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
votes
1answer
61 views

Proving that a matrix is diagonalizable

Let $ T $ be the linear operator on $ \Bbb R^3 $ which is represented by the matrix $$ A = \begin{bmatrix} 6 & -3 & -2 \\ 4 & -1 & -2 \\ 10 & -5 ...
1
vote
1answer
81 views

Help with a linear transformation problem

This is my matrix $A$: $$\begin{bmatrix} 1 & -3 & \hphantom{-}2 & \hphantom{-}5 & \hphantom{-}2 & \hphantom{-}0\\ 0 & \hphantom{-}0 & \hphantom{-}1 & -2 & ...
-4
votes
1answer
79 views

Can we conclude that this matrix is definite positive? [duplicate]

Let $A$ be a $n\text{-by-}m$ matrix. Suppose that columns of $A$ are linearly independent. Can we conclude that $A^TA$ is definite positive? Could you help me with proof? Thanks.
3
votes
2answers
60 views

Is this fact about matrices and linear systems true?

Let $A$ be a $m$-by-$n$ matrix and $B=A^TA$. If the columns of $A$ are linearly independent, then $Bx=0$ has a unique solution. If is true, can you help me prove it? If is false, could you give a ...
0
votes
1answer
63 views

Properties of the difference of two rotation matrices

Let $R_{1},R_{2}\in{SO}(3)$, be two rotation matrices. It must be due my ignorance but how can I calculate the rank of the difference of these two rotation matrices, i.e. $\mbox{rank}(R_{2}-R_{1})$?. ...
3
votes
1answer
78 views

How to analyze $A\cdot (\mathop{\rm tri} A)^{-1}$?

Suppose I have an upper triangular square matrix $A$, and $\mathop{\rm tri}A$ is the operator which takes the tridiagonal part of $A$. Assuming that we know ${\rm tri}(A)$ is invertible. I am trying ...
0
votes
1answer
322 views

Nullity of Kernel, Range of transpose

Define the linear transformation $T$ by $T(x) = Ax$, where $A=\left(\begin{matrix} \frac{9}{10} & \frac{3}{10}\\ \frac{3}{10} & \frac{1}{10} \end{matrix}\right)$. Find (a) $\ker(T)$, (b) ...
4
votes
3answers
422 views

Matrix which commutes with permutation matrix

I'm trying to show that if $A$ commutes with all $3\times 3$ permutation matrices, then $A$ has to be of the following form: $ A = \begin{pmatrix} a & b & b \\ b & a & b \\ b & b ...
2
votes
2answers
324 views

The second largest eigenvalue for Perron-Frobenius matrix

The Perron-Frobenius theorem is about the largest eigenvalue and eigenvector of a non-negative (irreducible) matrix. My question: Is there any estimation of the difference between the first and ...
4
votes
1answer
581 views

Cholesky for Non-Positive Definite Matrices

I am trying to approximate a NPD matrix with the nearest PD form and compute its Cholesky decomposition. I know that the usual method is to perform an eigenvalue decomposition, zero out the negative ...
0
votes
2answers
111 views

Inverse of Matrix

I have a doubt. When finding inverse of matrix, Let us take, $A$ be a matrix, and $A^{-1}$ exists, then to find $A^{-1}$, we write A=IA and we will apply a sequence of row operation(can we do ...
2
votes
0answers
34 views

Computing a particular finite set of quaternion matrices.

Let $B = \left(\frac{-1,-11}{\mathbb{Q}}\right)$ be a choice of quaternion algebra ramifying at $11$ and consider the maximal order ...
1
vote
1answer
155 views

Condition on a matrix sum with equal determinant and trace

Let $n$ be a positive integer, $J$ the matrix of all ones and $Q$ a symmetric positive semidefinite matrix such that $\det(nI-Q) = \det(Q+J)$ $\rm{tr}(nI-Q) = \rm{tr}(Q+J)$ and also $nI-Q \ne ...
4
votes
1answer
1k views

Are there Taylor series for functions of a matrix?

Say you have a scalar function $f(x,A)$ of a vector $x$ and a matrix $A$. Does there exist a Taylor series of sorts for the matrix $A$? I was thinking naively that this would simply be of the form ...
1
vote
2answers
80 views

Prove $A\in \mathbb R^{n\times n}$ is antisymmetric iff…

Prove that $A\in \mathbb R^{n\times n}$ is antisymmetric iff $ \forall v\in\mathbb R^n:\langle v,Av\rangle=0 $ $\langle \cdot,\cdot\rangle$ is just the dot product. I'm a little stumped by this ...
0
votes
1answer
45 views

A matricial process to assign different values to elements of a diagonal matrix

Consider having vector $$v = \begin{pmatrix} v_1\\ v_2\\ \vdots\\ v_n \end{pmatrix}$$ Consider the final result: $$ V = \begin{pmatrix} v_1 & 0 & \dots & 0\\ 0 & v_2 & \dots ...
2
votes
5answers
140 views

