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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2
votes
0answers
43 views

algebraic equation with trace

I have a problem which I don't know how to attack. Actually I am not even sure there is a way to do it. Is it possible to solve an equation of this form $$A²-\frac{tr(A)²}{4}Id_{4\times 4}=B$$ where ...
0
votes
1answer
22 views

Find basis for kernel and matrix representation

Problem 4 from https://math.berkeley.edu/~ogus/Math_54-07/Exams/midsol1.pdf $\beta$ is a basis of $P_3$, the set of all polynomials of at most degree 3.$\beta = (x^0,x^1,x^2,x^3)$. Let $T$ be a ...
1
vote
0answers
25 views

Self-adjoint operator and vertex conditions in quantum graphs

Let $\Gamma$ be a metric graph with finitely many edges. Consider the operator $H$ acting as $\frac{-d^2}{dx_e^2}$ on each edge $e$, with the domain consisting of functions that belong to $H^2(e)$ on ...
0
votes
0answers
17 views

Jacobian of an error function containing state space vector (for linearization)

1) I have a vector of robot poses, that indicates the state space --> $X = \{(x_1,y_1),(x_2,y_2)....(x_N,y_N)\} \equiv \{a_1,a_2,...a_n\}$. 2) I have odometry measurement values at each time step ...
0
votes
1answer
45 views

Find the matrix $A$

Let $A$ be a matrix such that $A\vec{x}=\begin{bmatrix}2 \\ 4 \\6 \end{bmatrix}$, where $\vec{x}=\begin{bmatrix}2 \\ 0 \\0 \end{bmatrix}+c\begin{bmatrix}1 \\ 1 \\0 \end{bmatrix}+d\begin{bmatrix}2 ...
-7
votes
0answers
51 views

can anyone help me in my homework in algebra [closed]

Consider the linear system: $$\begin{cases}x+y-z=1\\ 2x+3y+az=3\\ x+ay+3z=2\end{cases}$$ For what values of $a$ does the system have: a) No solution; b) More than one solution; c) A unique ...
0
votes
0answers
22 views

Self-adjoint linear operator $f_A$ of a scalar product $g_A$

Let $A$ be a real symmetric matrix of order $n$. Let $x,y \in \mathbb{R^n}$. Then, we have that $$g_A(u,v)=X^{\top}AY,$$ where X and Y are column vectors with at each entry one component of $x$ ...
2
votes
1answer
30 views

If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$.

Let $A,B\in \mathcal{M}_n(\mathbb{C})$ such that $AB=BA$ and $\det > B\neq 0$. a) If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$. b) Is the ...
0
votes
0answers
25 views

Adjoint of a 3x4 matrix

How do I find the adjoint of this matrix? I am familiar with finding the adjoint of an $n x n$ matrix, but this has thrown me. $$A= \left( \begin{array}{ccc} 1&-1&0\\ 0&0&1\\ ...
0
votes
0answers
18 views

Generate duplicate element from a matrix by formula $b(i:j)=A(i:j,:) \times A^{-1} \times b$

I have an interesting question about generate duplicate elements from matrix. I assume that I have a matrix A (such as the bellow example $5 \times 5$) and vector $b$ is $5 \times 1$. My goal is make ...
2
votes
0answers
81 views

Proving that the algebraic multiplicity of an eigenvalue $\lambda$ is equal to $\dim(\text{null}(T-\lambda I)^{\dim V})$

Without using induction, prove that the the algebraic multiplicity of an eigenvalue $\lambda$ is $$\dim (\text{null} (T-\lambda I)^{\dim V});$$ here, the algebraic multiplicity of an eigenvalue ...
0
votes
2answers
40 views

Matrix addition/multiplication with different sizes

I have the following two matrices: $$A=\begin{pmatrix}1 & -2\\3 & 1\end{pmatrix}\text{ and }B=\begin{pmatrix}1 & 3 & 2\\-1 & 0 & 2\end{pmatrix}$$ So I have two matrixes with ...
1
vote
0answers
41 views

How to calculate the inversion of a triangular matrix

Now I want to write a piece of code to calculate the inversion of a triangular matrix which do it in parallel. I know that the equation of the triangular matrix's inversion is like this: But I ...
1
vote
1answer
59 views

Properties shared by similar and unitary similar matrices.

