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
35 views

Is there any way to find out general form of proper 2 by 2 matrix?

2 x 2 orthogonal matrix $A$ is called proper if detA=1. I know this is a rotation matrix through an angle, and entries of this matrix is composed of $\sin$ and $\cos$. If you are only given the fact ...
1
vote
1answer
47 views

Easy way to calculate inverse of an LU decomposition.

I have a matrix A and a lower triangular matrix L (with 1's along the diagonal) and an upper triangular matrix U. These are constructed such that $A=LU$. I know that $A^{-1} = L^{-1}U^{-1}$ and I know ...
1
vote
1answer
81 views

Prove that if $\det(A) = 1$ or $-1$ and $A$ has all integer entries, then $A^{-1}$ also has all integer entries. [closed]

So this is a home work question I am stuck on: Prove that if $\det(A) = 1$ or $-1$ and $A$ has all integer entries, then $A^{-1}$ also has all integer entries. I would really be grateful of you ...
1
vote
1answer
33 views

How to find a real orthogonal matrix of determinant $1$?

A real orthogonal matrix $A$ is proper if $\det A=1 $. Find $2\times 2$ proper matrix $A$ I tried to use the fact that product of $A$ and its transpose is equal to identity. But, there ...
0
votes
0answers
39 views

Can I reconstruct Penney's game win probabilities from dominant strategy odds?

The probabilities of each strategy (row in the table below) in Penney's game (assuming the basic version played with a penny — no relation — and strategies consisting of a pattern the outcome of three ...
1
vote
1answer
56 views

How to find the determinant of this $(2n+2)$ x $(2n+2)$ matrix?

I need to calculate the determinant of the following matrix:$$\begin{bmatrix}0&0&-2x_1& \cdots &-2x_n&0& \cdots &0\\0&0&0& \cdots&0&-2x_1& ...
5
votes
1answer
64 views

Determinant of a $n\times n $ matrix

Let $n$ be a positive odd integer and let $A$ be a symmetric $n\times n$ matrix of integer entries such that $a_{ii}=0,i=1,2.....n$. Show that the determinant of $A$ is even. I tried using ...
0
votes
1answer
37 views

How to calculate the Matrix of a given Linear Transformation?

Let $V = F^3$ and $W = F^4$ and we define the following functions: $p\in {\cal L}(V,F)$ given by $p((x,y,z)) = 3x + 4y + 2z$ $q\in {\cal L}(W,F)$ given by $q((w,x,y,z)) = 2w + 5x + 7y + 11z$; $T\in ...
0
votes
1answer
27 views

A square matrix with some specific properties is always non-singular?

Let $A$ be a square matrix with the following properties: $a_{ii}=1$ $\ \forall i$, if $a_{ij}\neq 0$ then $a_{ji}= 0$ $\ \forall i,j$ and $i\neq j$; Is it possible to prove that $A$ is ...
3
votes
2answers
85 views

the difference of idempotent matrices

$\newcommand{\rank}{\operatorname{rank}}\newcommand{\diag}{\operatorname{diag}}$Let $A,B$ be two $n\times n$ real matrices with the property ($A^t$ is the transpose of $A$) $$A^t=A=A^2, B^t=B=B^2.$$ ...
1
vote
2answers
41 views

Linear systems of inequations

Ok so I have a systems with $6$ inequations and $3$ variables, and a point that may or may not solve this system. To check whether this point solves the inequations is straightforward, my problem is ...
0
votes
3answers
56 views

How did the equation $|A- \lambda I| = 0$ came for finding Eigen values of a Matrix

Does anyone know how this equation came.I was studying about Eigenvalues and a doubt that arised was how could the equation $(A- \lambda)V = 0$ become $|A- \lambda I| = 0$.
1
vote
1answer
40 views

