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.
53,572
questions
1
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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 &...
- 921
-1
votes
0
answers
12
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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
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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} = \...
- 2,552
-4
votes
0
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15
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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
35
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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 ...
- 1
0
votes
0
answers
19
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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 ...
- 21
0
votes
0
answers
11
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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
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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 ...
- 6,428
0
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0
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31
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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 ...
- 9
0
votes
3
answers
68
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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?
- 176
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0
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24
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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 ...
- 1
0
votes
1
answer
20
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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\...
- 388
0
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0
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32
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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
46
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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 ...
-4
votes
0
answers
42
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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 ...
- 1
0
votes
0
answers
14
views
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_{...
- 21
1
vote
1
answer
29
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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
0
answers
26
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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
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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 \\
...
- 484
0
votes
1
answer
15
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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 ...
- 2,044
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....
- 8,810
-1
votes
0
answers
21
views
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 ...
- 1,789
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 ...
- 253
-1
votes
0
answers
15
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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:
$$ \begin{aligned} N_{C}(\bar{x}) &= \left\{ v \mid \left<v,x-\...
- 45
-1
votes
0
answers
23
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Derive the differential equations defining the motion of the masses, via Newton’s 2nd law for a 2nd order coupled system [closed]
Tuned Mass Damper with 2 masses and 2 springs. x and y are the instantaneous displacements of the masses relative to the left-hand datum. The masses are supported on a frictionless horizontal plane.
...
3
votes
1
answer
55
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Left operator on $M_n(\mathbb{F})$ diagnolizable iff A is diagnolizable
Question: let $A\in V = M_n(\mathbb{F})$,left translation operator on V is defined as $T:V\rightarrow V,B\rightarrow T(B)=AB$, prove that $T$ is if and if only A is diagnolizable.
I have found a ...
- 31
0
votes
0
answers
30
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Moments of uniform distribution on Stiefel manifold
Suppose I have a $p \times n$ (with $p \geq n$) matrix $\bf U$ such that ${\bf U}' {\bf U} = {\bf I}_{n}$ and that $\bf U$ is uniformly distributed on the Stiefel manifold $V_{n,p}$. I would like to ...
-2
votes
0
answers
30
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Help finding the inverse of a 4x4 matrix? [closed]
$$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^...
- 1
1
vote
0
answers
27
views
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}
\...
- 292
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 ...
- 19
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 {$|r \rangle$} ...
- 147
1
vote
0
answers
55
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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 ...
- 113
-2
votes
0
answers
56
views
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 ...
- 97
0
votes
0
answers
21
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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 $A=D-E-F$...
- 306
2
votes
1
answer
42
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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 ...
- 4,051
2
votes
1
answer
22
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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:
$$P_n = \begin{pmatrix}
Q_0 & 0 & 0 & \ldots & 0 \\
Q_1 & Q_0 & 0 ...
- 513
2
votes
1
answer
25
views
Matrix series question
Let $A\in\mathbb{R}^{n\times n}$. When $\sum_{k=0}^{\infty}A^k$ converge, there exists a norm such that, for any positive integer $K$, $$\left\|(I-A)^{-1}-\sum_{k=0}^m A^k\right\| \le \dfrac{\|A\|^{m+...
- 306
3
votes
0
answers
46
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$(-1,0,1)$-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
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0
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15
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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 $A$ ...
- 388
0
votes
0
answers
20
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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
-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^...
- 11
-3
votes
0
answers
15
views
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.
- 1
0
votes
1
answer
29
views
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
$$e_{11}=s^*\mu s+t^* \nu t.$$ ...
- 1,690
-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 ...
- 2,103
0
votes
0
answers
24
views
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 \...
- 19
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 ...
- 13
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 ...
- 595