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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3
votes
1answer
114 views

I can't understand a step in the proof of the associativity of matrix multiplication

Matrix multiplication is proven by the following reasoning: Let there be matrices $A^{m \times n}$, $B^{n \times k}$ and $C^{k \times l}$. Then $$ \{(AB)C\}_{ij}=\sum\limits_{p=1}^k{\{AB\}_{ip}c_{pj}...
3
votes
2answers
149 views

Calculating a bound on the norm of a matrix exponential

The problem is this: Let A be a square $n \times n$ matrix, and define $$e^A=\sum_{k=0}^\infty \frac{1}{k!}A^k$$ Find a bound for $\lvert e^A \rvert$ in terms of $\lvert A \rvert$ and $n$. I was ...
0
votes
1answer
132 views

Over-specified linear system

Consider the matrix $A $ with RREF consisting of three of the 4, 4- dimensional standard vectors: $[\mathbb {e_1}, \mathbb {e_2}, \mathbb {e_3} ] $ Since the rank is 3 the matrix has one solution ...
0
votes
0answers
59 views

matrix with positive diagonal elements

I was wondering if a symmetric matrix with positive elements only in the diagonal (negative elsewhere) is any special beside the symmetry. Thanks in advance
0
votes
1answer
25 views

Generation of rank-$2$ matrices from a dictionary of rank-$1$ matrices.

I have a question about the construction of rank-$2$ matrices from a dictionary of rank-$1$ matrices. Consider the set $\mathcal{D} = \{\mathbf{A} \in \mathbb{C}^{2 \times 2} \mid rank(\mathbf{A}) = 1,...
3
votes
1answer
45 views

how to find the dimension of the image of $f$ in this case?

Let $A \in M_{m \times n}(\Bbb R)$ be fixed, and let $B \in M_{m \times l} (\Bbb R)$. Consider the map $f: M_{n \times l}(\Bbb R) \to M_{m \times l}(\Bbb R)$ defined by $f(X) = AX + B$ for all $X ...
1
vote
1answer
73 views

Determining the standard matrix from the images of the standard basis vectors

Let a linear transformation $T:$ $\mathbb{R}^3$ → $\mathbb{R}^3$ rotate a vector around the z-axis by $45^{o}$ followed by an orthogonal projection onto the x-axis. Determine the standard matrix for ...
0
votes
2answers
49 views

Relationship between eigenvalues of A symmetric matrices

Let $$A=\begin{pmatrix}a & b\\b & c\end{pmatrix} \in M_2\mathbb{(R)}$$ i) Find the eigenvalues of $A$ ii) If $\begin{pmatrix}1\\2\end{pmatrix}$ is an eigenvector of $A$, prove that $\begin{...
1
vote
2answers
52 views

Inverse of nonnegative Toeplitz matrice

Consider a right-hand circulant matrice of size $n$ (called also Toeplitz matrice) \begin{equation} T= \left( \begin{array}{ccccc} a_1 & a_2 & a_3 & \ldots & a_n \\ a_n & a_1 & ...
1
vote
2answers
240 views

Simultaneously diagonalization of two matrices.

Let $A$ be a real symmetric matrix and $B$ a real positive-definite matrix. Is it possible to simultaneously diagonalize of $A$ and $B$? Thank you very much.
6
votes
1answer
213 views

Eigenvalues of symmetric matrix with skew-symmetric matrix perturbation

If $A$ is diagonalizable, using the Bauer-Fike theorem, for any eigenvalue $λ$ of $A$, there exists an eigenvalue $μ$ of $A+E$ such that $|\lambda-\mu|\leq\|E\|_2$ (the vector induced norm). Here I ...
0
votes
4answers
98 views

$A.A^t$ is diagonal

Be $A$ a semidefinite nonnegative matrix. What kind of conclusions can we say about $A$ if $A.A^t$ is diagonal? Same question when $A$ is binary matrix. Thanks
8
votes
1answer
139 views

If $A$ and $B$ are $n×n$ matrices such that $AB=B$ and $BA=A$ then find the value of $A^{4} + B^{4} - A^{2} -B^ {2} + I$

