 & 0 & \cdots & 0 \\ 0 & -1 & 2 & -1 & \cdots & 0 \\ ...
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1 answer
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Indices without a given domain in sum notation.

I'm learning about normal modes right now and I have a question pertaining to the way the equations of motion are written vis-a-vis summation notation. The book defines the equations of motion of a ...
0 votes
0 answers
40 views

Name of symmetric multilinear function?

What's the name (if any) for the multilinear symmetric function that maps a square matrix over some commutative ring $R$ with $1$ to a value in $R$? It's basically like the determinant, but symmetric....
-1 votes
0 answers
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What should be the result of this Roto-translation? [closed]

Suppose I have a rotation matrix mat = | 1 0 0 | | 0 1 0 | | 0 0 1 | a translation vector vec = | 1 1 1 | and, I ...
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0 votes
1 answer
28 views

Expected value of dependent matrix product

Assume the matrix product term $AXBYC$ where each one is an $n\times n$ matrix. $A, B, C$ are constant, but $X, Y$ are variable and are not independent from one another, hence their covariance is not ...
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The normal and tangent cones of the positive semidefinite cone [closed]

For a convex set $C \subset \mathbb{R}^{n}$, the tangent cone and the normal cone at any $\bar{x} \in C$ are defined as following: $$ \begin{aligned} N_{C}(\bar{x}) &= \left\{ v \mid \left<v,x-\...
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Derive the differential equations defining the motion of the masses, via Newton’s 2nd law for a 2nd order coupled system [closed]

Tuned Mass Damper with 2 masses and 2 springs. x and y are the instantaneous displacements of the masses relative to the left-hand datum. The masses are supported on a frictionless horizontal plane. ...
3 votes
1 answer
55 views

Left operator on $M_n(\mathbb{F})$ diagnolizable iff A is diagnolizable

Question: let $A\in V = M_n(\mathbb{F})$,left translation operator on V is defined as $T:V\rightarrow V,B\rightarrow T(B)=AB$, prove that $T$ is if and if only A is diagnolizable. I have found a ...
0 votes
0 answers
30 views

Moments of uniform distribution on Stiefel manifold

Suppose I have a $p \times n$ (with $p \geq n$) matrix $\bf U$ such that ${\bf U}' {\bf U} = {\bf I}_{n}$ and that $\bf U$ is uniformly distributed on the Stiefel manifold $V_{n,p}$. I would like to ...
-2 votes
0 answers
30 views

Help finding the inverse of a 4x4 matrix? [closed]

$$A=\left[ \begin{matrix} (x^2)*(1+y^2) & (x^2)*y & 0 & 0 \\ (x^2)*y & (x^2)*(1+y^2) & 0&0 \\ 0 & 0 & (x^2)*(1+y^2)&(x^2)*y \\ 0&0&(x^2)*y&(x^2)*(1+y^...
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1 vote
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Approximate Diagonally Dominant Matrix

I have a following symmetric block matrix. \begin{equation} A = \begin{bmatrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{bmatrix} \...
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0 votes
1 answer
66 views

If A and B are positive semidefinite matrices, then $\mathrm{tr}( A^3B + AB^3)\ge 2\mathrm{tr}( A^2B^2)$. [closed]

We know the following relationship: If A and B are positive semidefinite matrices, then $\mathrm{tr}(AB)^2\le\mathrm{tr}(A^2B^2)$. How can I prove this relationship? If A and B are positive ...
2 votes
2 answers
63 views

Clarifying Bra-Ket Notation: Orthonormal Bases

I was asked to find the trace of $(A \in M_{n \times n})$, the matrix that can be written in the form:$$A=\frac{1}{n} \sum_{r, \, q \, = \, 1}^n (-1)^{r+q}|r \rangle \langle q|$$ where {$|r \rangle$} ...
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1 vote
0 answers
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Conditions for a representation of a group. (updated)

My lecturer wrote that a matrix representation of a group $G$ is a group homomorphism $\rho$ from $G$ to the set of invertible square matrices over a field. He proceeded to write that this is ...
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How can I transform nxm matrix A into nxm matrix B?

I have two numerical matrices of size 48x50, $A$ and $B$. I'd like to obtain the transformation of $A$ into $B$ and use the same transformation to transform $A'$ into $B'$. I don't know how to ...
0 votes
0 answers
21 views

Convergence of symmetric Gauss Seidel iteration

Consider $Ax=b$ where $ A\in \mathbb{R}^{n\times n}$ . Let $D$ be the diagonal part, $-E$ be the strictly lower triangular part and $-F$ be the strictly upper triangular part of $A$ such that $A=D-E-F$...
2 votes
1 answer
42 views

How many different determinants are possible by reordering entries?

I was preparing a worksheet about cofactor expansion of $3\times3$ determinants and decided to create only matrices of order 3 whose entries are among $\pm1,\pm2,\dots,\pm9$ such that the absolute ...
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2 votes
1 answer
22 views

Let $T,S$ be unilateral shifts in $H,K$ and $A\in B(H,K)$ a contraction. If $S^*A=AT^*$, then why is $A$ a transposed infinite Toeplitz matrix?

