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), ...

learn more… | top users | synonyms (2)

0
votes
1answer
87 views

Converting a matrix to the nearest positive definite matrix

I have a matrix $A = \begin{bmatrix} 634.156 & 0 & 755912.06 \\ 0 & 1426.8604 & 598151.25\\ 755912.06 & 598151.25 & 1.1517e9\\ \end{bmatrix} $ with eigenvalues ...
1
vote
2answers
76 views

Explain this relationship to me and prove it?

Let $T: \mathbb{R}_{\leq 3}[X] \rightarrow \mathbb{R}_{\leq 2}[X]$ be a linear map defined as $T(f(x)) = f'(x)$, and let $\beta$ and $\gamma$ be the standard ordered bases for resp. $\mathbb{R}_{\leq ...
1
vote
2answers
45 views

Question on positive semidefinite ordering of matrices.

We have symmetric, positive definite matrices $ {\bf A}, {\bf B},{\bf C}, {\bf D} \in {\bf R}^{n\times n}$ with $$\bf A \leq \bf B$$ and $$\bf C \leq \bf D$$ which means that the differences $(\bf B ...
3
votes
2answers
51 views

To show a matrix $A=I$ if all eigen values are $1$ and the set $\{A^n:n\in\mathbb{N}\}$ be bounded

$A\in M_n(\mathbb{C})$ with all eigenvalues equal to $1$. Suppose the set $\{A^k:k\in\mathbb{N}\}$ be bounded, then show that $A\equiv I$. I tried from spectral radius formuale ...
0
votes
1answer
44 views

Matrix representation from a linear function

I'm studying an opinion formation model [1] where the main rule is: $p^{(t)}_i = \frac{p^{(0)}_i + \sum_{j \in N(i)} w_{i,j} p^{(t-1)}_j}{1 + \sum_{j \in N(i)} w_{i,j}}$ Now, I'd like to represent ...
0
votes
1answer
37 views

Solving equations with matrices

Say I have $4$ simultaneous equations \begin{cases} 4.3S_1 - P = T \\ 8S_2 - P = T \\ 5.5S_3 - P = T \\ S_1 + S_2 + S_3 = T. \end{cases} I'm trying to solve these in Excel using MINVERSE and MMULT ...
2
votes
1answer
312 views

How to say if Eigenvectors of A are orthogonal or not? without computing eigenvectors

I am give matrix : $$A=\begin{bmatrix} 0&-1 & 2 \\ -1 & -1 & 1 \\ 2 & 1 &0 \end{bmatrix} $$ 1. Without finding the eigenvalues and eigenvectors, determine whether the ...
2
votes
2answers
174 views

Prove that exists matrices $B,C \in \mathcal{M}_{3}(\mathbb{R})$ such that: $A=B^2+C^2$ [closed]

Let $A \in \mathcal{M}_3(\mathbb{R})$. Prove that exists matrices $B,C \in \mathcal{M}_{3}(\mathbb{R})$ such that: $$A=B^2+C^2$$
1
vote
1answer
28 views

Block symmetric implies symmetric?

If a matrix is block symmetric then does that imply it is symmetric? If this claim is true, how can I prove it?
6
votes
5answers
4k views

Why is inverse of orthogonal matrix is its transpose?

So the question is in the title. It's easy to prove when we know that there are real numbers in it and the dot product is standard. But why this works in the general case - when there are complex ...
0
votes
1answer
29 views

Proving $(I+T)^k$ has positive entries for large k

This is mentioned in these slides. A non-negative square matrix $T$ is called primitive if there is a $k$ such that all the entries of $T^ k$ are positive. It is called irreducible if for any$ i, ...
1
vote
0answers
49 views

Proving that $A^2 x=x$ implies $Ax = x$ for a matrix of $n\times n$ [duplicate]

Let $A$ be an $n\times n$ matrix and $x\in \mathbb{R}^n$, both with positive real entries. Prove that if $A^2 x=x$ then $Ax = x$.
0
votes
3answers
131 views

Matrices, determinants, and applications to identities involving Fibonacci numbers

Preamble It is well known that since: $$ \begin{pmatrix} F_{n+1} \\ F_n \\ \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 1 & 0 \\ \end{pmatrix} \begin{pmatrix} F_n & F_{n-1} ...
0
votes
1answer
26 views

Uniqueness of LU decomp for non-singular matrices

Is the LU decomp for invertible matrices unique? I am reading a book that asks to prove it, but I can't because I don't think it makes sense that it could be unique. I looked online and at other posts ...
0
votes
1answer
59 views

Where does the rotation matrix come from?

