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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0
votes
1answer
134 views

Count the number of rational canonical form&find similarity classess

For a finite field $F=F_q$ having $q=p^d$ elements ($p$ a prime integer), compute the number of similarity classes in the vector space $M_n(F)$ of $n\times n$ matrices over $F$. (Maybe count the ...
0
votes
1answer
79 views

equivalent in cauchy integral for matrices

I don't know why $(zI-A)^{-1} = \frac{1}{z} \sum_{k=0}^\infty \frac{A^k}{z^k}$ in a link!
2
votes
0answers
67 views

How many rotation matrices with simple rational entries

This is a follow-on from this earlier question, which asked for examples of simple rotation matrices. I'm interested in rotation matrices whose entries are simple rational numbers, because these are ...
6
votes
4answers
154 views

Simple examples of $3 \times 3$ rotation matrices

I'd like to have some numerically simple examples of $3 \times 3$ rotation matrices that are easy to handle in hand calculations (using only your brain and a pencil). Matrices that contain too many ...
1
vote
5answers
197 views

Definition of determinant [closed]

Determinant is a certain function from the set of all $n\times n$ matrices to the set of scalars. How is the determinant defined? What characterizes the determinant function?
0
votes
1answer
60 views

matrix representation of $f$ with respect to the union of two ordered bases for $\ker f$ and $\ker (f-id_V)$

Can you help me with this problem? Let $f:V\longrightarrow V$ be a linear tansformation such that $f\circ f = f$. Let $B'$ be an ordered basis for $\ker f$ and $B''$ be an ordered basis for $\ker ...
2
votes
1answer
48 views

Would this be a free variable?

Let's say I have the RREF matrix $$ A= \begin{bmatrix} 1 & 3 & 0 & 0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}$$ If i'm looking for the solution of this ...
2
votes
2answers
174 views

How to express a matrix as a product of two symmetric matrices?

Let $A$ be a matrix and $J$ its Jordan canonical form. How can one express $A$ as a product of two symmetric matrices? I expressed $J$ as a product of two symmetric matrices: block by block in the ...
5
votes
2answers
185 views

Matrices such that $A^2=A$ and $B^2=B$

Let $A,B$ be two matrices of $M(n,\mathbb{R})$ such that $$A^2=A\quad\text{and}\quad B^2=B$$ Then $A$ and $B$ are similar if and only if $\operatorname{rk}A = \operatorname{rk}B$. The first ...
5
votes
3answers
614 views

Matrix derivative $(Ax-b)^T(Ax-b)$

I am trying to find the minimum of $(Ax-b)^T(Ax-b)$ but I am not sure whether I am taking the derivative of this expression properly. What I did is the following: \begin{align*} \frac{\delta}{\delta ...
0
votes
1answer
38 views

Is the matrix defined by $\bar{K}_{ij}=f(x_i)f(x_j)$ for a real valued function $f$ semi-positive-definite?

Let $f:\mathbb{R}^m \rightarrow\mathbb{R}$ be a real valued function and define $K: \mathbb{R}^m \times \mathbb{R}^m \rightarrow \mathbb{R}$ by $K(x,y)=f(x)f(y)$. For any vectors $x_1,x_2,...,x_n ...
25
votes
9answers
2k views

Are most matrices invertible? [duplicate]

I asked myself this question, to which I think the answer is "yes". One reason would be that an invertible matrix has infinitely many options for its determinant (except $0$), whereas a non-invertible ...
18
votes
1answer
207 views

How many pairs of nilpotent, commuting matrices are there in $M_n(\mathbb{F}_q)$?

As a follow-up to this question, I've been doing some work counting pairs of commuting, nilpotent, $n\times n$ matrices over $\mathbb{F}_q$. So far, I believe that for $n=2$, there are $q^3+q^2-q$ ...
0
votes
1answer
77 views

Find a counter example to the claim [closed]

I have a problem: Find a counter example to the claim: If A^2 = I, then A = I Thanks
1
vote
1answer
70 views

Show that $M$ is a submonoid of the group of $2\times 2$ matrices of integers mod $13$.

Define $$M=\left\{\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}\in(\Bbb Z/13)_{22}:a_{12}=0\right\}\;.$$ (This is the set of $2\times 2$ matrices with integers mod $13$ with the ...
0
votes
1answer
77 views

Free modules and free submodules. [duplicate]

Just finished my semester in advanced abstract algebra and there was one question that I could not answer. Let $R$ be commutative and let $(e_1,...,e_n)$ be a base for $R^{(n)}$. Put ...
0
votes
0answers
103 views

A=UL Doolittle method

A - symmetrical positively defined, matrx, 3-diagonal Make a modified Doolittle method A=UL U - upper triangular matrix L - lower triangular matrix with ones on the main diagonal I have to work on ...
1
vote
1answer
449 views

LU decomposition with zeros on diagonal

How to do LU decomposition with unit lower triangular matrix L, in case a decomposed matrix has zeros on diagonal? This is obviously possible for positive defined matrix. For example suppose instead ...
0
votes
3answers
55 views

Getting $x,y$ position on an image based on given value

This should be simple but my math skills are really bad ... I have an image of 36 images (6 by 6 matrix). These small images are 36 instances of a direction arrow (like from Google maps GPS), each ...
1
vote
1answer
68 views

Is this Jordan decomposition possible?