Let $H, K$ be Hilbert spaces. As the Toeplitz Matrix, I define an operator $P_n$ in the form: $$P_n = \begin{pmatrix} Q_0 & 0 & 0 & \ldots & 0 \\ Q_1 & Q_0 & 0 ...
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2 votes
1 answer
25 views

Matrix series question

Let $A\in\mathbb{R}^{n\times n}$. When $\sum_{k=0}^{\infty}A^k$ converge, there exists a norm such that, for any positive integer $K$, $$\left\|(I-A)^{-1}-\sum_{k=0}^m A^k\right\| \le \dfrac{\|A\|^{m+...
3 votes
0 answers
46 views

$(-1,0,1)$-square matrix has different line sums?

Let $A$ be a $n\times n$ matrix with coefficients from the set $\{-1,0,1\}$. Let $r_i$ and $c_i$ denote the sum of the elements of the $i$-th row and column of $A$ respectively. For which $n$ is it ...
1 vote
1 answer
40 views

paint a board with two colors without repeating the amount painted in each row rows

A child is playing coloring his chessboard and will paint each square either completely blue or completely red. To give it a personalized touch, he wants to paint the same number of red squares as ...
-1 votes
0 answers
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$Ax = y$ and $A'x = y'$ have the same solution set $L$, then $L = ∅$ or $\ker(l_A) = \ker(l_{A'} )$

Hey I have problems solving this exercise. Can someone help me? Let $A, A' ∈ Mat_{m,n}(K)$ and $y, y' ∈ K^m$. Let $l_A: K^n → K^m$ and $l_{A'} : K^n → K^m$ be the $K$-linear maps associated with $A$ ...
0 votes
0 answers
20 views

b = $(X’X)^-1$ alternate form [closed]

Prove that $b$ may also be written as: $b = B + (X^TX)^{-1} X^TY$ So far I’ve got this: $$(X^TX)^{-1}X^T[XB+u]$$ $$(X^TX)^{-1}X^TXB+(X^TX)^{-1}X^Tu$$ This is where I am stuck and don’t know how to ...
-1 votes
0 answers
26 views

Mixed strategy non zero sum game for n players

The game mentioned everywhere is for two players. In the case of more than two players, how the payoff matrix will change ? For 3 players, each having 2 options. The total scenario will be 2^3= 8. So ...
1 vote
0 answers
35 views

Spectrum of $(A^TA-B^TB)$ and $(AA^T-BB^T)$

I am wondering whether one can say anything useful about the relationship of $A^TA-B^TB$ and $AA^T-BB^T$ in terms of their spectra? In particular I know that their traces are the same since $tr(A^TA-B^...
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Showing that all matrices of a linear operator have the same rank. [closed]

Can someone help me with proof? Show that all matrices of a linear operator have the same rank.
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1 answer
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Question related to matrices

Let $\mu$ and $\nu$ be two positive $2 \times 2$ matrices. Suppose $s$ and $t$ are another $2 \times 2$ complex invertible matrices such that $s^*s+t^*t=I$ and satisfies $$e_{11}=s^*\mu s+t^* \nu t.$$ ...
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-1 votes
2 answers
40 views

Do row operations preserve any properties of matrices besides row space and null space?

Elementary row operations on a matrix preserve row space, null space, rank, nullity, and invertibilty. They do not preserve determinant, trace, or column space. Inspired by List of matrix properties ...
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0 votes
0 answers
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How to write Type III matrix as a product of Type I matrices?

I saw this question Prove that, over a Euclidean domain $R, I + cE_{ij}$ generate $SL_n(R)$. and in the comment section, someone mentioned that you can do something about type III matrices. I tried to ...
0 votes
0 answers
52 views

How to check whether a set is a subspace of a vector space?

Three vector spaces are given, and each one has a set that is potentially the subspace of the vector space. $$ (1) \quad V = \mathbb{R}^4 \quad W=\begin{Bmatrix} \begin{bmatrix} a \\ -b \\ 2a+b \\ a \...
0 votes
1 answer
80 views

Determinant of a polynomial: Exercise from chapter 1, "Linear Algebra" by Georgi E. Shilov

I am solving the exercise from "Linear Algebra" by Georgi E. Shilov. The question asks to prove the following: \begin{align*} \begin{vmatrix} 1 & 1 & \cdots & 1 \\ x_1 & x_2 ...
3 votes
2 answers
46 views

Find Jordan Basis of triple differentiation operator in $\mathbb{R}_8[x]$

General information and notation at first. Let $A$ be matrix of $T$ and let $T$ be such operator, that $(1, x, x^2, \cdots x^8) \to (0, 0, 0, 6, 24x, 60x^2, 120x^3, 210x^4, 336x^5)$ $T^2$: $(1, x, x^2 ...

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