We know that matrices $A$ and $B$ are similar if there exists an invertible matrix $P$ such that $A=PBP^{-1}$ and they are unitarily similar if $P$ is unitary ($PP^*=P^*P=I$). I want to know : What ...
1
vote
1answer
28 views

If $A $ is a square matrix of size $n$ with complex entries such that $Tr(A^k)=0 , \forall k \ge 1$ , then is it true that $A$ is nilpotent ? [duplicate]

If $A$ is a square matrix of size $n$ with complex entries and is nilpotent , then I can show that all the eigenvalues of $A^k$ , for any $k$ , is $0$ , so $Tr(A^k)=0 , \forall k \ge 1$ . Now ...
0
votes
0answers
13 views

Extra Operation required for Smith Normal Form over PID-Theoretical Justification

Why does one need an extra operation for performing smith normal form over a PID? One might suspect and say that it is because of the lack of Euclidean algorithm or just say that we need the ...
0
votes
0answers
35 views

Inverse Matrix in 3D

Suppose we have the following matrix in three dimensions $$ M_{ij} = g_{ij} + e_{ijk}z^{k} $$ where $e_{ijk}$ is an antisymmetric density, i.e. $e_{ijk} = \sqrt{\det g}\cdot\epsilon_{ijk}$ and $z^{k}$ ...
0
votes
1answer
26 views

Square coefficient matrix, matrix transpose, and solvability of the corresponding system of equations

Let $\mathbb{F}$ be a field and $n \geq 2$. I would like to prove that, for every $n \times n$ matrix $A$ over $\mathbb{F}$, there is a $b \in \mathbb{F}^{n}$ such that $Ax = b$ is unsolvable if and ...
0
votes
1answer
33 views

Exist a eigenvector $v=\begin{bmatrix} x \\ y \end{bmatrix}$ such that $x,y >0$

Problem: Let $\begin{bmatrix} a &b \\ c&d \end{bmatrix}$ is a real $2 \times 2$ matrix such that $a,b,c,d>0$. Prove that exist a eigenvector $v=\begin{bmatrix} x \\ y \end{bmatrix}$ ...
0
votes
0answers
21 views

Prove a matrix is Hermitian, if its eigenvalues are real and satisfy an orthogonality relation

Prove a matrix is Hermitian, if: (a) Its eigenvalues are real, and (b) the eigenvectors satisfy $ r_{i}^\dagger r_{j} = \delta_{ij} = \left<r_{i}|r_{j}\right> $ I can see this is the reverse ...
1
vote
1answer
34 views

Does $\lim_{N\rightarrow\infty}\frac{tr(A'A)}{N}=0$ imply $\lim_{N\rightarrow\infty}\frac{tr(A)}{N}=0$?

$A$ is an $N\times N$ matrix with bounded row and column norms. Does $\lim_{N\rightarrow\infty}\frac{tr(A'A)}{N}=0$ imply $\lim_{N\rightarrow\infty}\frac{tr(A)}{N}=0$? I know this is true for ...
0
votes
0answers
18 views

Prove that minimum of the matrix norm is achieved at certain parametres

Given matrix $A\in R^{n\times m}$ prove that minimum of the $||A-xy^T||$, $||B||=tr(B^TB)$, is achieved when $x$ is an eigenvector of $AA^T$, corresponding to its greatest eigenvalue, and $y$ is an ...
0
votes
0answers
34 views

Matrices - Inverse of the principal square root of a covariance matrix (^-1/2)

Say you have a square (variance)covariance matrix S How would one go about working S^-1/2 (inverse of the principle square)? Bearing in mind, I'm trying to understand a paper which states: I've ...
0
votes
2answers
38 views

