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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1
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2answers
32 views

Does the matrix logarithm always converge for exponential matrices?

We have defined functions on square matrices $X$: $$e^X := I + X + \dfrac{X^2}{2!}+\dfrac{X^3}{3!}$$ and $$log(X):= (X-I)-\dfrac{(X-I)^2}{2} + \dfrac{(X-I)^3}{3}-...$$ The exponential converges for ...
0
votes
1answer
59 views

The inverse of a matrix in which the sum of each row is $1$

Let $A$ be an invertible $10\times 10$ matrix with real entries such that the sum of each row is $1$. Then choose the correct option. The sum of the entries of each row of the inverse of $A$ is $1$. ...
0
votes
0answers
11 views

Conversion of network-like matrix

I have given a network in the following form (Example): x1 + x2 - x3 = 0 x3 + x4 - x5 = 0 x5 + x6 - x7 = 0 where = is something like a node, where flow needs to ...
0
votes
0answers
26 views

For any $n>1$ , $ Aut(\mathbb Z^n) \cong GL(n , \mathbb Z)$ ? And for any $m,n >1$ , $Aut(\mathbb Z_m ^n) \cong GL(n , \mathbb Z_m)$ ?

Is it true that for any $n>1$ , $ Aut(\mathbb Z^n) \cong GL(n , \mathbb Z)$ ? And is it true that for any $m,n >1$ , $Aut(\mathbb Z_m ^n) \cong GL(n , \mathbb Z_m)$ ? ( One problem I'm ...
1
vote
1answer
43 views

A question in numerical range of matrix polynomyal

Let $${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$$ a matrix polynomial(${A_j} \in {C^{n \times n}},j = 0,1,2....m$), and $$A = \left\{ {\lambda \in ...
-4
votes
2answers
27 views

Question about basis of Vector Space [closed]

Show that $B$ is a basis for $\mathbb{R}^2$:Where $$B=\{(-1,1) , (2,3)\}$$
1
vote
0answers
34 views

Generalized inverse of matrix product involving a positive semi-definite matrix

I have the following: A real square positive definite matrix $A$, and a real square conformable positive semi-definite matrix $B$. I form the product $$C = A^{-1}BA^{-1}$$ and I wonder, is it true ...
0
votes
1answer
23 views

Showing that $HTH = e^{-i \frac{\pi}{8} \sigma_x}$ (quantum gates)

I'm trying to prove that an arbitrary single qubit unitary (read: unitary two by two matrix, and thus rotation up to a phase) can be composed from Hadamard and T gates, given by $ H = ...
1
vote
2answers
229 views

What is the derivative of a vector with respect to its transpose?

I've already looked at Vector derivative w.r.t its transpose $\frac{d(Ax)}{d(x^T)}$, but I wasn't able to find the direct answer to my question in that question. What is the value of $$\frac{d}{dx} ...
0
votes
0answers
37 views

If $\mathcal{B}_1$ and $\mathcal{B}_2$ are two bases for $V$ then $\#\mathcal{B}_1 = \#\mathcal{B}_2$ [closed]

If $\mathcal{B}_1$ and $\mathcal{B}_2$ are two bases for $V$ then $\#\mathcal{B}_1 = \#\mathcal{B}_2$ I get the intuition, but how do you prove it with mathematical notations?
2
votes
0answers
43 views

Number of Arithmetic Operations in Gaussian-elimination/Gauss-Jordan Hybrid Method for Solving Linear Systems

I am stucked at this problem from the book Numerical Analysis 8-th Edition (Burden) (Exercise 6.1.16) : Consider the following Gaussian-elimination/Gauss-Jordan hybrid method for solving linear ...
-2
votes
0answers
27 views

$A = \left\{ {{P_\Delta }(\lambda ):\left\| {{\Delta _j}} \right\| \le \varepsilon ,j = 0,1,2…m} \right\} \Rightarrow$A is closed [closed]

