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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
1answer
34 views

Let $A$ be $2 \times 2$ nonzero real matrix.which of the following is true?

Let $A$ be $2 \times 2$ nonzero real matrix.which of the following is true? $(A)$ trace of $A^2$ is positive $(B)$ $A$ has non zero eigenvalue. $(C)$ All entries of $A^2$ can't be ...
4
votes
1answer
36 views

Is there any inequality between 2-norm condition number and Frobenius norm condition number for rectangular matrix?

What I have found in [1] Condition number inequality between Frobenius norm and 2-norm for square matrix, Consider a full rank matrix $X \in \mathbb{C}^{n \times m}$, $m=n$, then we can have, $$n - ...
6
votes
1answer
69 views

Fast way to Invert ADA' when D is a diagonal matrix that changes each iteration?

So I have a statistical learning algorithm in which D is a diagonal matrix that changes each iteration while A stays the same. I'm looking for a fast way to invert ADA' each iteration which ends up ...
1
vote
0answers
23 views

Gauss-Seidel: I can't get diagonally dominant equations

I've got the question: Use the Gauss-Seidel method to solve: $8x_1 - 16x_2 = 6$ $4x_1 + 8x_2 = 0$ I know that the equations have to be diagonally dominant, but I can't see how to determine this. I ...
1
vote
2answers
39 views

Can $\| \left( X'X + \lambda I \right) ^{-1} X'y \| = t$ be solved for $\lambda$?

In this post I suggested that the expression $$ \| \left( X'X + \lambda I \right) ^{-1} X'y \| = t $$ couldn't be easily solved for $\lambda$, because you need to "invert" the norm. But in general ...
0
votes
0answers
22 views

What kind of matrices can be visualized by graphing the column vectors?

In the 3Blue1Brown video “Linear transformations and matrices” linear transformations are visualized by overlaying gridlines which have a position determined by the values of the transformed basis ...
0
votes
1answer
20 views
0
votes
2answers
20 views

Let $v \in R^{k}$ ,with $v^{T} v \neq 0$ . Let $P=I-2 \frac{v v^{T}}{v^{T}v}$, where $I$ is the $k X k $ identity matrix.

Let $v \in R^{k}$ ,with $v^{T} v \neq 0$ . Let $P=I-2 \frac{v v^{T}}{v^{T}v}$, where $I$ is the $k \times k $ identity matrix. Then prove that eigenvalue of $P$ are $1,-1$ and $P^{2}=I$ $P^{2}=(I-2 \...
0
votes
2answers
45 views

Matrix equations, simplify them.

I have some equations and I don't know am I doing the simplification right. Can someone check it? For example, we have an equation $AX = 4X + B$, where $A$ and $B$ are matrices. So, what I have done ...
1
vote
1answer
22 views

Inverting variable size block matrix

Given a block matrix $M$ $$\bf M=\left(\begin{array}{cccc} a\ \mathbb{I}_{2} & \boldsymbol{\boldsymbol{A}}_{12} & \boldsymbol{A}_{13} & \boldsymbol{0_{(2,3)}}\\ \\ \boldsymbol{A_{21}} &...
0
votes
0answers
23 views

Differential equation for a vector : condition to be conservative

Assume there is a vector $\mathbf{N}(t)$ of elements $N_i$ and of dimension $n$ and there are matrices $\mathbf{B}_i$ of dimension $n\times n$, and that : $\begin{equation}\left\{ \begin{split}& \...
1
vote
0answers
38 views

Finding the the partial derivatives of a transformed surface, in the original space

I am trying to calculate the from the surface described by a function $P(x, y, z, w): \mathbb{R}^4 \to \mathbb{R}^4$. The surface is really only dependent on $x$ and $z$ but for computational reasons, ...
0
votes
0answers
14 views

Inverse of matrix expansion with negative exponents

The answer to this question shows that if I have a real nonsingular matrix $M$, such that its Taylor expansion in $\epsilon$ is $$M(x+\epsilon)= \sum_{n=0}^\infty M_n(x) \epsilon^n $$ its inverse ...
-1
votes
0answers
7 views

Why are points on opposite sides of the invariant line translated in different directions when being sheared?

