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
37 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
200 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
75 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
65 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
74 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
2answers
219 views

Modify a matrix so that entire row or column is set to zero [closed]

I have a matrix thats M x N in size, and it has a value of zero Ex: $$\begin{bmatrix} 1 & 2 &4 &3 \\ 0 & 5 &3 &7 \\ 5 & 8 &9 &2 \\ \end{bmatrix} $$ that zero could ...
0
votes
0answers
98 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
353 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
54 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
66 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
113 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
775 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
78 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
126 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
135 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
209 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
223 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
61 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
103 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
96 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
716 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
74 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
1answer
886 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
228 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
75 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 ...
7
votes
2answers
130 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
64 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
144 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
74 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
49 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
130 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
91 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
296 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
69 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
184 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 ...
1
vote
1answer
75 views

Given the inverse of a block matrix…

Given the inverse of a block matrix $X^{-1}$, where $$ X=\left(\begin{array}{cc} A & B \end{array}\right). $$ A is $m\times n$ and B is $m\times(n-m)$. Can I obtain the pseudo-inverse of A ...
1
vote
0answers
32 views

Matrix: Area of a Triangle, which point to choose for cross multiplication

When given 3 points(vertices), which one should you pick to do the your calculations with. E.g.: $P1=(1,-1,1) P2=(2,1,-1) P3=(1,-2,-1) $ I can pick P1 -> P2 and P1 -> P3. Then do my cross ...
0
votes
1answer
32 views

Get parametric bilinear form given the matrix?

I have a 2x2 matrix (over real numbers) which represents a symmetric bilinear form. Is there a way to get the parametric form (i just assume the std base)?
2
votes
1answer
34 views

Find a direct way to calculate recursive elements (simple problem with matrices)

I nearly solved this question, I just need a hand with the last part since it is a bit confusing. We are given the recursive sequences $\{a_n\}$ and $\{b_n\}$ like this: $a_1=1$, $b_1=2$ ...
5
votes
2answers
4k views

Simplified method for symmetric matrix determinants

Preparing for my final exams I have been doing all the exercises in my algebra book. I've seen that there are lots and lots of exercises about determinants of symmetric matrices. Some are easy and ...
2
votes
2answers
59 views

what would be the eigen vector for this value?

So I have a $3\times 3$ matrix $$A=\begin{pmatrix} 2&1&1\\ 1&2&1\\ 1&1&2 \end{pmatrix}.$$ My instructions are to find the eigenvalues and eigenvectors of the matrix. For each ...
1
vote
1answer
279 views

cube root of positive definite matrix

Suppose that $A$ is a real symmetric positive definite $20\times 20$ matrix with condition number $\kappa\le 1000$. I want to solve the system of linear equations $$A^{1/3}x=b$$ with $10$-digit ...
2
votes
2answers
27 views

$M=(a_{ij})$, with $1\leq x\leq3$ and $1\leq x \leq 3$, such that $a_{ij}=4i+2j-6$.

My exercise asks me to write the following matrix $M=(a_{ij})$, with $1\leq x\leq3$ and $1\leq x \leq 3$, such that $a_{ij}=4i+2j-6$. What I did not understand, and tried unsuccessfully was what ...
2
votes
1answer
58 views

Question about Jordan base - Why is finding the kernel and finding the generelized eigenvectors the same thing?

Quick question, perhaps a bit general but I don't understand something. When finding a jordan basis for a matrix $A$, what i do is for each eigenvalue $\lambda$ I find vectors such that: $(A-\lambda ...