Prove a matrix is not diagonalizable

To show that a matrix is not diagonalizable, I would just have to show that there are no eigenvalues present in the matrix. So, for example, if I want to prove that $$A=\begin{bmatrix} 0 & -1 ...
0
votes
1answer
78 views

How to prove that the matrix $A^k$ approaches $0$ as $k$ approaches infinity

First of all, what does it mean to say an eigenvalue is "less than unity"? I'm not exactly sure what this means. Secondly, how do I show that $\lim_{k\to\infty} A^k=0$ given that all eigenvalues of ...
0
votes
1answer
22 views

Eigenvalue Bound of Block Matrices

I have the following eigenvalue problem for block matrices A and B \begin{equation} \left[ \begin{array}{ccc} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & ...
3
votes
2answers
55 views

Determine the coefficient of polynomial det(I + xA)

Given matrix an n-by-n matrix $A$ and its $n$ eigenvalues. How do I determine the coefficient of the term $x^2$ of the polynomial given by $q(x) = \det(I_n + xA)$
0
votes
1answer
20 views

2D rotation matrix: express sin and cos in terms of the elements and the norm of an arbitrary non-zero vector

2D rotation matrix is used to derive the expressions for sin and cos in terms of $a_1 ,a_2 and ||\vec{a}|| $ with the following given I'm trying to figure out where the negative-sign comes from in ...
0
votes
0answers
23 views

Matrixproduct of A'A expressed as a sum

I have difficulties in proving (understanding, seeing) the following identity: $ \mathbf{A'A} = \mathbf{(a_1, a_2, ...,a_n)} \begin{pmatrix}\mathbf{a_1'\\a_2'\\ \vdots \\ a_n'}\end{pmatrix} = \sum ...
2
votes
2answers
56 views

Showing a linear combination of matrices is nilpotent for any constants

So I have three linear operators in a $3$-dimensional vector space $V$ over field $\Bbb k$ whose matrices w.r.t basis of $V$ are $$X= \left(\begin{matrix}1 & 0 & 1\\ 1 & 0 & 1\\ -2 ...
0
votes
1answer
48 views

If two invertible matrices A and B commute, then A^-1 and B^-1 must commute as well ??

If two invertible matrices A and B commute, so their inverse must commute as well or not ?
0
votes
1answer
24 views

Isomorphisms: $(Aut(V), \circ) \to (GL(n, \mathbb{R}), \cdot)$ and $(Or(V), \circ) \to (O(n), \cdot)$

Let $V$ be a $n$-dimensional $\mathbb{R}$-vector space. Let $Aut(V)$ be the set of the automorphism on $V$. I have shown that this is a group with respect to the composition of functions. However, I ...
0
votes
0answers
45 views

Characterizations of positive definiteness of a symmetric matrix of order $2$

Let $$M=\pmatrix{ a& b \\b&c }$$ be a symmetric matrix. In my textbook the following result is stated without proof, but I would like to know why it holds, but I cannot figure out what to do ...
0
votes
2answers
28 views

Some doubts and questions with the trace of a matrix

Let $\text{tr}A$ be the trace of the matrix $A \in M_n(\mathbb{R})$. I realize that $\text{tr}A: M_n(\mathbb{R}) \to \mathbb{R}$ is obviously linear (but how can I write down a formal proof?). ...
1
vote
0answers
25 views

Proximal operator of spectral norm of a matrix

How can I calculate the proximal operator of spectral norm for any general matrix, $X\in R^{m\times n}$ i.e., $X^* = \arg \min_X ||X||_2 + \frac{1}{2\tau} ||X-Y||_F^2$ I understand that the proximal ...
0
votes
0answers
17 views

understanding relative matrices

$Let \:T:\:\mathbb{R}^4\rightarrow \mathbb{R}^2,\:T\left(x_1,\:x_2,\:x_3,\:x_4\right)=\left(x_1+x_2+x_3+x_{4\:},\:x_4-x_1\right)$ $v\:=\:\left(4,\:-3,\:1,\:1\right)$ ...
0
votes
2answers
24 views

Convergence of a sequence of matrices

Let $A$ be a $n×m$ matrix with real entries, and let $B = AA^ t $and let $\alpha$ be the supremum of $x ^t Bx$ where supremum is taken over all vectors $x ∈ \mathbb R ^n$ with norm less than or equal ...
1
vote
2answers
46 views

When does $x^T (xy^T) y = x^T (x^Ty) y$?

