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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4
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2answers
342 views

What is the Jordan canonical form of $A^{2}$ if we know that of $A$?

Let $A\in M_{n}$ have Jordan canonical form $J_{n1}(\lambda_{1})\oplus\cdots\oplus J_{nk}(\lambda_{k})$. If $A$ is non-singular($\lambda\neq 0$), what is the Jordan canonical form of $A^{2}$? I can ...
1
vote
0answers
410 views

Rotate a 2D square by x degrees

Say I have a square represented by the following matrix: 0 0 10 0 10 10 0 10 0 0 Now, what would I multiply this matrix by to rotate the square (or what ...
4
votes
2answers
74 views

For what integers can this matrix be generated?

We call a positive integer $n$ "good" if there exists a $n \times n$ matrix such that: 1) Each element is either $0$ or $1$. 2) Sum of elements in each row is distinct. 3) Sum of ...
1
vote
2answers
325 views

matrix calculus equation - least squares minimization

Find a closed-form solution for vector $A$ minimizing the expression$$\frac{1}{2}\left|W(\Phi A-F)\right|^2$$ where $W$ is non-singular diagonal, $\Phi$ is full rectangular matrix, and $A$, $F$ are ...
0
votes
1answer
263 views

Determinant of order 4 question

I don't get this, maybe you can explain me how you solve it "Write down all the terms appearing in the determinant of order 4 which have a minus sign and contain the factor a23" Thanks a lot
4
votes
1answer
194 views

Prove $\text{rank}(A) \geq \frac{(\text{tr}(A))^2}{\text{tr}(A^2)}$ when $A$ is Hermitian

If $A \in \mathbb{C}^{n \times n}$ is a non-zero Hermitian matrix, I need to show that $$\text{rank}(A) \geq \frac{(\text{tr}(A))^2}{\text{tr}(A^2)}$$ and reason out when the equality is attained? ...
3
votes
1answer
2k views

When is a positive semi-definite matrix A positive definite?

Does it has something to do with the determinant of A? I saw two seperate websites - one which claims that when the determinant of A is zero, and the other claims that when the determinant of A is not ...
5
votes
3answers
723 views

Matrix multiplication: is C(AB) the same as (CA)B?

I would like to show that $(\mathbf{A} \mathbf{B})^{-1} = \mathbf{B}^{-1} \mathbf{A}^{-1}$, where $\mathbf{A}$ and $\mathbf{B}$ are $N \times N$ square matrices. I think that this can be done as ...
1
vote
1answer
101 views

Why does this algorithm work?

Given a matrix, $P$, why does finding its eigenvalues, say they are $\{\lambda_1, \lambda_2\}$ then the general form of $p_{ij}^{(n)}=A_{ij}\lambda_1^n+B_{ij}\lambda_2^n$? Thanks. Added: Context: $P$ ...
1
vote
1answer
43 views

An Embedding of $PGL_n \Bbb C$

I have a question about the projective general linear group. How does one realize it as a matrix group? Specifically, what is an embedding of $PGL_n \Bbb C \to GL_k \Bbb C$ for some $k$? In this case, ...
9
votes
1answer
787 views

Do these matrix rings have non-zero elements that are neither units nor zero divisors?

Let $R$ be a commutative ring (with $1$) and $R^{n \times n}$ be the ring of $n \times n$ matrices with entries in $R$. In addition, suppose that $R$ is a ring in which every non-zero element is ...
3
votes
1answer
138 views

A special eigenvalue problem

Let $A\in\mathbb{R}^{9\times 9}$, and $I_3\in\mathbb{R}^{3\times 3}$ is the identity matrix. Now I am going to find a matrix $\Lambda\in\mathbb{R}^{3\times 3}$ and $x\in\mathbb{R}^9$ such that ...
1
vote
3answers
2k views

Find values of a and b that make the system consistent

In the augmented matrix: $$\left(\begin{array}{rrr|r} 1 &-2 &4 & 7\\ 0 &a^2 - 1& a & 3\\ 0 &0 &b & -3 \end{array}\right).$$ How do I determine values for $a$ ...
6
votes
1answer
816 views

Cross product of vectors as a determinant: valid matrix operation?

"The definition of the cross product can also be represented by the determinant of a formal matrix." —Wikipedia This seems like a hack to me—something of much practical use but ...
2
votes
1answer
52 views

Relationship between $|a_{i,j}|$ and $|\alpha^{|i-j|} a_{i,j}|$

What is the relationship between the determinants of the square matrices of equal dimensions $\mathbf{A}$ and $\mathbf{B}$ where each element of $\mathbf{B}$ is equal to the corresponding element of ...
0
votes
1answer
99 views

Eigenvalues problem

Assume one knows the eigenvalues $(\lambda_i)$ of a real matrix $M$ of size $n \times n$. Let $b$ be a vector in $\mathbb{R}^n$ and construct the following matrix $J$ by: $$J_{ij} = M_{ij}b_j$$ Can we ...
4
votes
1answer
668 views

If a matrix is written with a double bar instead of square brackets, is there any significance?

