Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

3
votes
6answers
105 views

Proving $\left|\begin{smallmatrix}1&1&1\\a&b&c\\a^3&b^3&c^3\end{smallmatrix}\right|=(b-a)(c-b)(c-a)(a+b+c)$

Prove that$$\begin{vmatrix}1&1&1\\a&b&c\\a^3&b^3&c^3\end{vmatrix}=(b-a)(c-b)(c-a)(a+b+c)$$ My attempt: $$\begin{align}\begin{vmatrix}1&1&1\\a&b&c\\a^3&b^3&...
0
votes
0answers
8 views

How can I prove that I can make a RREF with finite amount of elementary row operations in induction?

I first concluded that for 1*1 it is trivial, let the thm be valid for mn 1) then for m(n+1) the n+1th column does not change the position of the leading 1s nor the echelon form thus trivial (how can ...
0
votes
0answers
50 views

Properties of $n \times n$ complex matrix with $A^m = I$

If $A^m = I_n$ then what can we say about the eigenvalues and diagonalizablity of $A$? The equation given above is an annihilating polynomial of $A$ and therefore minimal polynomial divides it. Since ...
0
votes
1answer
28 views

Does the following property of matrices hold for any commutative ring with identity?

I have recently gone through the proof of the following theorem given in my book $:$ Theorem $:$ Let $R$ be a commutative ring with identity with quotient field $K.$ Let $\alpha \in ...
4
votes
1answer
56 views

Suppose $U_1,\dots,U_k$ and $V_1,\dots,V_k$ are $n\times n$ unitary matrices. Show that $\|U_1\cdots U_k-V_1\cdots V_k\|\leq\sum_{i=1}^k\|U_i-V_i\|$

Let $V,W$ be complex inner product spaces. Suppose $T: V \to W$ is a linear map, then we define $$\|T\|:=\sup\{\|Tv\|_{W}:\|v\|_{V}=1\}$$ where $\|v\_{V}\|:=\sqrt{\langle v,v\rangle}$ and $\|Tv\|_{W}...
0
votes
1answer
22 views

Distance matrix properties

Are there sufficient conditions to say that a certain matrix is a distance matrix for a certain set of vectors? For example, for the Euclidean metric or Hamming distance. Suppose that all vectors are ...
1
vote
0answers
14 views

Some questions about bilinear forms and their non-degeneracy.

I was reading this post about bilinear maps and it raised some questions in my mind, for which I am not so sure about the answers. It is proven there that for any symmetric bilinear form, it is ...
3
votes
1answer
29 views

$a_k = rank(A^{k+1}) - rank(A^k)$ is increasing

$a_k = rank(A^{k+1}) - rank(A^k)$ is increasing. This is equivalent to $$ rank(A^{k+1}) + rank(A^{k-1}) \geq 2 rank(A^k) $$ which is a consequence of the Sylvester inequality: $$rank(XZY) + rank(Z) ...
2
votes
1answer
38 views

Let $A=[1 2 0 1 ]$. Find all $2×2$ matrices B, $B≠O_2$ and $B≠I_2$ such that $AB=BA$.

Let $A=[1 2 0 1 ]$. Find all $2×2$ matrices B, $B≠O_2$ and $B≠I_2$ such that $AB=BA$ Explain your answer. I know two ways to find B A) As det of A is 1, it is invertible. So A = B (inverse) B) Or ...
1
vote
0answers
15 views

Prove the inequality for Condition number of matrix

Let $ A \in \mathbb{R}^{n \times n}$ be a non-singular matrix. Let $\hat A=A+\delta A, \ \hat x=x+\delta x, \ \text{and} \ \hat b=b+\delta b$ with $Ax=b$ and $\hat A \hat x=\hat b \ $. Here $||.||$...
1
vote
2answers
37 views

If both $A-B$ and $B-A$ are positive semidefinite, then $A = B$

Let $A, B$ be two positive semidefinite matrices. Prove that if both $A-B$ and $B-A$ are positive semidefinite, then $A = B$. I can show that their diagonal elements are the same but for others I ...
0
votes
1answer
30 views