Finding the limit of a matrix

Suppose that $A=\begin{pmatrix}4&1 & 5\\ 2& 7& 1\\ 2& 2& 6\end{pmatrix}.\;$ How can I find $\;\displaystyle \lim_{n\to\infty}A^n$? What theorem(s) should I use to solve this? ...
1
vote
1answer
95 views

Determining $u=v \times w$ using the cross product

Let $v = (3,0,0)$ and $w=(0,1,-1).$ Determine $u = v \times w$ using the geometric properties of the cross product rather than the formula. What are the possible angles $x$ between two unit vectors ...
4
votes
2answers
531 views

An operator that commutes with another operator $T$ with distinct characteristic values is a polynomial in $T$

I'm trying to solve some problems in Hoffman and Kunze and I'm kind of stuck on this one. This is 6.5.3 on Hoffman and Kunze. Here is the question: Let $T$ be a a linear operator on an ...
4
votes
2answers
779 views

I am not sure how to calculate this norm?

I have the following matrix: $$A= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end{bmatrix} $$ What is the norm of $A$? I ...
2
votes
2answers
183 views

Optimize a log det function with respect to a matrix, and the saddle point analysis

Suppose I want to to find the local minima of a logdet function $\mathcal{L}$ with respect to a Matrix $\mathbf{A}$, $$ \mathcal{L} = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - ...
1
vote
1answer
64 views

Help with anti-image matrix

First of all, I am very sorry but I don't know the mathematics terminology in English, so I'll try to explain as good as i can but i will probably do some mistakes since it's not my native language. ...
2
votes
0answers
76 views

Ideals (one-sided ideals) of $n×n$ upper triangular matrices

Is there any characterization of ideals (one-sided ideals) of $n\times n$ upper triangular matrices? I have just seen in monthly journal about $2 \times 2$ matrices in the below article Left and Right ...
1
vote
1answer
342 views

Find an orthogonal matrix

I wonder if i can find for 2 unit vectors $v,w$ only one orthogonal matrix $Q$ such that $Qv=w$ is there any proof for that?
0
votes
2answers
55 views

Identity Matrix Question

Let $A$ and $B$ be a $n$ by $n$ matrix such that $A^2=I, B^2=I$ and $(AB)^2=I$. Prove that $AB = BA.$ Any hints on how to attempt this question? I'm stuck. But my first approach would be, $A^2B^2=I ...
5
votes
2answers
477 views

How does one obtain the Jordan normal form of a matrix $A$ by studying $XI-A$?

In our lecture notes, there's the following example problem. Find a Jordan normal form matrix that is similar to the following. $$A=\begin{bmatrix}2 & 0 & 0 & 0\\-1 & 1 & 0 ...
2
votes
1answer
89 views

Basis of kernel and image of a linear transformation - verification

The transformation matrix I found is: $$\begin{pmatrix} 1 & -1 \\ 1 & 1 \\ 0 & 0\end{pmatrix}$$ Is this how a basis for $\ker$ and $\mathrm{im}$ is calculated? $$\begin{pmatrix} 1 & ...
5
votes
2answers
481 views

$A$ is some fixed matrix. Let $U(B)=AB-BA$. If $A$ is diagonalizable then so is $U$?

This is from Hoffman and Kunze 6.4.13. I am studying for an exam and trying to solve some problems in Hoffman and Kunze. Here is the question. Let $V$ be the space of $n\times n$ matrices over a ...
0
votes
1answer
104 views

Could someone help me to prove that this symmetric matrix is definite positive?

Let $a_{ij}\in\mathbb{R}$ for all $i,j\in\{1,...,n\}$ and $m\in\mathbb{N}$. Consider the matrix below. $$B=\begin{bmatrix} \sum_{k=1}^n(a_{1k})^2 & \sum_{k=1}^na_{1k}a_{2k} & \cdots & ...
2
votes
1answer
105 views

I would like a hint in order to prove that this matrix is positive definite

Let $a_{ij}$ be a real number for all $i,j\in\{1,...,n\}$. Consider the matrix below. $$B=\begin{bmatrix} \sum_{k=1}^n(a_{1k})^2 & \sum_{k=1}^na_{1k}a_{2k} & \cdots & ...
5
votes
2answers
705 views

Why must the determinant of a matrix with with integer entries be an integer?

