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
97 views

Partial derivative with matrices

I have reforumulated my problem of computing some quantities $\mathbf{a}\in R^{m}$ from $\mathbf{b}\in R^{n}$ in a matricial form: $$\mathbf{b} = (C\odot(\mathbf{1}_{n}\cdot \mathbf{a}^{T}))\cdot ...
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1answer
78 views

Matrix Derivative of this Equation

I'm trying to solve this minimization problem: $$ \min_{\Theta} \frac{C_1}{2} \sum_{j=1}^{N-1} \|\vec{\theta_{j+1}} - \vec{\theta_j}\|^2 ,$$ where $\Theta = (\vec{\theta_1}, \vec{\theta_2}, \ldots, ...
1
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1answer
50 views

How to find all $B$ that commute with $A$?

Let $$ A=\left(\begin{matrix} \lambda_1 I_{n_1} & & \\ & \ddots & \\ & & \lambda_r I_{n_r} \\ \end{matrix}\right) \in ...
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3answers
88 views

Preimage of non-invertible matrix

I am given the matrix $$\begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{-1}{\sqrt{2}} \\ \frac{-1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}$$ Apparently this one is not invertible. ...
3
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0answers
194 views

Row-normalized and column-normalized matrix notation

I'm searching for the mathematical, algebraic notations of a row-normalized and column-normalized matrix. For example, let us consider the following matrix A: $$ A = \begin{pmatrix} 2 & 7 \\ 4 ...
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0answers
58 views

How do I solve this matrix?

How do I expand the following matrix? I have no experience in linear algebra (well 35 years ago..) Its the equation just after equation 16b in this link Here are the equations in question: $$ ...
3
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0answers
87 views

An inequality concerning non-negative integer matrices with constant row and column sums

I'd appreciate any suggestions for how to prove (or disprove) the inequality described below. Some notation first: for positive integers $k$ and $M$, let ${\mathcal D}_{k,M}$ denote the set of all $k ...
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1answer
31 views

Prove $N(R^TR)=N(R)$

Suppose R is m by n with rank r and pivot columns first: (strang 4 ed. 3.3, problem 27) $$ R = \begin{bmatrix} I & F \\ 0 & 0 \end{bmatrix} $$ Prove that $R^{T}R$ has the same ...
0
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1answer
61 views

distance-measure method to measure the distance between two matrixes(probability distribution)

I should find a suitable distance-measure method to measure the distance between two matrixes. The elements of such matrix is 0 to 1, and the sum of the all element is 1, so I think I could treat it ...
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2answers
2k views

Block Diagonalizing an antisymmetric matrix

I was wondering how to block diagonalize a $10\times10$ antisymmetric matrix into block matrices along the diagonal. Can I just diagonalize each non-diagonal block? Thanks!
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1answer
26 views

Finding the matrix of linear transformation

What is the orthogonal projection on the line of equation $x = y$ of the point $\begin{pmatrix} 3 \\ -1 \end{pmatrix}$? Assume this is a linear transformation. The matrix for this linear ...
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1answer
48 views

Prove that a set of matrices is a linear space

Prove that the set of matrices $$v:=\left\{ \begin{pmatrix} 2x-y+z & x-2y-2z \\ x+y-z & 3x+y+2z \end{pmatrix} \middle|\, x,y,z \in R\right\}$$ Is a linear space above $R$ and find it's base. ...
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0answers
70 views

Induced matrix p-norm

Let $\|\cdot\|_p$ denote the $p$ norm $(p≥1)$ defined for every vector $x=(x_1,x_2,\ldots,x_n)^t\in\mathbb C^n$ by $\|x\|_p=(\sum|x_j|^p)^{1/p}$ and let $|||\cdot|||_p$ denote the matrix norm defined ...
7
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1answer
279 views

Is $(tr(A))^n\geq n^n \det(A)$ for a symmetric positive definite matrix $A\in M_{n\times n} (\mathbb{R})$

If $A\in M_{n\times n} (\mathbb{R})$ a positive definite symmetric matrix, Question is to check if : $$(tr(A))^n\geq n^n \det(A)$$ What i have tried is : As $A\in M_{n\times n} (\mathbb{R})$ a ...
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2answers
118 views

How can linear a operator have more than one matrix representation?

Let $A$ be linear operator on a vector space $V$. That is $A : V \to V$. How can a linear operator have more than one matrix representation ? ( As suggested in the Book Neilson and Chuang that matrix ...
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2answers
44 views

$M^{-1}$ has at most $n^2-2n$ coefficients equal to zero

Let $M\in GL_n(\mathbb{R})$ such that all its coefficients are non zero. How can one show that $M^{-1}$ has at most $n^2-2n$ coefficients equal to zero ? I have no idea how to tackle that problem, ...
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6answers
157 views

Prove that $\operatorname{Trace}(A^2) \le 0$

Let $A \in M_n(\mathbb{R})$ is a antisymmetric matrix such as $A^T=-A$. Prove that $\operatorname{Trace}(A^2) \le 0 $ I see that, for some matrix such as, their terms in diagonal are negative ?
2
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1answer
192 views

find a matrix that satisfies $A^6= I$…

How to solve this type of questions .....please explain.... I'm not getting how to start?
3
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1answer
229 views

For $A(t)$ differentiable, taking positive matrices as values, how show $\sqrt{A(t)}$ is differentiable?