Urgent help needed: The image shows a figure being sheared with a shear factor of 1 and the x-axis being the invariant line. A transformation matrix is used for this to happen. Why is it that the ...
0
votes
0answers
24 views

showing that Symplectic matrix is diagonalizable [on hold]

please how to Show that any symplectic matrix is ​​diagonalizable by using the fact that every unit matrix is ​​diagonalisable.
0
votes
0answers
6 views

On $QR$ factorization of upper Hessenberg matrices with real entries

Let $A$ be an $n\times n$ invertible upper Hessenberg matrix (https://en.wikipedia.org/wiki/Hessenberg_matrix) with real entries. I am trying to prove the following three statements : (1) There ...
0
votes
0answers
38 views

Numerator layout for derivatives and the chain rule

We have three matrices $\mathbf{W_2}$, $\mathbf{W_1}$ and $\mathbf{h}$ (technically a column vector): $$ \mathbf{W_1} = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} \;\;\;\;\;\;\;\;...
1
vote
1answer
31 views

Showing that a matrix F is a matrix representation of $\mathbb{R}$

The question is given below and its answer are given below: 1-I know that according to Vinberg book which is called "Linear representations of groups" all finite dimensional differentiable ...
1
vote
2answers
41 views

$A^{2n}=I$ but $A^{n}\neq I, -I$ [duplicate]

Let $A$ be a $n\times n$ real matrix. such that $A^{2n}=I$ but $A^{n}\neq I, -I$, $n\geq 2?$ I have a example for $A^2=I$ but $A\neq I, -I$ but could not find a similar example for this question. I ...
-1
votes
0answers
32 views

Is $A$ is diagonalizable/non-diagonalizable? [on hold]

Let $A$ $\in $$\mathbb{M}_5({\mathbb{C}})$ matrix such that $(A^2-I)^2=0$. Assume that $A$ is not a diagonal matrix. Which of the following statement is true ? $1)$$A$ is diagonalizable. $2)$ $A$ ...
1
vote
1answer
51 views

Properties of upper triangular matrix

let $A$ be the set of all invertible upper triangular matrics in $\mathbb{M}_n (\mathbb{R})$ where $n \ge 2$ then $A $ is choose the correct option $1.$ dense $2.$Nowheredense $3.$open ...
2
votes
0answers
22 views

Probability for a boolean matrix from a certain class to have full rank

While considering a certain type of computational problems, I have encountered the following probabilistic problem over the two element field $GF_{2}$. For $c\in \mathbb{Q}_{>0}$, let $p_{c}(n)$ ...
0
votes
0answers
9 views

On positive definiteness of a sub-matrix after first step Gaussian elimination to a symmetric positive definite matrix

Let $A=[a_{ij}]\in M_n(\mathbb R)$ be a symmetric positive definite matrix (i.e. all eigenvalues of $A$ are real and positive ) with $a_{11}\ne 0$ . Now let $A_1=[a' _{jk}] \in M_{n-1}(\mathbb R)$ ...
0
votes
1answer
18 views

Markov matrices with <=1 absorbing states have all but one eigen values <1?

I found plenty of proofs online that a Markov matrix will have all its eigen values of modulus <=1. In the book on Introduction to Matrix analysis by Bellman, he also shows in section 8 of chapter ...
1
vote
1answer
23 views

If $A$ is a real Symmetric Matrix of order $n (\geq 2)$ , then there exists a symmetric Matrix $B$ such that $B^{2k+1} = A$.

If $A$ is a real Symmetric Matrix of order $n (\geq 2)$ , then there exists a symmetric Matrix $B$ such that $B^{2k+1} = A$. Is the statement true? I think the statement is true. My Attempt : I ...
3
votes
1answer
65 views

How to show Von Neumann Trace inequality $ \text{Tr}(AB) \leq \sum_{i=1}^n \sigma_{A,i}\sigma_{B,i} $?