Is this Jordan form possible? $$J=\begin{pmatrix} \lambda & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & \lambda & 0 & 0 & 0 & 0 & 0\\ 0 ...
8
votes
2answers
116 views

Matrix exponential: $e^A\ge 0\iff a_{ij}\ge 0$ when $i\neq j$

Let $A$ be a $n$ by $n$ matrix. Prove that that $$a_{ij}\ge 0 \text{ whenever }i\neq j\iff e^A\text{ has all entries }\ge 0.$$ I'd like just a hint for now please.
0
votes
0answers
866 views

Linear algebra Pruning X, linear combinations and Spans

Consider the following subset of the vector space $\mathbb{P}_4(\mathbb{R})$ (real polynomial functions of degree at most 4): $X := \{f_1,f_2,f_3,f_4,f_5 \}$ with $f_1(x) = 1 + x^3 + x^4$, ...
1
vote
1answer
82 views

Orthogonal projection of a vector $x$ onto $\operatorname{Col}(A)$

Let $A$ be a matrix. The orthogonal projection of a vector $x$ onto $\operatorname{Col}(A)$ is unique if and only if the columns of $A$ are linearly independent. True or False?
10
votes
1answer
131 views

How many matrices in $M_n(\mathbb{F}_q)$ are nilpotent?

I have strong computational evidence to think that the answer is $q^{n(n-1)}$, although a proof eludes me. Any ideas?
0
votes
1answer
166 views

Is there a simpler, more abstract proof of the Cayley-Hamilton theorem for matrices?

The Cayley-Hamilton theorem is equivalent to: Let $R$ be a ring and let $M_n(R)$ be $n\times n$ matrices over $R$. Then the minimal polynomial of $A \in M_n(R)$ over $R$ divides the characteristic ...
2
votes
2answers
258 views

Einstein Notation for product of stacked matrices

Background Information: I recently started using the Einstein summation notation to express certain operations over an "image" $\mathbf{A}$ where to each pixel a square matrix is attached. That is, ...
8
votes
1answer
227 views

Given the inverse of a block matrix - Complete problem

Given $X$ a block matrix $$\pmatrix{A&B}$$ where $A$ is $m \times n$ and $B$ is $m \times (n−m)$. I know a priori the value of $X \times (X^{T} \times X)^{-1}$. Substituting $X$: ...
4
votes
1answer
73 views

Is anybody here able to construct this example?

I need to construct an example for such a situation: Let $x_1,x_2$ and $v_1,v_2$ be four vectors in $\mathbb{C}^2$, so that they are mutually different from each other. Further, there have to be ...
4
votes
1answer
30 views

Decompose symmetric matrix to scaling factors

I have a symmetric square matrix $P$ composed by left- and right-multiplying another symmetric square matrix $Z$ with a diagonal matrix $Λ$: $$P = ΛZΛ$$ i.e. ($λ_i$ means $λ_{ii}$): $$ ...
4
votes
2answers
48 views

Powers of matrices equality

let $A$ be a $3$ by $3$ matrix with two eigenvalues $\lambda _1, \lambda _2$ such that $\lambda _1$ has algebraic multiplicity $2$ and $\lambda _2$ has multiplicity $1$. I want to prove that ...
1
vote
0answers
65 views

Integrate the ratio of quadratic forms

Please, help me to solve the folowing problem. Given two positive-definite $n$-dimensional matrices $A$ and $B$, need to integrate its ratio over unit ball: ...
3
votes
1answer
111 views

Linear transformation whose $n$th power is identity

Let $V$ be a vector space over field $F$ with $\dim_FV=2$. Suppose $T:V\longrightarrow V$ is a linear transformation with $T^n=Id$ for some positive integer $n$ (the finite $n$ is the order of $T$). ...
1
vote
1answer
125 views

How to calculate a matrix with its orthogonal complement known?

(1)$\mathbf{Q}$ is a matrix with orthonormal columns, $\mathbf{Q}\in\Bbb{R}^{4\times 3}$. (2) $\mathbf{Q}^T\mathbf{q}=0$. Then the column space of $\mathbf{q}$ is the orthogonal complement of the ...
3
votes
2answers
810 views

If N is elementary nilpotent matrix, show that N Transpose is similar to N

If $N$ is a $k \times k$ elementary nilpotent matrix, i.e. $N^k = 0$ but $N^{k-1} \ne 0$, then show that $N^\top$ is similar to $N$. Now use the Jordan form to prove that every complex $n \times n$ ...
1
vote
2answers
79 views

Using transformations and basis to find standard matrices

Let $A =\{(1,3), (2,5)\}$ be a basis of $\mathbb{R}^2$. Let $M =\left[\begin{array}{rr} 1 & -2\\ 3 & 0\end{array}\right]$ be the standard matrix for the linear transformation from ...
2
votes
2answers
1k views

A problem with the geometric series and matrices?