The given question is If $A$ and $B$ are $n×n$ matrices such that $AB=B$ and $BA=A$, then find the value of $A^{4} + B^{4} - A^{2} -B^ {2} + I$. Any hints?
7
votes
3answers
175 views

Prove that $\det(A^2 + A + xI) = x$

Let $x$ be a positive real number and $A$ a $2\times2$ matrix with real values satisfying the following property $\det(A^2 + xI) = 0$. Prove that $\det(A^2 + A + xI) = x$ I have tried something with ...
0
votes
2answers
64 views

Finding all matrices for which the homogeneous system has a given solution space

Find all $3\times 3$ matrices for which the homogeneous system has a solution space as the line $x = 2t$, $y = t$, $z = 0$. (Hint: Write the row reduced augmented matrix from given information.) ...
0
votes
3answers
82 views

Rank of matrices and their product

Let $\operatorname{rank}(A_{3 \times 3})=\operatorname{rank}(B_{3 \times 3})=2$. I need to figure out whether $AB=0$ is possible. On the one hand, I know that $\operatorname{rank}(AB) \leq \min(\...
3
votes
2answers
76 views

Given $A$ is $6×6 $ real symetric matrix of rank $5$ , then to determine rank of $A^{2}+ A+I $

Given $A$ is $6×6 $matrix of rank $5$ , then to determine rank of $A^{2}+ A+I $. I knowthat rank of matrix doesnot change when we square it , but how to proceed in this question.Any hints ? Thanks
6
votes
4answers
564 views

Proof If $AB-I$ Invertible then $BA-I$ invertible.

I have these problems : Proof If $AB-I$ invertible then $BA-I$ invertible. Proof If $I-AB$ invertible then $I-BA$ invertible. I think I solve it correctly, But I'm not so sure, I'll be glad to ...
1
vote
3answers
61 views

Matrix multiplication ambiguity

From this source here, it says that matrix multiplication is given by this: $AB = \begin{bmatrix} a_{1,1}b_{1,1}+a_{1,2}b_{2,1}+...+a_{1,n}b_{p, 1} & ...\\ \vdots & a_{m,1}b_{1,p}+a_{m,2}b_{...
0
votes
1answer
232 views

How do row operations affect the column space?

I've been curious about this: Row operations do not affect the row space, but they affect the column space. Is there any way to 'systematically' perform row operations to make the column space the ...
4
votes
1answer
84 views

Matrix equation solution

Does anybody know how to solve this matrix equation: $$P = P P^T R + X,$$ where $P, R,$ and $X$ are vectors with $n$ elements, and $P$ is the unknown vector?
0
votes
1answer
34 views

Relationship between type of matrix and eigenvalues

Prove that if the eigenvalues of a diagonalizable matrix $A\in M_n(\mathbb{R})$ are all $1$ or $-1$, then $A^{-1}=A$ What I tried to reverse the way to get the rough idea. $$A^{-1}=A\implies A^2=I\...
-1
votes
1answer
72 views

True / False about a matrix

Let $A= \begin {pmatrix} x & y \\ -y & x \end {pmatrix}$ where $x,y \in \mathbb{R}$ such that $x^2+y^2=1$. 1) For any $n \ge 1$, $$A^n= \begin {pmatrix} \cos\theta & \sin \theta \\ -\sin ...
1
vote
2answers
46 views

Eigenvalues of $6 \times 6$ matrix?

Which of {$\pm1,\pm i$} are the eigenvalues of matrix A, $$A=\begin{pmatrix} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & ...
1
vote
1answer
51 views

$P$ is an matrix invertible Proof $|\lambda I-PBP^{-1}|=|\lambda I -B|$

I have this problem : $P$ is an matrix invertible Proof : $|\lambda I-PBP^{-1}|=|\lambda I -B|$ I'm not so sure about my answer, since I don't think I could use "double" determinant for example $||...
3
votes
0answers
150 views

Second structural equations in lorentzian space $\Bbb L^3$.