Let $A,B$ have the appropriate size. How can we show Von Neumann Trace inequality $ \text{Tr}(AB) \leq \sum_{i=1}^n \sigma_{A,i}\sigma_{B,i} $? Also, what is the intuition behind this inequality?
2
votes
2answers
40 views

How to show $\sqrt{\text{Tr}(A^2)} \leq \text{Tr}(A)$?

Let $A$ be a positive semi-definite matrix. How to show that Frobenius norm is less than trace of the matrix? Formally, $$\sqrt{\text{Tr}(A^2)} \leq \text{Tr}(A)$$ Also, show when $A$ is an $n \times ...
1
vote
1answer
29 views

Rewriting the Hermitian transpose of a normal matrix as a polynomial of that matrix

In one of my tutorials today, the tutor said that he read somewhere that the Hermitian transpose of a complex matrix $A^{H}$ can always be rewritten as a polynomial of $A$, i.e. $$A^H = \sum_{i = 1}^...
-1
votes
3answers
48 views

Can a matrix be not a multiple of identity, have repeated eigen values and still be diagonalizable?

The question: Diagonalisability of 2×2 matrices with repeated eigenvalues suggests that if a matrix has all its eigen values distinct, it must be diagonalizable. However, any multiple of the ...
0
votes
2answers
33 views

When does a matrix have a non-trivial solution?

Can someone please explain why this theorem is true? Theorem: If A is the matrix of coefficients of a system of linear equations, then the system has a solution if and only if the rank of the ...
1
vote
3answers
55 views

Ranks of matrix

Find the rank of the following matrix $$\begin{bmatrix}1&-1&2\\2&1&3\end{bmatrix}$$ My approach: The row space exists in $R^3$ and is spanned by two vectors. Since the vectors are ...
0
votes
0answers
22 views

Projection of a vector into the nullspace of a matrix

I need a clarification about the correct way to compute the projection of a vector into the nullspace of a matrix. For sake of clarity, let's call $A$ the matrix, $N(A)$ it's kernel and $A^\sharp$ ...
0
votes
0answers
14 views

Are the row vectors in a row reduced echelon matrix always independent?

Are the row vectors in a row reduced echelon matrix always independent? I'm thinking that since the first row is the only row with a non-zero coefficient, then it must be independent of all the ...
1
vote
1answer
21 views

quadratic programming /symmetric matrix

I have a quadratic program with $ F: \mathbb{R^n} \rightarrow \mathbb{R}, F(x)=x^TQx$ I want to find a symmetric matrix M for Q, such that $F(x)=x^TMx$ holds for all x. I can write Q as sum of ...
1
vote
2answers
20 views

How to show the trace inequality of two P.S.D matrices $\text{Tr(X)}\leq\text{Tr(Y)}$ when $X \preceq Y$?

Let $X,Y$ be two Positive Semi-Definite matrices. How can we show the following in the most elegant and shortest way? Because I know how to prove it but I think there is a better way? Alos, MaoWao ...
3
votes
1answer
57 views

Can I take a derivative of any complex function so long as I treat the complex numbers as matrices?

Complex numbers can be represented as matrices, for example $$ a+bi \leftrightarrow \pmatrix{a &b\\-b&a} $$ Only some functions of a complex variable have a derivative that is a complex ...
1
vote
1answer
30 views

How to show $\text{Tr}(AB) \leq \text{Tr}(AC)$ where $B \preceq C$?