Let $n$ be a positive integer. Let $A$ be a square matrix. Let $I$ be the identity matrix with the same size as $A$. I want to simplify $f_n(A) = I + A + A^2 + A^3 + A^4 + \cdots + A^n$ Now I know ...
2
votes
1answer
269 views

Linear Algebra: Finding the matrix representation with respect to standard basis

I would appreciate some help with a linear algebra practice question, I'm studying for my final and I am stuck, this is a screenshot of the question: Are my answers correct? a) $P_{2}$: $ ...
0
votes
1answer
86 views

Solving Linear Systems by hand

My professor said for our final we would have to solve linear systems by hand on our final. Some of our questions for interpolation and finding splines involve large 6x6 or 12x12 matrices. What is the ...
6
votes
2answers
160 views

Trace of matrices inequality

If I have two matrices, $\mathbf{A}$ which is symmetric and postive definite, and $\mathbf{B}$ symmetric, positive definite, and all entries in $\mathbf{B}$ are between 0 and 1, with the diagonal ...
3
votes
3answers
66 views

An old test question proving $\|\mathbf{B} - \mathbf{A}\| \lt \frac{1}{\|\mathbf{A}^{-1}\|}$ implies invertiblity of $\mathbf{B}$

I have an old test question that I am not sure about and would like some idea. It is from a Numerical Analysis class. Suppose that $A$ is an invertible $n$-by-$n$ matrix. Prove that for every ...
4
votes
1answer
151 views

If matrix A is invertible, is it diagonalizable as well?

If a matrix A is invertible, then it is diagonalizable. Is it true or false?
2
votes
1answer
50 views

If $A=LL^T$, is $A\otimes I_3 = (L \otimes I_3)(L \otimes I_3)^T$?

$A$ is a symmetric positive definite matrix and $LL^T$ its Cholesky factorization. $A \otimes I_3$ is the Kronecker product of $A$ with the 3x3 identity matrix. Is the relation $A\otimes I_3 = (L ...
2
votes
3answers
75 views

if $A$ is a square matrix of order 3 s.that $A^5=0$ then A is diagonlizable or not?

if $A$ is a square matrix of order 3 s.that $A^5=0$ then A is diagonlizable or not? please someone help me i dont know how i start this.
0
votes
1answer
50 views

Find the standard matrix of $T$ with respect to$S=\left \{ 1,x \right \}$ and $S'=\left \{ 1,x,x^{2} \right \}$

$B =\left \{1+x,3+2x \right \}B'=\left \{ 2,3-x,5+x^{2} \right \}$ you are given the matrix of a linear transformation $T:\mathbb{P}^{1}\rightarrow \mathbb{P}^{2}$ with respect to $B$ and $B'$ is: ...
6
votes
2answers
142 views

show that the characteristic polynomial of this matrix has negative coefficients

Let $n\geq 2$, $A$ be the $n\times n$ matrix $A=(a_{ij})$ where $a_{ij}=\max(i,j)$. Can anybody show that the characteristic polynomial $P(x)=\det(xI-A)$ has all its coefficients negative except the ...
0
votes
1answer
97 views

Representation of Linear Transformation with respect to basis please helppp

Let $A = (1,3) (2,5)$ be a basis of $\mathbb{R}^2$. Let $M =\left[\begin{array}{rr} 1 & -2\\ 3 & 0\end{array}\right]$ be the standard matrix for the linear transformation from $\mathbb{R}^2$ ...
2
votes
1answer
324 views

Check if matrix determinant is zero

What's the simplest way to check if a NxN Matrix determinant is zero ? Using Gauss Jordan to calculate the determinant first is to complicated (took N^3 calculation), is there any way to know it in at ...
0
votes
1answer
73 views

Proving Invertibility and Eigenvalues

If matrix $A$ is an $n\times n$ matrix such that $A^2 -A -2I=0$. How can I show that $A$ is invertible and that $A^{-1} = \frac12(A-I)$? Also, how do i show that one of the eigenvalues of $A$ is 2 or ...
1
vote
2answers
236 views

Diaonalized Matrix of the form $S^2=D$

If $D$ is a diagonal matrix, with non-negative eigenvalues, prove that there is a matrix $S$ such that $S^2 = D$
0
votes
1answer
22 views

Matching infinite matrices

Can any one solve this problem that I have. I have been sitting with this problem for a while now. Completely confused. "For inifinite matrices a complete matching may not be possible even though ...