For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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1answer
32 views

$\kappa(B^{-1}A)=\kappa (AB^{-1}) = \frac{\lambda_{max}(B^{-1}A)}{\lambda_{min}(B^{-1}A)}$

Prove or disprove if $A,B$ are symmetric positive definite (s.p.d) matrices then operator 2-norm condition number $\kappa(B^{-1}A)=\kappa (AB^{-1}) = ...
0
votes
0answers
13 views

Is this block matrix Hurwitz?

Let $A\in\mathbb{R}^{n\times n}$ be lower-triangular and Hurwitz. (Hence, the eigenvalues of $A$ are all real and strictly negative). Let $k_1,k_2>0$. Consider the matrix $$ M = \begin{bmatrix} 0 ...
0
votes
3answers
36 views

Can't understand this question from Matrices

Let $P=\frac{\underline{x}\underline{x}^{T}}{\underline{x}^{T}\underline{x}}$ be an $n×n (n>1)$ matrix, where $\underline{x}$ is a nonzero column vector. Then which one of the following statement ...
1
vote
3answers
33 views

Semimagic square Matrix

How to prove this? Basically I have no idea at all as how to proceed in this particular question. Please look into this.
1
vote
3answers
91 views

Calculating determinant of 100x100 matrix

I was trying to calculate the determinant of 100x100 matrix: ...
1
vote
1answer
25 views

Does matrix norm change under an equivalence transformation? [closed]

Consider $||.||_2$ matrix norm. Let A, B be symmetric matrix, is ||A|| and $||BAB^{-1}||$ equal? Thank you in advance.
0
votes
0answers
11 views

How to solve this complicated matrix multiplication operation

Assume we have vectors of size $1\times N$ $${\bf h_1} = [ h_{1a}, h_{1b}, h_{1c}, \cdots h_{1N}]$$ $${\bf h_2} = [ h_{2a}, h_{2b}, h_{2c}, \cdots h_{2N}]$$ Define a matrix $2 \times N$ $${\bf H}= ...
0
votes
1answer
41 views

What is the enclosed volume of an irregular cube given the x,y,z coordinates of the 8 corners?

I have the xyz coordinates of 8 points that forms an irregular-shaped cube. This is an animation, so the cube is undergoing periodic or cyclical shape-change. The co-planarism of any group or set ...
0
votes
0answers
13 views

“Canonical” choice of parity check matrix from generator matrix

Let $\mathcal{C}$ be a binary linear code of length $n$ and dimension $k$, given as the left-image of an $k \times n$ generator matrix $G$: $$ \mathcal{C} = \{ i G : i \in \mathbf F_2^k \} \subset ...
1
vote
2answers
23 views

Matrix rank considering a real parameter

Assuming the linear system: \begin{align} & (1+\lambda)x+y+z=1 \\ & x+(1+\lambda)y+z=\lambda \\ & x+y+(1+\lambda)z=\lambda^2 \end{align} for which I need to determine the rank of the ...
1
vote
1answer
26 views

Finding change of basis matrix given a linear map's matrix representation in each basis

Let $V$ and $W$ be vector spaces with (standard, orthonormal) bases $\mathcal{V} = (v_1, \dots, v_k)$ and $\mathcal{W} = (w_1, \dots, w_n)$, and let $T : V \to W$ be a linear map. With respect to the ...
0
votes
0answers
19 views

Uniqueness of a Row-Reduced Echelon Form

Prove that the row-reduced echelon form obtained by row reduction of a matrix $A$ is uniquely determined by $A$. So, I don't understand why the row-reduced echelon form is uniquely determined by ...
6
votes
3answers
368 views

What does echelon mean?