On p. 150 of Lax's Linear Algebra, he mentions that is is not hard to show that if $R(t)$ is a differentiable matrix-valued function of a single variable, whose values are positive matrices, then the ...
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2answers
29 views

What does the matrix derivative of this equation look like?

I'm trying to solve this minimization problem: $$ \min_{\Theta} \frac{C_1}{2} \sum_j^N ||\vec{\theta_j}||^2 $$ where $\Theta = (\vec{\theta_1}, \vec{\theta_2}, ..., \vec{\theta_N})$. (FYI, it's ...
9
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1answer
20k views

What does double vertical-line means in linear algebra?

I have a formula, which I have no idea how to solve, because I don't know that double vertical-line sign: ||Ax|| What does it mean? How should I solve this?
5
votes
1answer
57 views

Left invertible matrices over rings with some special property

Suppose $R$ is a ring in which every left invertible element is invertible. Does this condition imply that every left invertible matrix in $\mathrm{M}_{n\times n}(R)$ is necessarily invertible?
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3answers
85 views

Find the set of all $\alpha$ such that Matrix A is invertible and calculate the inverse for all $\alpha$

$A=\begin{pmatrix} 0 & 1 & -1 & 2\\ 2 & -1 & 3 & 0 \\ \alpha & 0 & 1& 0 \\ 3 & -1 &4 & 0 \end{pmatrix}$ I know that a ...
1
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1answer
47 views

What can we say about $m_A(x)$ with respect to $m_{A^2}(x)$ when $A$ is diagonalizable?

Suppose $A$ is a real $n\times n$ matrix and diagonalizable over $R$. Is one of the following propositions is true? (1) $m_{A^2}(x)$ divides $m_A(x)$ (2) $m_A(x)$ divides $m_{A^2}(x)$ I think ...
2
votes
2answers
175 views

Normal, Real, and Square matrix which is diagonalize over C but not over R

I'm trying to find out a normal, real and $\boldsymbol n\times \boldsymbol n$ ($n\ge3$) matrix $A$ which is diagonalize over $C$ but isn't diagonalize over $R$. I know that the following matrix ...
2
votes
3answers
469 views

How to solve a system of 3 equations with Cramer's Rule?

I am given the following system of 3 simultaneous equations: $$ \begin{align*} 4a+c &= 4\\ 19a + b - 3c &= 3\\ 7a + b &= 1\end{align*} $$ How do I solve using Cramers' rule? For one, I ...
0
votes
1answer
62 views

Upper bound on the largest singular value

If I have any matrix $W \in R^{nxm}$, and matrices $U, V$ where the following properties hold: 1) $U^{T}W =0$ 2) $WV = 0$ I want to show that the upper bound of the largest singular value of the ...
2
votes
1answer
55 views

matrix representations of linear tranformatitons

I am having trouble with this problem. I have to find the matrix representation of a linear transformation. The example in my book got me this question below.Can someone explain this question?
2
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3answers
263 views

diagonalizable matrix such that sum of every column is the same number

Let A be a square matrix which is diagonalizable over field $\mathbb{F}$ and the sum of the entries of any column is the same number $a\in\mathbb{F}$. Show that $a$ is eigenvalue of the matrix A. ...
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2answers
1k views

Are eigen spaces orthogonal?

Let $A$ be a $N$ x $N$ matrix which has $k < N$ distinct eigenvalues. Are eigenspaces corresponding to different eigenvalues orthogonal in general ? I know it is true if $A$ is normal matrix. But ...
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1answer
40 views

Singular values of rectangular matrix

can any one explain me the need for singular values of a matrix. Explanation with a practical example will be appreciated
2
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1answer
190 views

identity of $(I-z^nT^n)^{-1} =\frac{1}{n}[(I-zT)^{-1}+(I-wzT)^{-1}+…+(I-w^{n-1}zT)^{-1}]$

I am trying to understand the identity $$(I-z^nT^n)^{-1} =\frac{1}{n}[(I-zT)^{-1}+(I-wzT)^{-1}+...+(I-w^{n-1}zT)^{-1}] \quad (*),$$ where $T \in \mathbb{C}^{n\times n},z\in \mathbb{C}$ and the ...
1
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1answer
186 views

Conditions for diagonalizability of $n\times n$ anti-diagonal matrices

Let $A$ be an $n\times n$ anti-diagonal matrix: $a_{i,j}=0$ unless $i+j=n+1$. A) When is $A$ diagonalizable (what are the conditions on the $a_{i,n+1−i}$)? B) Find the eigenvalues and eigenvectors ...
4
votes
1answer
176 views

Is $\text{Trace}(e^{XA+A^TX})$ a convex function of X?