Matrix conditioning and eigenvalue conditioning

I cannot understand the difference between the two kinds of conditioning in the title. During a lecture,our prof said that a well-conditioned matrix can have ill-conditioned eigenvalues and vice ...
0
votes
1answer
63 views

why the Identity matrix have 1's at the Main Diagonal

Could anyone explain why the 1's in the identity matrix are present in the main diagonal. $ A=\begin{bmatrix} 4 & 2 \\ 8 & 0 \end{bmatrix} $ $ B=\begin{bmatrix} 1 & 0 \\ 0 & 1 ...
0
votes
1answer
24 views

Finding eigenvalues for second equilibrium

f 1 (T,D,C)=λ−μT−βTC, f 2 (T,D,C)= βTC - αD, f 3 ( T , D , C ) = kD - γC Given ...
0
votes
2answers
63 views

If $A$ is real skew-symmetry. $I-A$ and $I+A$ are nonsingular

I asked these questions to my TA during workshop but none of them figured it out. Can anyone help me out?
0
votes
1answer
12 views

Relationship between hermitian matix and hermitian transformation

My TA said that every hermitian matix implies transformation is hermitian because you can find orthonormal basis for every hermitian matrix and therefore transformation is hermitian. Is that true?? ...
1
vote
1answer
53 views

Finding the transformation matrix $T$ so $T^{-1}AT$ will be diagonal

I have a matrix $$ A = \begin{pmatrix} 3 & 1 & -1 \\ -1 & 1 & 1 \\ 2 & 2 & 0 \end{pmatrix}. $$ For this matrix, I have found eigenvalues of $2$ and $0$ and ...
0
votes
1answer
27 views

xA=0 sufficient condition for zero determinant?

Let A be a symmetric n by n matrix and x be a 1 by n vector. If I find one x such that xA=0, does it mean A is singular?
3
votes
2answers
95 views

An annoying Pell-like equation related to a binary quadratic form problem

Let $A,B,C,D$ be integers such that $AD-BC= 1 $ and $ A+D = -1 $. Show by elementary means that the Diophantine equation $$\bigl[2Bx + (D-A) y\bigr] ^ 2 + 3y^2 = 4|B|$$ has an integer ...
0
votes
1answer
11 views

$|E| = Θ(|V|^2)$ better to use adjacency matrix?

If $|E| = Θ(|V|^2)$ would it be better to use an adjacency list or an adjacency matrix? I'm leaning towards adjacency matrix since the graph seems to be dense. Sorry for the basic question but can ...
0
votes
2answers
42 views

Intuition of the column space of a vector $4\times3$ matrix $\mathbf{A}$

Suppose I have this matrix: $\mathbf{A} = \begin{bmatrix} 1&1&2\\ 2&1&3\\ 3&1&4\\ 4&1&5\\ \end{bmatrix}$ My understanding that $\mathbf{A}$ is the equivalent of this ...
1
vote
1answer
70 views

Solving a matrix equality

I have this Algebra problem... I've just learned basic stuff (trace, transposed matrices, symmetric matrices, etc). Must solve for $X$: $$ AX - \operatorname{tr}(C)X + X^T = B^*A $$ I know that $A$ ...
0
votes
0answers
10 views

Bounds on Interpolated Transformation Matrices Times Constant

Summary/TL;DR: Given: 4x4 transformation matrices (one of translation, scaling, rotation), each a function of $t$:$$ M_0(t), M_1(t), M_2(t), \cdots, M_{n-1}(t) $$ Given: ...
1
vote
1answer
28 views

How to make sense of this simple complex number question

I was asked the following question: Let $A: \mathbb R^2 \to \mathbb R^2$ be a linear transformation, represented by the matrix $\begin{bmatrix}a & b \\ c & d\end{bmatrix}$ in the standard ...
0
votes
1answer
34 views

Can Polynomials be positive definite?