When you solve a system of linear equations, you write down the augmented matrix and reduce this to reduced row echelon form. What is the meaning of the word echelon?
0
votes
3answers
42 views

Compute the square norm $||\cdot||_2$ of matrix [closed]

Compute the square norm $||\cdot||_2$ of matrix : \begin{bmatrix} O & A_n^{-1} & \dots & O \\ A_n^{-1} & O & \ddots & \\ \vdots & \ddots & \ddots ...
0
votes
1answer
22 views

Sum of columns matrix

I found this exercise: Let $A\sim\begin{bmatrix}1&3&6&1\\0&0&1&3\\0&0&-5&3\\0&0&0&0\end{bmatrix}$ where the equivalence was accomplished solely ...
0
votes
1answer
33 views

Derivative of determinant. [duplicate]

I have the following identity which I want to prove: $$\frac{d}{dt} det(A+tB)|_{t=0} = Tr( Cof(A)^TB)$$ where $Cof(A)$ is the cofactor matrix of $A$, and $A$ is an $n\times n$ complex matrix. The ...
1
vote
0answers
29 views

Relationship between row space and orthogonal component of kernel of complex vector space

When we consider the real vector space, row space is equal to orthogonal complement of the null space (kernel). This fact can be proved as follows. Let's consider linear map $A : \mathbb{R}^n ...
0
votes
1answer
28 views

$4\times 4$ Matrix determinant (For computer graphics)

So Opengl and other graphics Api's use Matrices that are $4\times 4$, because they have to include affine transformations (translation). The 4th row and column are included for this reason. The ...
1
vote
1answer
75 views

Find $||\cdot||_2$ norm of block matrix

Compute the square norm $||\cdot||_2$ of matrix : \begin{bmatrix} O & I_n & \dots & O \\ I_n & O & \ddots & \\ \vdots & \ddots & \ddots & I_n ...
0
votes
0answers
16 views

Convergence of Adjacence Matrices

I have a graph, with some nodes, and weighed edges with positive weights only. The sum of the weights from one node is less or equal to 1. There will be at least one node, that the sum is less then 1. ...
0
votes
1answer
50 views

How to prove that the inequality holds for any nonzero x?

The inequality is given by $$x^{H}(\Phi(x))^{-1}x-x^{H}\dfrac{aa^H}{a^H\Phi(x)a}x\ge0,\text{ for any }x\ne0,$$ where $\Phi(x)$ is positive definite and is a function of $x$, $a$ can be any nonzero ...
0
votes
1answer
36 views

Prove or disprove that $A$ is compact.

Let $\mathbb{A} : = \{L \in \mathcal L(\mathbb{R^2},\mathbb{R^2}): L(x)=Ax$ where $A=(a_{ij})$ is a matrice $2 \times 2$ such that $|a_{ij}| \leq i + j$ for $i,j = 1,2 \}$ (Note : $\mathcal ...
0
votes
0answers
21 views

Normalizing disturbed rotation matrix

I was playing with simulations of Euler's equations of rotation in this question. This involves integrating an ordinary differential equation of a rotation matrix, $R$, which is calculated for all of ...
-1
votes
0answers
31 views

Commutative and Orthogonal matrices

Given an $n \times n$ matrix $M$ and the identity I, show that the matrices (I+M) AND $(I-M)^{-1}$ commute. For a real skew- symmetric matrix A, the matrix N is defined by $N=(I+A)(I-A)^{-1}$ ...
1
vote
1answer
93 views

What does $\bigotimes$ and $X^*$ mean?

Can someone explain / link me to a linear algebra worked problem where I can see how these work. I've searched and given their statistics and matrix specialty uses, can't find any ready examples.
-1
votes
1answer
31 views

Anti-symmetric and non-singular matrix [closed]

$A$ is an anti-symmetric matrix and S is a symmetric matrix, such that the matrices $I+AS$ is non-sigular if $B=(I-AS)(I+AS)^{-1}$ Prove that $B^{T}SB=S$ ...
10
votes
2answers
214 views

A tricky problem about matrices with no $\{-1,0,1\}$ vector in their kernel

A Hankel matrix is a square matrix in which each ascending skew-diagonal from left to right is constant. Let us call a matrix partial Hankel if it is the first $m<n$ rows of some $n$ by $n$ Hankel ...
1
vote
1answer
25 views