I'm rewriting O'Neill's "Elementary Differential Geometry"'s section on connection forms in Lorentz-Minkowski space $\Bbb L^3$, and I'm having trouble proving the second structural equations $${\rm d}\...
1
vote
0answers
90 views

A question on matrices

Let $M\in\Bbb \{0,2\}^{n\times n}$ be a rank $t\leq n$ matrix and we know that it can be rewritten as $A+B$ where $A$ has $\{0,1\}$ entries and symmetric and $B$ has $\{-1,0,+1\}$ entries and skew ...
2
votes
1answer
120 views

A matrix version of L'Hopital's Rule?

Is there a version of L'Hopital's Rule for matrix calculus? For example: let $A$ be a symmetric $n\times n$ positive definite matrix and $b$ be an $n\times 1$ vector. As $b$ converges to $0_{n\times ...
10
votes
1answer
2k views

Powers of adjacency matrix doesn't seem to correspond to observed number of paths on graph

I would really appreciate some help on this! $A^n$ represents $n^{th}$ power of the adjacency matrix of a graph. I keep reading that the $A^n_{ij}$ entry equals "the number of paths of length n ...
0
votes
1answer
53 views

Proof If $A_{2x2}$ and $\lambda$ real number then $|\lambda I-A|=\lambda^2-(\operatorname{tr}A) \lambda+|A|$

I have this problem : Proof If $A_{2x2}$ and $\lambda$ real number then $|\lambda I-A|=\lambda^2-(\operatorname{tr}A) \lambda+|A|$ This is what I did : I took an arbitrary $A$ $$ A= \left( {\...
4
votes
1answer
223 views

Perturbation theory of the eigenvalues about the symmetric matirx

From Weyl's theorem, i.e.: Let $A$ and $E$ be $n\times n$ real symmetric matrices. Let $\alpha_1\geq\ldots\geq\alpha_n$ be the eigenvalues of $A$ and $\hat{\alpha}_1\geq\ldots\geq\hat{\alpha}_n$ be ...
0
votes
1answer
46 views

Rank of a general matrix

Given some scalars $a_1,a_2,...,a_m \in F$ not all zero and $b_1,b_2,...,b_n \in F$ not all zero, what is the rank of the matrix $M=(a_i b_j)_{\begin{matrix}1 \leq i \leq m \\ 1\leq j \leq n \end{...
2
votes
0answers
54 views

LU factorization of a modify matrix

Suppose you know $L$, $U$, decomposition LU of a matrix $M+I$ ($M+I=LU$). Lets $J$ a diagonal matrix whose elements are $0$ or $1$. Is there any relation between the factorization LU of $M+I$, and ...
1
vote
0answers
42 views

Prove the Basis of Column Vectors

Let $V$ be a vector space over field $\mathbb{F}$. Let $B=\{u_1,...,u_n\}$ be an ordered basis of $V$. Show that $\{[u_1]_c,...[u_n]_c\}$ is a basis of $M_{n,1}(\mathbb{F})$ for every ordered basis $...
1
vote
2answers
215 views

Prove an upper bound for the determinant of a matrix A

Let $A$ be a $3 \times 3$ real matrix with all $0\le a_{ij} \le 1$. Show that $\det(A) \leq 2$ and find such matrices with $\det(A) = 2$. Let $A$ be a $n \times n$ matrix with all $0\le a_{ij} \le 1$....
0
votes
1answer
46 views

Dot “power” of a matrix

By analogy with the matrix product is there a name for the matrix "power" operation defined by $$y_i = \prod_j x_j^{a_{ij}}?$$ For example: $$\left( \begin{array}{lll} x_1 & x_2 & x_3\end{...
0
votes
2answers
85 views

What is the determinant of matrix?

Find determinant of the $n \times n$ permutation matrix $$ M= \left[ {\begin{array}{cccc} 0 & 0 & \ldots & 0 & 1\\ 0 & 0 & \ldots & 1 & 0\\ \vdots & \...
1
vote
1answer
326 views

Schur product theorem

The theorem states that the Hadamard product of two positive definite matrices $ A \circ B$ is also positive definite. Can I make any statement about a the Hadamard product of a positive definite ...
1
vote
1answer
26 views

Prove that $A$ is similar to $B$ probably using Jordan form

Let $A, B \in M_n(\mathbb{F})$ such that: $a_{ij} = 0 \iff b_{ij} = 0$. $a_{ij} = b_{ij}$ for all $i \ne j+1$ $\exists \lambda \in \mathbb{F}$ such that $a_{ii} = b_{ii} = \lambda$. Prove that $...
1
vote
2answers
76 views