It seems to me that polynomial functions are ,trivially, not positive-definite (for definition )because of growth property of p.d functions. Am I right?
0
votes
2answers
133 views

Give a counterexample to show that $(AB)^{-1} \neq A^{-1}B^{-1}$

Give a counterexample to show that $(AB)^{-1}$ doesn't equal $A^{-1}B^{-1}$ I'm not sure how to approach this, so I just used the idea that the matrix multiplication is not commutative. so it goes: ...
0
votes
2answers
35 views

Linear operator $A$ on a finite -dimensional vector space

Let $A$ and $B$ be linear operators on a finite dimensional vector space $V$ over $\mathbb{R}$ such that $AB=(AB)^2$. If $BA$ is invertible then which of the following is true - (a) $BA = AB$ on $V$ ...
0
votes
2answers
25 views

quadratic matrix derivation using chain rule

Giving $$f(x) = x^T A x$$ with $x \in \mathbb{R}^{n} $ and $A \in \mathbb{R}^{n \times n}$, than $$ \frac{\partial f(x)}{\partial x} = (A + A^T) x$$ I tried to prof this with the chain rule. With ...
0
votes
1answer
23 views

Determining the dimension of span$\{AB-BA : A,B \in M_{n \times n}(\mathbb R) \}$

Let $S$ be the subspace , of $M_{n \times n}(\mathbb R)$ (the vector space of all $n \times n$ real matrices ) , generated by matrices of the form $AB-BA$ , where $A,B \in M_{n \times n}(\mathbb R)$ ...
0
votes
2answers
130 views

Does a overdetermined system always has no solution [closed]

What is the problem with over-determined systems in linear algebra? why does it have no solution? can you show me the proof for that?
0
votes
1answer
81 views

Need help finding a specific transition matrix with polynomials.

I am not told it is a transition matrix till later and the book never describes a transition matrix, it just tells me to find $A$ from the equation $[w]_{B}=A[w]_{C}$ where $w$ is an arbitrary matrix ...
0
votes
2answers
37 views

Inverse of a triangular matrix of special form

How should I begin when I want to get an inverse matrix from this one? Progress I have tried to do it explicitly for small $n$. But now I am not able to deduce the general pattern with $n$ from ...
0
votes
0answers
16 views

Finding Rank and Null space of a Matrix [duplicate]

Find rank and null space of the matrix A with different values of parameter λ \begin{pmatrix}- \lambda & 1&2 &3 &1 \\ 1 & -\lambda & 2&3&1 ...
2
votes
1answer
29 views

$AM=I$, where $M$ is a rectangular matrix with full column rank, prove that $A=M^+$?

$AM=I$, where $M$ is a rectangular matrix with full column rank, then $A=M^+ $(Moore-Penrose pseudoinverse)?
1
vote
2answers
59 views

Matrix algebra involving exponents

I tried it with choice A and it worked but why is C correct? I see if you multiple both sides of the equation in choice C by the inverse (A^-1), then I'll get the condition from the question. Why ...
1
vote
0answers
36 views

Integral of a matrix exponent

What is the analytic closed form expression of $\int e^{A_1+A_2s} \ ds \tag 1$ where A and B are constant skew symmetric matrices NB $A_1=\left( \begin{array}{ccc} 0 & -c_0 & b_0 ...
0
votes
1answer
67 views

left eigenvector and similarity matrix of Jordan canonical form

Let $\mathbf{L}$ be a $N\times N$ matrix and $\mathbf{L}\mathbf{P}=\mathbf{P}\mathbf{J}$ where $\mathbf{J}=[j_{ik}]$ is Jordan block matrix. If $~j_{NN}=0$ is a "simple'' eigenvalue of ...
0
votes
1answer
24 views

Matrix Transformation across multiple planes

Let $T_1$ be a reflection of $\Bbb{R}^3$ in the xy plane, $T_2$ is a reflection of $\Bbb{R}^3$ in the xz plane. What is the standard matrix of transformation $T_2T_1$? Here's my thinking so far: ...
2
votes
1answer
38 views

Is $\|A\|=\lim_{n\to\infty}|tr A^n|^{1/n}$ a norm?