Generator matrix of $E_5$

Let $E_5$ denote the binary even weight code of length 5. Write down a generator matrix of $E_5$. So I know the length $n = 5$ is the number of rows in the generator matrix and the number of ...
1
vote
2answers
37 views

How to prove that $ A^TA$ is singular for $2\times 3$ matrix $A$

I was trying to find the determinant for $A^TA$ where $$ A = \left( \begin{array}{ccc} a & b & c \\ d & e & f \\ \end{array} \right) $$ I tried out with some numbers in place of $a, ...
0
votes
2answers
47 views

How do you express a $3\times3$ semi magic square(same sum for each row&col) in form of a set?

By set, I mean like a subspace $W=\{[] \in M_{3\times3} (\mathbb{R})| $something$\}$. Since matrix such as $ \begin{pmatrix} 0 & 0&3 \\ 3 & 0&0 \\ 0 & 3&0 \end{pmatrix}$ $ ...
0
votes
1answer
19 views

Showing the matrix 1-norm equals the maximum of a sum

I want to show: $$\|A\|_{1} := \max_{x\neq 0}\frac{\|Ax\|_{1}}{\|x\|_{1}} = \max_{1\leqslant j \leqslant n}\sum_{i=1}^{n}|a_{ij}|$$ My attempt to show it directly by applying the definition of ...
1
vote
1answer
34 views

Cosine of matrix and matrix of cosines

Suppose I have cosine of the matrix $$ \tag 1 \cos\left( \begin{pmatrix} a & b \\ c & d\end{pmatrix}\right) $$ May I write it in a form $$ \tag 2 \begin{pmatrix} \cos(a) & \cos(b) \\ ...
3
votes
5answers
53 views

Non-singular matrix

Assuming $B$, $I+B$, $I+B^{-1}$ are all non-singular, show that $$(I+B)^{-1}+(I+B^{-1})^{-1}=I$$ All I know is that determinants not equal to $0$ and that the inverse of $B$ exists.
0
votes
2answers
40 views

Proving matrix properties: [closed]

Prove: (i) $A(I+BA)^{-1}=(I+AB)^{-1}A$ (ii) $(I+AB)^{-1}=I-A(I+BA)^{-1}B$ (i) Consider $A(I+BA)=(A+ABA)=(I+AB)A$ Taking inverse on both sides (invert) ...
0
votes
2answers
50 views

Defining Unitary Matrices

I have got a question and I would appreciate if one could help. I start with an example to explain what I am looking for. Assume a scaled unitary matrix like $U_2 = \begin{bmatrix} 1 & 1 \\ 1 ...
0
votes
0answers
21 views

trace inequality trivial on kernel

Let $x_1, x_2>0$ such that $x_1+x_2=0$. Then the concativity theorem claims that for any $n \times n$ matrix $ K$ and any positive matrices $A_1, A_2$ following inequality holds for all ...
1
vote
1answer
42 views

Finding out whether two matrices are similar? [duplicate]

$\mathbf{B = P^{^-1} A P} \iff$ ($\mathbf{B}$ is similar to $\mathbf{A}$) I'm a little confused about matrix similarity. Let's say we have the following matrix $A$: ...
0
votes
1answer
16 views

Show that: the column rank of a matrix $A ∈ Mat(n × m; F)$ equals the rank of the linear mapping $(A◦) : F^m → F^n$

Show that: the column rank of a matrix $A ∈ Mat(n × m; F)$ equals the rank of the linear mapping $(A◦) : F^m → F^n$ The column rank of a matrix $A$ is the dimension of the subspace of $F^n$ ...
0
votes
1answer
36 views

Prove or disprove if $µ_0(Bx, x) ≤ (Ax, x) ≤ µ_1(Bx, x), ∀x ∈ R^n$, then $κ(B^{−1}A) ≤ \frac{µ_1}{µ_0}$