Determinant of the Adjoint of a matrix

The determinant of the adjoint of a matrix $A$ is given by $|A|^{n-1}$ where $n$ is the order of the square matrix. So, for odd $n$, the determinant is always positive. Is there an intuitive ...
1
vote
1answer
54 views

Show that $T_H$ is a linear subspace of $T_G$ so that $\dim H\leq \dim G$

Let $H$ be a subgroup of a matrix group $G$. Show that $T_H$ is a linear subspace of $T_G$ so that $\dim H\leq \dim G$ Definition: Let $\phi:G\rightarrow H$ be a smooth homomorphism of matrix ...
1
vote
0answers
18 views

Extinction problem for a generalisation of Galton-Watson branching processes

Suppose $A$ is a random $n \times n$ matrix with integer entries in respect to the orthonormal basis $(e_i)_{i=1}^{n} \subset \mathbb{R}^n$. Define $\{v_t\}_{t=1}^{\infty}$ as a sequence of random ...
3
votes
1answer
27 views

Trace of product of semidefinite matrices is nonnegative

I want to prove this: $A$ is a symmetric positive semi-definite matrix $\Leftrightarrow$ $tr(AB) \geq 0$ $\forall $ B positive semi-definite. I tried using eigenvalues, because they all have to be ...
0
votes
3answers
40 views

Let $T:P_3\left(\mathbb{R}\right)\rightarrow P_3\left(\mathbb{R}\right)$ be the linear map defined by $T(P(x))=x^2P’’(x)$. Create the map

I haven't learned to create a map using a derivative, how would I do that (sorry if its a stupid question). $T:P_3\left(\mathbb{R}\right)\rightarrow P_3\left(\mathbb{R}\right)$ is defined for $P \...
0
votes
1answer
24 views

Factorized trace of matrix product

Are there any particular types of matrices for which: $tr(AB)=tr(A)tr(B)$.
2
votes
1answer
52 views

Noninvertible matrix mod s from an invertible matrix

Let $A\in M_2(\mathbb{Z})$ with nonzero determinant. Show that there exist infinitely many numbers $s\in\mathbb{N}$ such that $$ \exists a_s\in\mathbb{N}^*:A^{a_s}-I \equiv O\mod s. $$ My attempt is ...
1
vote
0answers
25 views

Showing the matrix is non-negative definite

Let $X$ be full column rank. I am trying to show that the matrix $$(X^TX)^{-1}(X\beta)^T(X\beta)-\beta\beta^T$$ is non-negative definite. Here, $\beta$ is a parameter vector with intercept. To ...
4
votes
1answer
86 views

Let $A, B$ be two positive definite $2 \times 2$ matrices. Prove or disprove: $AB+BA$ is positive definite.

I know that $AB+BA$ is not necessarily positive definite, as this question has been asked before on here. What I don't understand is how one would go about constructing counter-examples. Previous ...
-1
votes
0answers
40 views

Null space of a matrix [closed]

A is a matrix which has two special solutions to $Ax=0$, all other solutions are linear combinations of the special solutions. They are $(3,1,4,0,5)$ and $(2,0,2,1,2)$. $C$ is a matrix that is the ...
0
votes
0answers
8 views

Cosine Similarity Between Words and SVD

Suppose we have a $m \times n$ matrix $M$ of $m$ words and $n$ co-occurrences. That is, each element of the matrix quantifies how often a word $w_i$ ($i = 1, \dots, m$) occurs with another word $c_j$ (...
0
votes
1answer
39 views

the augmented matrix of a system of equations

We have the augmented matrix of a system of equations: $$ \begin{pmatrix} 1 & 0 & 1 & 4 \\ 0 & 1 & 0 & -7 \end{pmatrix} $$ The system has infinitely many ...
0
votes
0answers
41 views

Prove that there's only one matrix which is involutory and idempotent at the same time.