Given three positive semi-definite matrices $A, B, C$. Show $\operatorname{Tr}(AB) \leq \operatorname{Tr}(AC)$ where $B \preceq C$? This inequality is the matrix form of multiplying a positive ...
1
vote
0answers
22 views

Existence of a QR factorization

Let $A \in \mathbb{R^{n \times n}}$ be an nonsingular matrix and $LL^T$ the Cholesky decomposition of $A^TA$. How to show that it exists a QR factorization with $Q=A(L^T)^{-1}$? I tried this: $A^TA=...
1
vote
0answers
17 views

quadratic eigenvalue problem reformulation with eigenvalues on unit cycle

Suppose a quadratic eigenvalue problem is $$(A_2\lambda^2+A_1\lambda+A_0)x=0$$ where $\lambda$ is the eigenvalue and $x$ is the eigenvector. the quadratic form can be rewritten by $$\left[ {\begin{...
0
votes
1answer
41 views

Can we simplify $ A^{-1}Bx = x$ where $A$ is a block matrix with each block being diagonal and half the blocks of $B$ are zero?

I have the following eigenvalue problem involving block matrices $A$ and $B$: $$ A^{-1}Bx = x. \quad \quad \quad \quad (*) $$ $A$ and $B$ have special structures. I would like to reduce/simplify this ...
3
votes
0answers
51 views

What is the number of elements in $\{A=(a_{ij}) \in Gl_n(\mathbb{F}_q):a_{11}=1\}$

Let $\mathbb{F}_q$ be the finite field with $q$ elements. I want to calculate the number of elements of the set $S=\{A=(a_{ij}) \in Gl_n(\mathbb{F}_q):a_{11}=1\}.$ We know that $|Gl_n(\mathbb{F}_q)|=(...
1
vote
2answers
36 views

Solve the equation with the help of determinants

The following set of equations is given: $$\begin{cases}{{x^2z^3\over y}=e^8\\{y^2z\over x}=e^4\\{x^2y\over z^4}=1}\end{cases}$$ I have solved problems before with three variables in three equations ...
2
votes
1answer
22 views

Similarity of two tridiagonal matrices

I am considering two complex symmetric tridiagonal matrices. First, A is a tridiagonal matrix with identical non-diagonal elements : A = $\begin{pmatrix} ig_1 & \kappa & 0 & 0 \\ \kappa &...
2
votes
0answers
44 views

Is the differential of left multiplication still left multiplication?

Let $A$ be a matrix in the real n-dimensional general linear group $G$. Let $l: G \to G$, $l(B)=AB$ be left multiplication by $A$. Consider the differential of $l$ at the identity matrix $I$ of $G$, $$...
6
votes
1answer
451 views

Matrices with all non-zero entries.

I am reading a paper and it uses one of these facts, I would like to know if it has a simple proof: Let $F$ be an infinite field and $n\ge2$ and integer. Then for any non-scalar matrices $A_1,A_2,......
1
vote
3answers
50 views

Eigenvalue of (some) $ 4 \times 4 $ symmetric matrices

$$A=\pmatrix{ 0 & 3 & 2 & 0 \\ 3 & 0 & 0 & 2 \\ 2 & 0 & 0 & 3 \\ 0 & 2 & 3 & 0 \\ }$$ Is there a quicker way to compute eigenvalues of this ...
0
votes
1answer
31 views

Prove $M(M+I)^{-1}M \succeq \frac{1}{2}I$ for any PSD matrix $M \succeq I$

For any $n \times n$ symmetrix matrices A, B, we define $A \succeq B$ if and only if $v^TAv \succeq v^TBv$ for all $v \in \mathbb{R}^n$
2
votes
4answers
73 views

Number of solutions $X$ to $AX=XB$ in $\mathbb F_2$

It is a well-known theorem that in an arbitrary field $F$, if $A$ is an $m\times m$ square matrix and $B$ is an $n\times n$ square matrix, then there is a unique $m\times n$ solution $X$ to the ...
1
vote
0answers
29 views

How to determine if the augmented matrix has no solution?

"Convert the following system of linear equations into an augmented matrix and then solve it" $$x_1 + x_2 - x_3 - x_4 = 7$$ $$2x_1 - x_2 + 3x_3 + x_4 = 1$$ $$x_1 - 5x_2 + 9x_3 - x_4 = 3$$ My ...