Let $A, B ∈ \mathbb{R}^{n×n}$ symmetric. Show that conditional number $$κ(B^{−1}A) ≤ \frac{µ_1}{µ_0}$$ holds, if $B ∈ \mathbb{R}^{n×n}$ is a symmetric positive definite matrix satisfying $$µ_0(Bx, x) ...
1
vote
0answers
31 views

Is there any easiest way to find the determinant? [duplicate]

Suppose, $ M=\begin{bmatrix}\begin{array}{ccccccc} -x & a_2&a_3&a_4&\cdots &a_n\\ a_{1} & -x & a_3&a_4&\cdots &a_n\\ a_1&a_{2} & -x ...
2
votes
3answers
75 views

Result of matrix $A^{2016}$

I want to find the result of $A^{2016}$ but I cannot find any pattern except for the zeros in the middle row and column. $$A=\begin{bmatrix}1 & 0 & {-2}\\0 & 0 & {0}\\3 & 0 & ...
0
votes
2answers
32 views

Matrices and unique and infinite solutions

If a matrix has a row of zeros but the number of variables is equal to the number of nonzero rows, does that mean the matrix has an infinite amount of solutions or a unique solution?
0
votes
0answers
40 views

Write the Following in matrix notation

Write the following in matrix notation: $$\matrix{i' &=& \cos(wt)i - \sin(wt)j\\ j' &=& \sin(wt)i + \cos(wt)j\\ k' &=& k}$$ Note: $i, j$ and $k$ are all vectors! Show ...
0
votes
0answers
34 views

Is dot product defined for matrices for other than terms being $n \times 1$ or $1 \times n$ vectors?

Is dot product defined for matrices for other than terms being $n \times 1$ or $1 \times n$ vectors? That is, is $Ax\cdot b$ defined for other than $Ax\in M_{1 \times n} \text{ or } M_{n \times 1}$? ...
4
votes
0answers
62 views

Low-degree “determinant” for non-square matrices?

Consider a matrix $A\in \mathbb R^{n\times n}$ of indeterminates. The determinant of $A$ is a degree $n$ polynomial in the $n^2$ entries satisfying $\det A\ne0\iff A$ is nonsingular. What about when ...
0
votes
2answers
35 views

$A-B$ and $B-A$ positive semidefinite $\implies$ $A=B$

I am trying to show that $A-B$ and $B-A$ positive semidefinite $\implies A=B$, but I'm having some trouble. $A-B$ PSD iff $x'(A-B)x \geq 0 \forall x$, and $B-A$ PSD iff $x'(B-A)x \geq 0 \forall x$. ...
0
votes
2answers
41 views

If a matrix $\boldsymbol{\mathrm {A}}$ is orthogonal, its determinant is $\pm 1$. Is the converse also true?

I know that an orthogonal matrix satisfies $$~~~~~~~~~~~~~~~~~~~~~~~~~\boldsymbol{\mathrm {AA}}^T=\boldsymbol{\mathrm {A}}^T\boldsymbol{\mathrm {A}}=\boldsymbol{\mathrm {I}}~~~~~~~~~~~~~~~~~~~~~~(*)$$ ...
0
votes
0answers
29 views

Typos in Gentle's Matrix Algebra?

In Gentle's (2007, p.180) Matrix Algebra, he writes It seems to me that this requires $u$ to be orthonormal (i.e. unit length), not just orthogonal to $v$ as hypothesized in the text, because ...
2
votes
2answers
39 views

Finding the Jordan Form of a matrix

Let $A$ be a 7*7 matrix with a single eigenvalue $q\in C$. It is know that $\rho (A-qI) = 2$ and that $\rho (A-qI)^2 = 1$. How can I find the Jordan form of A (+ the minimal polynomial)?
0
votes
1answer
19 views

Finding solutions using Gauß-Jordan-Algorithm

I have matrix A = \begin{bmatrix}1&2&3&4&5\\2&4&3&5&4\\3&6&5&8&7\end{bmatrix} and\vec b = \begin{bmatrix}1\\2\\3\end{bmatrix} I expanded the matrix ...