I looked through many different books, but none of them explain how they derive the matrix below. They just state it as is and move on. Can you explain, please. $ \begin{bmatrix} R(\vec e_1), R(\vec ...
3
votes
2answers
120 views

How to show that $B^{-1}\cdot A^{-1}=B\cdot A$?

I can't find the solution of this problem: Given two $n\times n$ square matrices $A,B$ such that $A^2\cdot B^2=I_n$, show that $B^{-1}\cdot A^{-1}=B\cdot A$. Thanks in advance.
3
votes
1answer
146 views

Relation between Frobenius norm and eigenvalues

I'm considering a stochastic multivariate process, the stability of which implies that all eigenvalues $\lambda_i$, $i = \overline{1,n}$ of a certain square real-valued matrix $A$ lie within the ...
0
votes
1answer
34 views

Bound a Lyapunov storage function

How to effectively bound the following entity to deduce its definite negativeness $\dot{v} = -k_1 e_1^\top A e_1 + k_1 e_1^{\top} A e_2- k_2|e_2|^2$, with A a positive definite square matrix, $e_1$ ...
2
votes
2answers
98 views

Derivative of a vector

Let $p, v :$ real, positive $1\times n$ vectors, $c^T:$ real, non - negative $n\times 1$ vector, $I:$ the identity matrix. Assume that the following relationship holds true: $$p(v) = v\cdot ( I - ...
0
votes
1answer
41 views

Composition of linear transformations different

In $\mathbb{R} ^3$ a base $A=\{\alpha_1,\alpha_2,\alpha_3\}$ and in $\mathbb{R}^2$, $B=\{\beta_1,\beta_2\}$ are given, where $\alpha_1=[1,1,1],\alpha_2=[1,1,0],\alpha_3=[1,0,0]$ and ...
0
votes
2answers
34 views

Find matrix such that:

Find matrix such that: $$\begin{pmatrix} 3 & 2 & 3 \\ 3 & 6 & 3 \\ 1 & 2 & 4 \end{pmatrix} X+ \begin{pmatrix} 1 & 1 & 0 \\ 1 & -1 & 2 \\ 1 & 0 ...
0
votes
1answer
62 views

Matrix norm properties: inequalities

Is the fololwing relationship always true: $x^{\top} (A - \frac{1}{2} \|A\|_F \, I_3) x > 0$, knowing that matrix A is definite positive?
1
vote
1answer
59 views

A question from Golan's linear algebra:

A question from Golan's linear algebra: Let $A\in M(n,\mathbb R)$ (which denotes the set of all $n\times n$ matrices, for some $n\geq 2$) be symmetric. Does there exist a symmetric matrix $B$ such ...
2
votes
2answers
192 views

Simulating simultaneous rotation of an object about a fixed origin given limited resources.

Sorry if the title is a bit cryptic. It's the best I could come up with. First of all, this question is related to another question I posted here, but that question wasn't posed correctly and ended ...
1
vote
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 ...
1
vote
3answers
86 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
votes
0answers
184 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 ...
1
vote
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
votes
0answers
85 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 ...
6
votes
1answer
156 views

If $f(AB) =f(A)f(B)$, then $A$ is inversible iff $f(A)\neq 0$

Let $f:\mathscr M_n(\mathbb K) \to \mathbb K$ be a non constant function such as $f(AB) = f(A)f(B)$ for all $A,B$ in $\mathscr M_n(\mathbb K)$. The question is to show that $M\in GL_n(\mathbb K)$ iff ...
0
votes
1answer
30 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
votes
1answer
58 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 ...
7
votes
1answer
437 views

Eigenvalues of Kronecker Product

Maybe it's simple but I can't see the solution of this problem (Russell Merris, Multilinear Algebra, CRC Press, 1997, chapter 6, p.202, exercise 4): Let $\lambda_1,\ldots,\lambda_p$ be the ...
1
vote
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 ...
1
vote
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. ...
1
vote
2answers
113 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 ...
1
vote
6answers
154 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
votes
1answer
183 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?
0
votes
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, ...
0
votes
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 ...
0
votes
0answers
143 views

Is a family of commuting self adjoint operators simultaneously diagonalizable?

Let $V$ be a finite-dimensional inner product space over $\mathbb{R}$. Let $\mathscr{A}$ be a family of self-adjoint operators on $V$ such that $ST=TS$ for all $S,T\in \mathscr{A}$. Then, ...
1
vote
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 ...
1
vote
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
161 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 ...
1
vote
3answers
82 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 ...
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?
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 ...
1
vote
1answer
44 views

LU factorisation

I am studying the LU factorisation. What I have learned is that with this technique we start with a matrix $A$ and result into two matrices $L$ and $U$ where $L$ is a Lower Triangular matrix and $U$ ...
5
votes
2answers
963 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 ...
0
votes
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