Suppose ${P_\Delta }(\lambda ) = ({A_m} + {\Delta _m}){\lambda ^m} + ....... + ({A_1} + {\Delta _1}){\lambda ^1} + ({A_0} + {\Delta _0})$ is a matrix polynomial, and $\lambda $ is a complex ...
3
votes
2answers
92 views

Computing $\operatorname{Tr} \bigl( \bigl( (A+I )^{-1} \bigr)^2\bigr)$

Suppose that $A \in \mathbb{R}^{n \times n}$ is a symmetric positive semi-definite matrix such that $\operatorname{Tr}(A)\le n$. I want a lower bound on the following quantity $$\operatorname{Tr} ...
0
votes
1answer
22 views

Manipulating block matrices

Is there an easy way (maybe a similarity or congruence transform) to turn the matrix $$ \begin{pmatrix} A & B \\ B^T & D \end{pmatrix} $$ into the matrix $$ \begin{pmatrix} A & -B \\ ...
4
votes
3answers
110 views

$A$ is normal matrix.If $A{A^T}$ has $n$ distinict eigenvalue,why $A$ is symmetric?

Let $A \in {M_n}(R)$ and $A$ is normal matrix.If $A{A^T}$ has $n$ distinict eigenvalue,why $A$ is symmetric?
0
votes
1answer
935 views

Difference between rotation and pure rotation

Hi i am trying to understand my teacher's assignment. I have 2 write 2 Matlab functions ...
-4
votes
0answers
42 views

Linear Transformations? [closed]

I have three linear transformation problems which I have no idea of how to start solving. If possible could someone guide me through this? Links to videos would be great so I can solve future problems ...
0
votes
0answers
19 views

Scaling Matrix?

I have two matrix problems which I have no idea of how to start solving. If possible could someone guide me through this? Links to videos would be great so I can solve future problems myself 1) Find ...
8
votes
1answer
69 views

Find all $n\times n$ matrices $A$ satisfying $\det(I+A^n)=(1+\det(A))^n$

Problem: Find all $n\times n$ matrices $A$ satisfying $$\det(I+A^n)=(1+\det(A))^n.$$ Clearly, the identity matrix $I$ works because $$\det(I+I^n)=\det(2I)=2^n=(1+\det(I))^n.$$ Are there any ...
14
votes
1answer
167 views

Bound on the difference of two determinants

Let $A$ and $B$ be two real, $n\times n$ matrices. Using Hadamard's inequality, it is not hard to show that $$ \left|\det A - \det B \right| \leq \|A-B\|_{2} \frac{\|A\|_{2}^n -\|B\|_{2}^n}{\|A\|_2 ...
-1
votes
1answer
20 views

SVD of matrix whose columns are orthogonal vectors [closed]

Write the SVD of a matrix $A$ whose columns are orthogonal vectors ($w_1, ...,w_n$). Can please someone give me a tip how to start writing the SVD ? Thanks
2
votes
2answers
47 views

$\log (A + \delta A) = ?$ (as an expansion in $\delta A$), where $A$ and $\delta A $ are matrices

$A$ and $\delta A$ are two non-commuting matrices and I am seeking a power series expansion to 2nd order in $\delta A$. After writing it as $\log (A (1 + A^{-1}\delta A) )$, I am unable to figure out ...
1
vote
1answer
45 views

Prove $\begin{pmatrix} a&b\\ 2a&2b\\ \end{pmatrix} \begin{pmatrix} x\\y\\ \end{pmatrix}=\begin{pmatrix} c\\2c \\ \end{pmatrix}$

Prove that $$ \begin{pmatrix} a & b \\ 2a & 2b \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix} = \begin{pmatrix} c \\ 2c \\ \end{pmatrix} $$ has solution $$ \begin{pmatrix} x \\ y \\ ...
1
vote
1answer
30 views

Find $x$ and $y$ by Cramer from $\left\{\begin{aligned} \frac{2x+2}{2y-3}&=\frac{x-2}{y+3}\\ \frac{x+2}{3y-2}&=\frac{x-1}{3y+8} \end{aligned}\right.$