For matrix A, define $$\|A\|=\lim_{n\to\infty}|\operatorname{tr}{(A^n)}|^{1/n}.$$ Is this a norm defined on some specific subspace of $Gl(d,\mathbb{R})$? Does it have a name?
1
vote
2answers
34 views

Transpose of Eigenvectors Properties

Given $v$ an eigenvector of an $m \times m$ matrix $A$ with eigenvalue of $5$, where $|v| = 1$, is it the case that $v'*A'*A*v = 5$? $A*v = 5*v \Rightarrow$ $A'*A*v = 5*A'*v \Rightarrow$ ...
1
vote
1answer
61 views

Conjecture that $A^{T}BA = ABA^{T}$ for any symmetric matrix $B$ in $\mathbb{R}^n$

While trying to understand the Kalman filter, and by experimentation with Python I came up with the conjecture in the title. First of all is it true? Second, if it is, how can I prove this? I would ...
1
vote
3answers
86 views

Solve for $\textbf{x}(t)$ from the system of differential equations $\textbf{x}' (t) = A \textbf{x}(t)$.

Solve for $\textbf{x}(t)$ from the system of differential equations $\textbf{x}' (t) = A \textbf{x}(t)$, where $$ A =\begin{bmatrix} 0 & -1 & 0\\1 & 0 & 0\\ 0 & 0 & ...
1
vote
1answer
27 views

How can hermitian matrix have orthornormal eigen vectors with same eigenvalues?

Suppose hermitian matrix has same eigen values There is a thereom saying that every hermitian matrix has is similar to siagonal matrix. If hermitian matrix has same eigen values, it seems to me that ...
3
votes
3answers
359 views

Can basis vectors have fractions?

So I was diagonalizing a matrix in a book, and one of the basis vectors was [3/2, 1], after doing the problem, the answer in the book was different than mine. It came with an explanation, and in it ...
0
votes
1answer
48 views

If A is some invertible $n \times n$ matrix then show $\det(A^n) = (\det(A))^n$ for all $n\in \mathbb{Z}$

So there exists $A^{-1}$. I am assuming $\det(AB)=\det(A)\cdot\det(B)$ and $(A^d)^f=(A^{df})$ I know the proof for $\det(A^{-1})=(\det(A))^{-1}$ is: $\det(I_n)=1$ $\det(A\cdot A^{-1})=1$ ...
0
votes
1answer
37 views

minors and rank of a matrix

When reading a text, I came across a statement saying "the rank of an $m\times n$ matrix is $r$ if and only if all $(n-r+1)\times(n-r+1)$. minors vanish" Could anyone explain what it means by a ...
0
votes
1answer
54 views

Permutation Matrices for n = 5

Let $\sigma \in S_n$ denote the permutation given by $$ \sigma = \left( \begin{matrix} 1 & 2 & 3 & \ldots & n \\ n & 1 & 2 & \ldots & n-1 \end{matrix} \right) $$ and ...
1
vote
2answers
62 views

Determinant matrix proof

Let $A$ be an $n\times n$ matrix and $i,j,k$ be $1\leq i,j,k\leq n$ and $\alpha,\beta \in \mathbb{R}$. I am supposing that $\bf{a}_k$(the $k$-th row) is equal to $\alpha \bf{a}_i+\beta \bf{a}_j$. ...
0
votes
1answer
23 views

$H=[h_1, h_2, \cdots, h_n]$, $h_i\in \mathbb{C}^m, m>n$. prove the orthogonal complement problem

$H=[h_1,h_2,\cdots,h_n]$, where $h_i\in \mathbb{C}^m, m>n$. Let $Q_i$ be the matrix whose columns are formed by the orthonormal bases of the orthogonal complement of the subspace spaned by ...