Why must the determinant of a matrix with integer entries be an integer? Note: I know what a determinant of a matrix is, not sure how to explain this question. Is that because if the matrix is made ...
3
votes
1answer
283 views

Parallel rank computation of huge sparse matrices over the binary field

I'm looking for a parallel algorithm to compute ranks of huge, sparse binary matrices over F2 (say 10^5 x 10^5 with 10^7 ones in total). Currently I'm doing this by packing 64 bits in a long and ...
2
votes
0answers
51 views

counting symmetric nilpotent matrices

In a recent paper [ Counting symmetric nilpotent matrices , by A. Brouwer], the author states that the number of 3x3 symmetric nilpotent matrices over the field of q elements is given by the ...
2
votes
2answers
45 views

How to calculate these matrices? - explanation of the procedure

Can you please help me solve this problem? I have got these matrices $A=\left(\begin{array}{ccc} 2 & 1 & 0 \\ 0 & 1 & 2 \\ 3 & 0 & 1 \end{array}\right) $, ...
2
votes
1answer
39 views

Regularity of matrix with coefficients from GF$(p)$

I have matrix $A$ (its size is $n \times n$) with coefficients from GF($p$), where $p$ is prime. How can be proven that this matrix has all lines linearly independent iff det$(A)\neq 0 $(mod $p$). I ...
3
votes
0answers
76 views

The intuition behind a matrix of a Hamiltonian?

We have derived an elegant partition function for a problem which resembles a quantized model taking the particles to be Bosons. The related Hamiltonian for every $i$th ensemble is there: ...
2
votes
0answers
253 views

Pseudo inverse of matrix: SVD vs $A^{T}(A.A^{T})^{-1}$

For a C++ implementation I have to calculate Moore Penrose Inverse (AKA pseudo inverse) of non squared matrices. I was wondering ...
4
votes
1answer
143 views

Examples of how to calculate $e^A$

I'm trying to learn the process to discover $e^A$ For example, if $A$ is diagonalizable is easy: $$A =\begin{pmatrix} 5 & -6 \\ 3 & -4 \\ \end{pmatrix}$$ Then we ...
2
votes
2answers
80 views

Is there a quick way to compute the matrix whose column space is the basis of the null space of another matrix?

Is there a quick way to compute the matrix whose column space is the null space of another matrix? I can do this by hand, but if I wanted a computer to do it, is there a quick, efficient way for me ...
2
votes
1answer
99 views

Proving $\|e^A\|\le e^{\|A\|}$

I'm trying to prove this inequality: $\|e^A\|\le e^{\|A\|}$, where $A$ is a matrix and $\|A\|:=\sup_{|x|=1} |Ax|$. My attempt of solution: Since $e^A:=I+A+A^2/2!+A^3/3!+\ldots$ we have ...
2
votes
0answers
43 views

Meaning of nonlinear vectorial equation

I am trying to apply some methods in a paper and I have to solve the following fixed point equation from Proposition VIII.4.3 in Asmussen (2000): $$\mu_+ =\mu ...
4
votes
1answer
287 views

Matrix Chain Multiplication?

The following are questions about using dynamic programming for matrix chain multiplication. Pseudocode can be found in the Wikipedia article on matrix chain multiplication. 1) Why is the time ...
2
votes
3answers
468 views

What are the dimensions of the product of two matrices?

A simple question is a (5x2)*(2x5) = a (5x5) matrix?
5
votes
1answer
237 views

Let A and B be $n \times n$ real matrices with same minimal polynomial.

Let $A$ and $B$ be $n \times n$ real matrices with same minimal polynomial. Then (i) $A$ is similar to $B$. (ii) $A-B$ is singular. (iii) $A$ is diagonalizable if $B$ is so. (iv) $A$ and $B$ ...
3
votes
0answers
33 views

Squares of random binary matrices

Are there results/techniques pertaining to the analysis of squares of random matrices ? More specifically, let $A$ be an $n\times n$ matrix such that each entry is $1$ or $-1$ independently and with ...
6
votes
2answers
1k views

How to show if $ \lambda$ is an eigenvalue of $AB^{-1}$, then $ \lambda$ is an eigenvalue of $ B^{-1}A$?

Statement: If $ \lambda$ is an eigenvalue of $AB^{-1}$, then $ \lambda$ is an eigenvalue of $ B^{-1}A$ and vice versa. One way of the proof. We have $B(B^{-1}A ) B^{-1} = AB^{-1}. $ Assuming $ ...
0
votes
0answers
47 views

interpolation and Vandermonde

Looking at a problem of interpolation, I find a Vandermonde type matrix. To be precise I consider the following, let $$A(z)= \sum_{i=1}^p \sum_{j=2}^{n_i+1}\frac{a_{i}^j}{(z-z_i)^j}$$ where the $z_i$ ...
4
votes
1answer
56 views

solvable subalgebra

I want to show that a set $B\subset L$ is a maximal solvable subalgebra. With $L = \mathscr{o}(8,F)$, $F$ and algebraically closed field, and $\operatorname{char}(F)=0$ and $$B= ...
10
votes
1answer
181 views

$\mathcal{O}(n,\mathbb R)$ spans $\mathcal{M}(n,\mathbb R)$

Let $n\geq 3$. One can show that the orthogonal group of degree $n$ over the real field, $\mathcal{O}(n,\mathbb R)$, spans the entire vector space of real $n\times n$ matrices, $\mathcal{M}(n,\mathbb ...
1
vote
2answers
57 views

Any neat solution for determinants?

I have been struggling to find the solution of the following system, though I do not want to write everything explicitly. I know that there is some trick which might be useful, but I could not find ...