An involutory matrix $\mathbf{A}$ is such that $\mathbf{A}^{2}=\mathbf{I}$. An idempotent matrix $\mathbf{A}$ is such that $\mathbf{A}^{2}=\mathbf{A}$ So we'd have to satisfy both conditions ...
1
vote
1answer
13 views

Scale vector based on previous scaling

I’m writing peace of software that is able to uniformly scale set of vertices. To do that I have a buffer where I keep my vertices untouchable. When I need to resize them, I create copy of that ...
2
votes
1answer
18 views

3x5 Matrix whose column span is R3

Hello everyone, I would like to confirm my understanding of these two problems because I believe I have the answer, but I may be incorrect. For the first question, I know that there can be at most 3 ...
0
votes
0answers
8 views

Is there a name for this sort of mod(n)-Diagonal Matrix?

I've been tinkering around with some linear algebra lately and have stumbled across some problems where matrices keep popping up (even for matrices of larger sizes) with the form: $$ X =\begin{pmatrix}...
0
votes
1answer
11 views

Why isn't $[P]_{C \leftarrow B}[T(x)]_{B}$ equal to $[T(x)]_C$ ? $P$ is the change of basis matrix from $B$ to $C$ and $T$ is a linear transformation

Why isn't $[P]_{C \leftarrow B}[T(x)]_{B}$ equal to $[T(x)]_C$ ? ($P$ is the change of basis matrix from $B$ to $C$ (both vector spaces) and $T$ is a linear transformation). It would make a lot of ...
0
votes
2answers
30 views

If all the entries of a matrix on and below the diagonal are zero, then it is nilpotent

Show that, if all the entries of $A$ on and below the diagonal are zero, then $A$ is nilpotent. I know this has been asked before , but I want to solve this question without the use of Cayley-...
0
votes
0answers
36 views

Is it possible for a sequence of matrices to have pointwise but not uniform convergence?

Is it possible for a sequence of matrices to have pointwise but no uniform convergence? The norm for the matrices is the operator norm.
0
votes
0answers
12 views

Lipschitz constant for a function of matrix

I have the following function in $\Lambda \in R^{k \times k}$ $$f(X) = (-2X^T\Lambda_1Z^T + 2X^T X\Lambda YY^T)\circ I $$ where $X \in R^{n \times k}$,$Z \in R^{k \times n}$ $Y \in R^{k \times k1}$, ...
0
votes
0answers
10 views

Hessian matrix over the complex field

I compute the Hessian matrix for a cost function which defined on a complex Riemannian manifold, so i find that this matrix is an hermitien matrix not symmetric. It is possible to obtain in this case ...
2
votes
1answer
45 views

Some questions about additive groups of matrices

,I'm trying to solve this problem. I have two additive groups $G, H$. The first is the group of matrices $4x1$ with coefficients in $\Bbb Z_{11}$. The second is the group of matrices $3x1$ with ...
1
vote
1answer
48 views

If $A$ is nonsingular matrix such that $A^T=A^{-1}$, then $|A|=1$ (T/F)?

If $A$ is nonsingular matrix such that $A^T=A^{-1}$, then $|A|=1$ (T/F)? I am not sure if this statement is true or false. All i have got is: $A^T=A^{-1}$ $A^T= \dfrac{adj (A)}{|A|}$ $|A|=adj (A)/...
1
vote
1answer
46 views

Existence of Latin Squares without symmetry around main-diagonal

A Latin square of order $n$ is a matrix $L$ with entries from $[n] \equiv \{0, \dots, n-1\}$ such that each row and column contains every symbol from $[n]$ once. For which orders does there exist a ...
0
votes
1answer
39 views

The matrix A=[-2 2 1 3 ] is invertible with A^(-1)=1/8 [-3 2 1 2 ].