Shilov's Linear Algebra writes a matrix with two bars on each side rather than square brackets. I didn't find any mention of it with a quick Google search, and I can't see any other examples this ...
0
votes
1answer
707 views

Eigenvalues problem in Mathematica

The simple problem eigenvalues problem defined as: $A-(x^2)*B = 0$, where $A$ and $B$ are matrices. How to use command Eigenvalues[] in Mathematica to find $x$, ...
2
votes
1answer
189 views

Bounding determinants of the following form from above

To bound a determinant of a matrix from above it's quite common to apply Hadamard's inequality. Unfortunately, in the following problem Hadamard's inequality isn't good enough. Are there other methods ...
8
votes
2answers
743 views

Span of Permutation Matrices

The set $P$ of $n \times n$ permutation matrices spans a subspace of dimension $(n-1)^2+1$ within, say, the $n \times n$ complex matrices. Is there another description of this space? In particular, ...
2
votes
1answer
58 views

Expressing the number of non-zero rows of a binary matrix as a polynomial

Let $X$ be an $m\times n$ matrix, such that all of its elements are binary, i.e., for every $1\leq i\leq m$ and $1\leq j\leq n$ holds $x_{ij}=(X)_{ij}\in\{0,1\}$. Is there any possible way to express ...
1
vote
2answers
94 views

The vectorial ODE $ D\left( X \right) = AX$, with $A$ a constant coefficient matrix

Let $ X(t) $ a vector function in $ R^n $, and let A be an $ n \times n $ matrix with constant coefficients. Let us use $ D(X) $ to denote the derivative with respect to $t$ of the function $X$ (this ...
1
vote
1answer
144 views

The form of a $2\times2$ real orthogonal matrix

I am reading about unitary matrices in Horn and Johnson's Matrix Analysis. On page 68, the exercise asks, letting $T(\theta)=\begin{pmatrix}\cos\theta & \sin\theta \\-\sin\theta & \cos\theta ...
1
vote
2answers
289 views

State transform from one state space representation to another

I have a state space representation, system S1, in the form of: $$ \frac{dx}{dt} = Ax + bu $$ $$y = c^Tx$$ This system is transformed with the state transform $$x=T z$$ into the system S2: $$ ...
2
votes
2answers
133 views

Choosing set of best estimators for linear least squares

I have a measured experimental dataset which is well approximated by the sum of several basis functions in linear combinations. Linear least squares of course gives me the optimal weight for each ...
1
vote
1answer
134 views

Kronecker Product

Is this right $$\mathbf{A}\left(\mathbf{B}\otimes\mathbf{C}\right)\mathbf{D}=\left(\mathbf{A}\mathbf{B}\mathbf{D}\otimes\mathbf{C}\right)$$ Thanks in advance for your help.
3
votes
1answer
579 views

How to obtain a possible state space representation of this 2nd order transfer function?

I have this 2nd order transfer function: $$G(s) = \frac{2}{s} + \frac{1}{s+2}$$ And I need to find a possible state space representation in the form of: $$ \frac{dx}{dt} = Ax + bu $$ $$y = c^Tx$$ ...
5
votes
3answers
1k views

Understanding direct sum of matrices

I read the definition of direct sum on wikipedia, and got the idea that a direct sum of two matrices is a block diagonal matrix. However this does not help me understand this statement in a book. In ...
1
vote
1answer
292 views

How to obtain the state matrix of this trajectory?

Continuous-time LTI case. I have a problem getting the state matrix of this trajectory. One element of the state matrix is known. $$ A = \begin{pmatrix} a & 4 \\c & d \end{pmatrix} $$ I ...
2
votes
1answer
74 views

Prove K is a real matrix

This is a question from my friend. It should be easy. But I have no knowledge about complex matrix. Let $A,K$ be two invertible complex n-by-n matrices satisfying the following conditions: $A' ...
3
votes
1answer
159 views

Is this determinant equal to 1

Let $V$ be a finite dimensional vector space over $\mathbf{C}$ with a hermitian inner product. Let $e=(e_1,\ldots,e_n)^t$ and $f=(f_1,\ldots,f_n)^t$ be orthonormal bases for $V$. There is a matrix ...
1
vote
1answer
490 views

elementary matrices and row operations

I am studying for an exam tomorrow and this is one of the problems given. The instructor gave the solution but I do not understand how he found the solution. The question is "write down the elementary ...
2
votes
0answers
306 views

Inverse function theorem for matrices and vectors

I am trying to figure out why a result invoked in a proof might be proved. Basically, the following equation holds \begin{equation*} ...
5
votes
2answers
230 views