Find $x$ and $y$ with Cramer rule from these equations $$\left\{ \begin{aligned} \frac{2x+2}{2y-3}&=\frac{x-2}{y+3} \\ \frac{x+2}{3y-2}&=\frac{x-1}{3y+8} \end{aligned} \right. $$ I'm ...
0
votes
1answer
48 views

Find $ax^5 + by^5$ by matrix from $ \left\{ \begin{aligned} ax+by&=3\\ ax^2+by^2&=7\\ ax^3+by^3&=16\\ ax^4+by^4&=42 \end{aligned} \right.$

Find $ax^5 + by^5$ if the real numbers $a$, $b$, $x$, and $y$ satisfy the equations $$ \left\{ \begin{aligned} ax+by&=3 \\ ax^{2}+by^{2}&=7 \\ ax^{3}+by^{3}&=16 \\ ...
6
votes
1answer
140 views

3D rotation group

It is known that the group $\text{SO}(3)$ of rotation-matrices (matrices $A$ with $\det(A)=1$) are generated from three parameters. This can be expressed by the fact, that any rotation matrix is a ...
3
votes
1answer
52 views

$S = \left\{ x^* Ax\mid x \in C^n ,\ x^*x = 1 \right\} \implies S\;$ is compact and convex

Let $\,A \in {\mathbb{C}^{n \times n}}\,$ and $\,S = \left\{ {{x^*}Ax \mid x \in \mathbb C^n,\ {x^*}x = 1} \right\}.\,$ Why is $A$ compact and convex?
0
votes
2answers
678 views

How to compute the eigenvalue condition number of a matrix

How to compute the eigenvalue condition number, $\kappa(4,A)$, of a matrix $A$ $$A = \begin{bmatrix} 4 & 0 \\ 1000 & 2\end{bmatrix}$$ I am a bit stuck on how to proceed solving this problem ...
-1
votes
0answers
13 views

Three State Model [closed]

Okay hey everyone I am given 6 transition intensities are go as follows : \mu _12 = 0.1 \mu _13 = 0.6 \mu _22 = 0.4 \mu _23 = 0.5 \mu _31 = 0.2 \mu _32 = 0.6 Okay now 'using the fact that you can ...
0
votes
1answer
34 views

Help Finding Elementary Matrix.

I apologise in advance if this is simple, but I'm losing my brain over this question. I'm unsure how to make the matrix format work either. I'm trying to find elementary matrices so that ...
0
votes
1answer
30 views

Proof that transpose of Hadamard Matrix is also a Hadamard matrix

The question is self explaining from the title, but let me elaborate it. In most of the articles/books I've read, fact that the transpose of Hadamard matrix is also a Hadamard matrix is used, but I ...
-1
votes
1answer
30 views

matrix multiplication questions [duplicate]

$A$ and $B$ are two matrices, when is $(A-B)(A+B)=A^2 - B^2$
0
votes
1answer
33 views

Reduced-row echelon form associated to three lines in the plane

Let $\ell_1,\ell_2$ and $\ell_3$ be three lines in the plane $\mathbb{R}^2$. For $i = 1, 2, 3$, let the line $\ell_i$ have equation $a_i x + b_i y = c_i$. Is it possible for the matrix $$ ...
3
votes
1answer
2k views

Why does spectral norm equal the largest singular value?

This may be a trivial question yet I was unable to find an answer: $$\left \| A \right \| _2=\sqrt{\lambda_{\text{max}}(A^{^*}A)}=\sigma_{\text{max}}(A)$$ where the spectral norm $\left \| A \right ...
1
vote
0answers
8 views

Request reflection matrix about these types

Supposed there's $(a,b)$ point and going to be reflected and find the mapping. The baseline formula will I use is $\begin{pmatrix} x' \\ y' \end{pmatrix}=M_{R} \begin{pmatrix} x \\ y \end{pmatrix}$. ...
-4
votes
0answers
11 views

Matrix Transformation (REflection) [closed]

Reflect triangle with vertices $A(2,2), B(4,1)$ and $C(5,3)$ along the line $x=5$.
3
votes
1answer
49 views

Product of a Finite Number of Matrices with a Cosine Entry

Does any one know how to prove the following identity? $$ \mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix} 2\cos\frac{2j\pi}{n} & a \\ b & 0 \end{pmatrix}\right)=2 $$ when $n$ is ...
1
vote
2answers
34 views

Why does similarity with a diagonal matrix imply that the Jordan normal form must also be diagonal?