Is $\text{Trace}(e^{XA+A^TX})$ a convex function of $X$? $X$ is diagonal and positive definite, $A$ is symmetric negative definite definite. And by the way, what is the best way to solve a problem of ...
1
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2answers
596 views

Factorise a matrix using the factor theorem

Can someone check this please? $$ \begin{vmatrix} x&y&z\\ x^2&y^2&z^2\\ x^3&y^3&z^3\\ \end{vmatrix}$$ $$C_2=C_2-C_1\implies\quad \begin{vmatrix} x&y-x&z\\ ...
1
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3answers
59 views

Why if matrix $A$ is invertible and $A(\mathbf{x-y})=0$ then $\mathbf{x-y}=0$?

When browsing through my algebra textbook on examples of isomorhic linear transformation in one of the proofs there was a statement that if matrix $A$ is invertible and for vectors $\mathbf{x,y}$ it ...
1
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4answers
145 views

Prove that $A^3-3A^2+4A-5=0$ for a given matrix $A$.

Consider the following matrix $A$: $$A= \begin{bmatrix} 0 &1 &-1\\ 1 &1 &1\\ -1 &0 &2\\ \end{bmatrix} $$ $$\text{Prove that }\;A^3-3A^2+4A-5=0$$ I have no idea how to solve ...
0
votes
2answers
67 views

Does the eigenvectors of a sub-block matrix are contained in the eigenvectors of the original matrix?

Given a $2 \times 2$ matrix $A>0$, one of its eigenvectors is $[x_1 \space x_2]^T>0$, and another $3 \times 3$ matrix $B=[A \space C, \space D \space e]$, where $C= [c_1 \space c_2]^T$ and ...
3
votes
0answers
241 views

simple formula for det(AB-BA)?

This is not homework. I was wondering whether there exists a simple formula for $det(AB-BA)$, given two square matrices $A$ and $B$ with coefficients in your favorite commutative ring.
5
votes
1answer
181 views

Moscow puzzle. Number lattice and number rearrangement. Quicker solution?

I have already considered chains of numbers like $4-19, 19-9, 9-22$, to solve the problem and got the answer. However just out of curiosity, can anyone think of a better/quicker solution? (answer ...
3
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2answers
150 views

Transforming a matrix A into a zero matrix using finitely many steps.

Let $A$ be a $m\times n$ matrix whose entries are positive integers. A step consist of transforming the matrix either by multiplying every entry of a row by $2$ or subtracting $1$ from every entry ...
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0answers
105 views

Tools to bound the singular values of a finite sum of random matrices from below?

Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower ...
0
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2answers
62 views

An infinite matrix series

Playing around with my CAS, I found that apparently $$\sum_{n=1}^\infty\begin{bmatrix}1/2 & 1/3\\1/4 & 1/5\end{bmatrix}^n = \begin{bmatrix}29/19 & 20/19\\15/19 & 11/19\end{bmatrix}$$ ...
0
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0answers
52 views

How to interpret some matrix lemmas on Wikipedia - the number 1 vs. the matrix I

I'm reading some lemmas on Wikipedia, eg, the Matrix determinant lemma, and the Sherman-Morrison formula, and both of these formulas have a 1 added to a product of column vectors and matrices. How ...
3
votes
1answer
384 views

Questions about matrix rank, trace, and invertibility,

(a) Prove that a square matrix $T$ of rank one has $\text{tr}(T)=0$ if and only if $T^2=0$. (b) Consider a matrix $A$ of the form $A=aI+T$, where $a\ne0$, $I$ is the identity matrix, and $T$ has ...
0
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1answer
518 views

Can i Find the Matrix from Eigenvalues and Eigenvectors?

If i given eigenvector: $$V_1=\begin{pmatrix} {1\over \sqrt{3}}\\{1\over \sqrt{3}}\\{1\over \sqrt{3}}\end{pmatrix} , V_2=\begin{pmatrix} {1\over \sqrt{6}}\\{-2\over \sqrt{6}}\\{1\over ...
0
votes
2answers
103 views

what is the convex hull of such a matrix cone?

A matrix cone is in the following form: $M: = \begin{pmatrix} 1 \\ x\end{pmatrix}\begin{pmatrix}1 & x^T\end{pmatrix}$ where $x\in F$ , let $F = \{x: x\in [l,u]^n\}$ How to express the convex ...
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1answer
131 views

solve by Gauss-Jordan elimination method [closed]

I am not able to solve this following problem, using Gauss-Jordan elimination method: $$\begin{cases}x_1 -2x_2 + x_3 = 3\\ -2x_1 + x_2 + 3x_3 =2\\ -3x_1 - x_2 + 2x_3 = 3 \end{cases}$$ I would be ...
1
vote
1answer
44 views

If $A \in M_{3\times 3}(\mathbb{R})$ is normal, there is an orthogonal $O$ such that $O^TAO$ is either diagonal or in this form

Given an $n \times n$ normal matrix $A$ over $\mathbb{R}$, show that there is an orthogonal matrix $O$ such that $O^TAO$ is either diagonal or in the form ...