Help with commutative property of matrices problem

Given matrices A and B, where that AB = A + B, prove AB = BA. I keep coming up with AB = AB. It seems like basic algebra, but for the life of me, I'm getting nowhere :/. Someone help please?
2
votes
1answer
605 views

Cholesky decomposition and permutation matrix

Dear all, Let's assume that I have a symmetric matrix $\Sigma$ and a permutation matrix $A$. Is there a relationship between the Cholesky decompositions of $\Sigma$ and of $A^T \Sigma A$ ? Many ...
1
vote
1answer
452 views

Iterating over all matrices with fixed row and column sums

Can anyone suggest an algorithm for iterating once through all matrices with non-negative integer entries which are $2$ by $n$ with fixed row sums ($r_1$ and $r_2$) and fixed column sums ($c_1, c_2, ...
3
votes
1answer
163 views

Why is the determinant of the following matrix zero

Let $a_1,\ldots,a_n$ be complex numbers and let $b_1,\ldots,b_n$ be complex numbers. Let $A$ be the matrix whose $(i,j)$-th entry is $A_{ij} = a_i b_j$. Then I think $\det A = 0$ when $n>1$. This ...
3
votes
2answers
192 views

eigenvalues of a matrix under a polynomial

Let $A\in M_{n}(\mathbb{C})$, and suppose $A$ has eigenvalues $\lambda_{1},\lambda_{2},...,\lambda_{n}$ counting multiplicities. Let $f(\cdot)$ be a polynomial. How can we show that $f(A)$ has ...
0
votes
0answers
86 views

How to get all matrices of the given one by only using swapping?

I've created a program which solves any matrix (rows contain one element more than columns) by using Gaussian method. Matrix can be as small as $6\times5$ and as large as $101\times100$. So far I have ...
1
vote
0answers
74 views

The largest size of a subgroup of a general linear group with the property that all elements have the first $k \times k$ block invertible

I wanted to see if there is any connection between the invertibility of a matrix and the invertibility of a particular block of the matrix. Particularly I want to find out the largest size of a ...
1
vote
3answers
165 views

Good computer programs for dealing with sparse matrices

I'm looking for some FOSS/GPL programs (or Python libraries) for dealing with sparse matrices. I haven't found much online about these. Can someone please point me in the right direction?
4
votes
3answers
380 views

What are the requirements for a rotation matrix?

Generally speaking, what are the necessary and sufficient properties of a matrix to make it a rotation matrix? Is det(A) = 1 enough?
3
votes
3answers
322 views

notation for real matrices

Is this a valid notation for real $m \times n$ matrices: $\mathbb{R}^{m,n}$. $m$ and $n$ are known. If it is not, then what would be the right notation for the set of such matrices?
1
vote
2answers
480 views

Conditions for Stationary Distributions in Markov Chains?

The book by Durrett "Essentials on Stochastic Processes" states on page 55 that: If the state space S is finite then there is at least on stationary distribution. How can I find the ...
6
votes
4answers
216 views

Given $A^2+cA+cI=0$, how to find inverse of $A+(c-1)I$?

Suppose a square matrix $A$ such that $A^2+cA+cI=0$ for all $c \in \mathbb{Z}$. How can I show that $A+(c-1)I$ is invertible and find its inverse? I started off this way: $A+(c-1)I = A+cI-I$ Then ...
2
votes
3answers
355 views

Optimizing Cholesky factorization for multiple sparse matrices with same nonzero pattern

I'm using a Cholesky factorization to solve the linear step in a nonlinear system of equations (nonlinear finite element analysis). In the PETSc library, one can specify a parameter for ...
2
votes
1answer
907 views

Subordinate matrix norm

I have the following matrix norm: $$\Vert A \Vert = \max_{1\leq i, j\leq n} \vert a_{ij} \vert \>.$$ I have to decide if this is a subordinate matrix norm or not. I have tried to use the ...
0
votes
1answer
97 views

If $A\vec{x}=\vec{b}$ and $B\vec{x}=\vec{b}$ inconsistent, then $(A+B)\vec{x}=\vec{b}$ inconsistent?

The matrices $A$ and $B$ are of $n \times m$. Suppose there is a $\vec{b}$ such that both $A\vec{x}=B\vec{x}=\vec{b}$ have infinitely many solutions. Is it true that $(A+B)\vec{x}=\vec{b}$ also has ...
3
votes
2answers
111 views

Proving $S+T$ is a subspace of $\mathbb{R}^{n}$

$S$ and $T$ are subspaces of $\mathbb{R}^{n}$ and is defined as $S+T = \{v+w \mid v \in S \; and \; w \in T\}$. I need to show that $S+T$ is a subspace of $\mathbb{R}^{n}$. Instinctively, $S+T$ is ...
1
vote
2answers
1k views

2D elastic collision equation: How does it work?

Hey so I recently started learning physics, and came upon this wonderful site that taught me how to calculate 2D collisions between two circles. The only part I'm confused about is how the velocity ...