The matrix $A=\begin{pmatrix} -2 & 2 \\ 1 & 3 \end{pmatrix}$ is invertible with $A^{-1}=\frac{1}{8} \begin{pmatrix}-3 & 2\\ 1 & 2 \end{pmatrix}$. (TRUE/FALSE)? In my opinion the ...
1
vote
0answers
15 views

Eigenvalues under Hadamard product

Consider $\mathbf{A,B}$ two matrices which are unitary and/or Hermitian, What can we say about eigenvalues of their Hadamard product $(\mathbf{A \circ B})$? Can we bound the eigenvalues in relation ...
1
vote
0answers
26 views
+100

Smith normal form of rectangular matrix in MATLAB

Suppose I've got a nonsquare integer matrix, say $\begin{pmatrix}3 & 1 & 1 & 1\\1 & 1 & 1 &1\end{pmatrix}$ and want to compute its Smith normal form--in this case, $\begin{...
0
votes
1answer
41 views

What do you call a matrix with these properties?

What do you call a matrix, when multiplied from left and right by vectors $x$ and $v$, then it produces the same result as multiplying it from left and right with $v$ and $x$ respectively. Basically: $...
0
votes
0answers
14 views

Optimizing next guess

I have some trouble implementing an algorithm which i want to try out, it simply doesn't work as intended. Let's say we ( i implement this in matlab ) have two fields $P(t)$ and $R(t)$ representing ...
2
votes
3answers
84 views
+300

Minimum Elementary row/column transformations to find Inverse of given Matric

While working out some elementary transformation to find Inverse of matrix, it get in my mind, what is Minimum such transformations needed to find Inverse.
1
vote
0answers
25 views

Intermediate matrices in the positive definite ordering.

Let $$ M= \left[ {\begin{array}{cc} a & b \\ b & c \\ \end{array} } \right],$$ be a positive definite matrix such that $M \succeq xI_d$, for some $x > 0$. Here '$\succeq$' means ...
2
votes
0answers
26 views

Is there a way to describe the structure of $Aut(UT(3, p))$?

Is there a way to describe the structure of the automorphism group of $$C_{p^2} \rtimes C_p \cong \langle x, y, z | [x,y]=z, [x,z]=[y,z]=x^p=y^p=z^p=e \rangle \cong UT(3, p)?$$ Here $p$ is an odd ...
0
votes
1answer
35 views

How do I find the bases of the Jordan Canonical Form of $C$?

Let $$C = \left[ {\begin{array}{cccc} 0 & -1 & -2 & 3 \\ 0 & 0 & -2 & 3 \\ 0 & 1 & 1 & -1 \\ 0 & 0 & -1 & 2 \end{array} } \right].$$ What ...
3
votes
0answers
30 views

Matrix Transformation for 2D, how do I tell what this matrix does geometrically?

Given a 2x2 matrix, $$\begin{bmatrix}1&-1\\-1&\frac12\end{bmatrix}$$ What geometric effect does it have? So a way I did to solve this was to simply apply it to a unit square that I drew on a ...
1
vote
0answers
24 views

Matrices over quaternions make Hopf Algebra or not?

I am learning Hopf algebra now a days. I am still confused about it’s axioms. I don’t know how to define antipode structure. What are the basic rules to define it. ? Can any one help me to solve this ...
3
votes
1answer
57 views

Matrices Inequality Proof

Recently, I read a paper and there is a step which turns out not obvious to me. The statement is as follows: All matrices here are real matrices. $F$ is an arbitrary square matrix. $\Psi$ is a ...
0
votes
0answers
14 views

How to prove that a matrix (linear function ) is a symmetry for a space vectorial? [closed]

How can I prove that a linear function (which is presented by a matrix) is a symmetry for a vectorial space parallel to another one (it's complementary). Actually I tried raising it (the matrix) to ...
0
votes
1answer
35 views

Given two square matrices $A_{nxn}$ and $B_{nxn}$, prove that $trace (\mathbf{A}^\intercal \mathbf{B}) = trace (\mathbf{A} \mathbf{B}^\intercal)$.

I checked with an example of random matrices and I noticed the sum of the resulting diagonals are indeed the same. Also, I have that $trace (\mathbf{A}^\intercal \mathbf{B}) = trace(\mathbf{B} \...
-1
votes
1answer
42 views

I dont want an Answer Just to know what i would need to answer this question

I have this question for an assignment in mathematics for computing, im not looking for someone to answer the question just for some advice on how to answer it. This is The Assignment question