Finding inverse linear transformation

I'm solving a homework question and I'm stuck with it's last part. The question goes like this: Let $\displaystyle T:M_n(\mathbb{R})\to M_n(\mathbb{R})$ be a transformation defined as $T(A)=A+2A^...
2
votes
1answer
158 views

How to continue on proving that rank (A+B) ≤ Rank A + Rank B? [duplicate]

Theorem: $rank(A+B) \leq rank (A) + rank(B)$. Proof: Let $U = Im(A)$ and $W = Im(B)$. By dimension theorem, we know that: $Dim(U+W) = Dim(U) + Dim(W) - Dim (U \cap W)$. By substituting $U$ ...
1
vote
3answers
77 views

Prove that if $\mathbf A$ is an $n\times m$ matrix, then $\text{tr}(\mathbf A \mathbf A')=\text{tr}(\mathbf A' \mathbf A) $

If $\mathbf A$ is an $n\times m$ matrix, then $\text{tr}(\mathbf A \mathbf A')=\text{tr}(\mathbf A' \mathbf A) \text{ where } \mathbf A'\text{ is transpose of }\mathbf A\text{ and tr}(\mathbf A )\...
3
votes
1answer
69 views

Prove that $A$ is similar to $B$

Let $A, B \in M_n(\mathbb{F})$ such that $m_A(x) = m_B(x)$ and $f_A(x)=f_B(x)=(x-\lambda_1)^{d_1}\cdots (x-\lambda_k)^{d_k}$ for different $\lambda_1, \ldots, \lambda_k$ such that $1 \le d_l \le 3$ ...
2
votes
1answer
54 views

Endomorphism ring as a set of matrices

Let $A=\mathbb Z[\sqrt{-5}]$, and let $I=(2,1+\sqrt{-5})$ (which is known to be a non-principal ideal of $A$ with $I^2=2A$). If we put $P=A \oplus I$, my question is: Why the endomorphism ring of $...
0
votes
1answer
24 views

Can $LL^T$ decomposition of a matrix be computed by the same algorithm as $LU$-one?

I know that's the silly question. But if I perform $LU$ decomposition on a symmetric positive definite matrix, will this decomposition be the same one as $LL^T$ one?
1
vote
0answers
48 views

Prove that $V_i$ are $T$-invariant for $1\le i\le k$ and $V=\bigoplus_{i=1}^{k}V_i$

Let $T$ be a linear operator over $V$ with dim$(V)=n$ and let the ordered set $B=${$v_1,v_2,...v_n$} be a basis for $V$. Furthermore, let $A=[T]_B$ be block-diagonal matrix (that is $$A=\bigoplus_{i=1}...
2
votes
7answers
138 views

Prove that a matrix equals to its transpose

Let $A$ be a $(n\times n)$ matrix that satisfies: $AA^t=A^tA$ Let $B$ be a matrix such that: $B=2AA^t(A^t-A)$ Prove/disprove that: $B^t=B$ I started with: $$\begin{align} B &=2AA^t(A^t-A) \\ B^...
1
vote
2answers
33 views

What row-operations allow this $\operatorname{Mat}_{2\times2} (\mathbb{R})$

$$ A = \begin{pmatrix} 1 & r \\ s & 1 \\ \end{pmatrix} \Rightarrow \begin{pmatrix} 1 & r \\ 0 & 1-s \cdot r \\ \end{pmatrix} = B \quad\quad r,s \in \mathbb{R} $$ Matrix B is ...
0
votes
0answers
32 views

Matrix operation repeat matrix members

I am going to use C++ Armadillo library which handles matrices to generate matrix $B$ and $C$ from matrix $A$. $$ A=[M_0,M_1,\ldots,M_{n-1}]^T $$ $$ B=[M_0,M_0,...
3
votes
2answers
85 views

At least one diagonal element of any real symmetric matrix of rank $1$ is non-zero ?

If $A$ is a real symmetric matrix of rank $1$ then is it true that at least one diagonal element is non-zero ?