If a matrix representation of a linear transformation is similar to a diagonal matrix, why does this imply that the Jordan normal form must also be diagonal?
1
vote
1answer
117 views

Proof using the Rank Theorem

Let $v$ and $w$ be non-zero column vectors in $\Bbb R^n$ and let $A = vw^t$ so that $A$ is an $n\times n$ matrix. Use the rank theorem to show dim Nul $A = n − 1$ Here Nul(A) represents the Null ...
2
votes
1answer
51 views

Find matrix dimensions satisfying a strange condition.

I came across this question and was wondering how it could be proven. Find all pairs $(m,n)$ of positive integers for which there exists an $m\times n$ matrix $A$ and an $n\times m$ matrix $B$, both ...
2
votes
1answer
23 views

Cayley transform a matrix that is invertible when added to the identity

Let A be an nxn matrix such that (I+A) is invertible. I need to prove that the Cayley Transform of A, denoted by $A^c$, is such that $(I+A^c)$ is invertible. The Cayley Transform is defined as ...
0
votes
1answer
314 views

Solution of a system of linear equations with n variables

I have a system of linear equations with n variables \begin{cases} a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n = \frac{1}{2}x_1\\[4pt] a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n = \frac{1}{2}x_2\\[4pt] ...
0
votes
4answers
433 views

Given a $4\times 4$ symmetric matrix, is there an efficient way to find its eigenvalues and diagonalize it?

I have a $4\times 4$ matrix $$A=\left(\begin{array}{cccc}8 & 11 & 4 & 3\\11 & 12 & 4 & 7\\4 & 4 & 7 & 12\\3 & 7 & 12 & 17\end{array}\right).$$ I want to ...
1
vote
3answers
67 views

Find the values of 'a' in a $4\times 4$ matrix(A) when the determinant is less than 2012

The matrix is $A \ =\begin{pmatrix} 7 & 1 & 3 & -2\\ -2 & 1 & -12 & -1 \\ 1 & 16 & -4 & a \\ ...
0
votes
0answers
21 views

Calculating the Span of a Matrix in MATLAB

If I have a matrix (or a set of vectors) say A=[1 2 4] [2 9 8] [7 9 3] how can I calculate its span in MATLAB? There is no direct command for it? Do I have to form a set of linear ...
1
vote
0answers
26 views

Does one row zero means major diagonal zero? [closed]

Let $A$ be a symmetric positive semidefinite matrix. Show that the $i$-th row of $A$ is zero if and only if $A_{ii} = 0$. This is what I have thought as of now From the what Surb has just said ...
-1
votes
0answers
14 views

A question in perturbation of $P(\lambda )$

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
-1
votes
0answers
12 views

Inner Product of Square Matrices

Let K$^{n*n}$ & M$^{n*n}$ be two square matrices, and K$\cdot$M= \begin{matrix} t_{11} & \cdots & t_{1n} \\ \vdots & \ddots & \vdots \\ t_{n1} & \cdots ...
2
votes
1answer
31 views

Tensors in matrix multiplication algorithms

Fast matrix multiplication algorithms, be it the Winograd and Coppersmith algorithm or any further improvement of it, extensively use tensors. In fact, the entire construction is based on tensor ...
0
votes
0answers
12 views

Dot Product of Square Matrices & Inner Product

I need some help! Thank you in advance. Let K$^{n*n}$ & M$^{n*n}$ be two square matrices, and K$\cdot$M= \begin{matrix} t_{11} & \cdots & t_{1n} \\ \vdots & \ddots ...