$x$ and $y$ are column vectors. When does $x^T (xy^T) y = x^T (x^Ty) y$? After a few trial and errors, I found that if at least one of $x$ and $y$ is a zero matrix then the equality is true. The ...
0
votes
2answers
87 views

How to prove that tr(A) = tr(B) given that B is similar to A [duplicate]

If A and B are similar, how does one prove that tr(A) = tr(B)
1
vote
1answer
29 views

Determinant and submatrices

I have an m x n matrix that has the rank at most one. What I am trying to show is that the determinants of all 2 x 2 matrices is zero. My idea is that I can row reduce the main matrix to one row ...
0
votes
1answer
31 views

Deducing a formula for multiplying a tri-diagonal symmetrical matrix with vectors

This is more like a math-programming problem, dealing with memory efficiency, but I thought it would be nice to expose it here. Let $A \in \mathbb{R}^{n \times n}$ be a tri-diagonal symmetrical ...
1
vote
2answers
31 views

Finding Eigenvalues and Eigenvectors for Leslie Matrix

A Leslie Matrix is given by $$L =\begin{pmatrix}0 & (3/2)a^2 & (3/2)a^3\\1/2 & 0 & 0\\ 0 & 1/3 & 0\end{pmatrix}\cdot$$ Find the Eigenvalues and determine the dominant ...
0
votes
2answers
45 views

What is the characteristic polynomial?

Let $A\in M_4(\mathbb{F})$, such that the minimal polynomial is $m_A = (x-3)(x^2+6x+10)$. What is $f_A(x)$ (the characteristic polynomial)? I'd be glad for help. By the way, I just proved a ...
-2
votes
1answer
33 views

Diagonalizing a matrix in C

A question for homework asked to show that the matrix $[T]^{\alpha}_{\alpha}$ is diagonalizable, and find a basis $\alpha$, for $[T]^{\alpha}_{\alpha}$, where $T:C^{3}\to C^3$ ...
-1
votes
0answers
21 views

Derivative of a matrix function by a matrix

How can I obtain the derivative of a matrix function $f(X)=X^TX$ by matrix $X$? Does the derivative organized in matrix form? Thanks in advance.
0
votes
1answer
24 views

How do I change this basis for a transformation?

I have $$\left[ L\right]_\mathcal{B}^\mathcal{B} = \begin{pmatrix}2&2&-1\\7&4&-2\\8&5&2\end{pmatrix}$$ and I want to get $[L]_\mathcal{E}^\mathcal{E}$ where the ...
0
votes
1answer
50 views

Cayley–Hamilton theorem and the characteristic polynomial

Let $A$, an invertible matrix and $f_A(x)$ to be the characteristic polynomial. By Cayley–Hamilton theorem we know that $f_A(A) = 0$. More detailed: $$0 = f_A(A) = a_0 + a_1A + \ldots + ...
1
vote
1answer
39 views

Shortcut when finding D when diagonalizing matrices when encountered with tedious matrices

P is given as P = $\left(\begin{array}{rrr} 1 & 1 & 1\\ 1 & 0 & -2\\ 1 & -1 & 1 \end{array}\right).$ It is known that P is invertible. I is a 3x3 identity matrix Supposed ...
2
votes
0answers
22 views

Some maximization over stochastic matrix

I am writing some applied assignment which leads me to the following problem. I will be very grateful if anyone can provide a solution or even some thoughts. Thanks a lot! Consider a (row-)stochastic ...
0
votes
1answer
8 views

Notation to define a function mapping from a vector to a two-dimensional matrix

I have a set $\mathcal{D}$, and I'm trying to define a mapping from that set to a two-dimensional matrix where each location contains either a $1$ or $0$. The notation I am using is ...