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
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2answers
23 views

It is true that $rank(xy^T)=1$?

Let $x,y\in \mathbb{C}^n$. It is true that $rank(xy^T)=1$?
-1
votes
0answers
17 views

Gaussian Elimination vs matrix inversion

Why Gaussian Elimination is better than matrix inversion in therms of FLOPS? Also how LU decomposition improves the shifted inverse power method?
1
vote
0answers
7 views

Is it possible to write the Hadamard product of two matrices in tensor notation?

Say I have two $4 \times 4$ matrices $(A^{\alpha \beta})$ and $(B^{\mu\nu})$ and want to compute the Hadamard (entry-wise) product. Is there an elegant way of writing this down in the common ...
0
votes
0answers
13 views

Eigen vectors and determinant of a block matrix

I have two questions regarding matrix $A$. The matrix $A$ can be partitioned into four tridiagonal matrices $A_1$, $A_2$, $A_3$ and $A_4$. $$A=\begin{pmatrix} A_1&A_2\\A_3&A_4 \end{pmatrix}$$ ...
1
vote
0answers
26 views

Best algorithm to compute the first eigenvector of symmetric matrix

Assume that we have a real symmetric matrix $\mathbf{A}\in\mathbb{R}^{n\times n}$ obtained as following : $$\mathbf{A}=\mathbf{N}-\mathbf{P},$$ with $\mathbf{N}\in\mathbb{R}^{n\times n}$ and ...
0
votes
2answers
19 views

Null space is an Invariant subspace

Let $\lambda$ be an eigenvalue of a square matrix $A$. Show the null space of $(A-\lambda I)^j$ is an A-Invariant subspace of $\mathbb{C}^n$ for all positive integers $j$. Proof without requiring ...
2
votes
0answers
27 views

SVD of a Matrix Product

Suppose we have a matrix $A$ with dimensions $m$ by $n$ and a column-wise permutation matrix $R$ (re-orders columns) with dimensions $n$ by $n$. Then we have a matrix $X$ which is constructed as $X ...
1
vote
1answer
35 views

inequality with a positive matrix

Let $$ A=\left[ \begin{array}{cc} a & b\\ \overline{b} & c\\ \end{array} \right]$$ be a positive semi-definite positive of $M_2(\mathbb{C})$. How prove the inequality $ac \geq ...
0
votes
1answer
29 views

What is meant by In-Place Matrix Inversion?

I come across the term "In Place Matrix Inversion" a lot in numerical libraries like NumPy and ND4J. What does it mean ? How is it different from the normal matrix inversion ? What are the advantages ...
0
votes
1answer
18 views

Multiplicity of Jordan blocks between $B$ and $-B$

Let $B$ and $-B$ be square complex matrices such that they are similar. If there is $m$ Jordan block $J_k(\lambda)$ in $B$, the Jordan block $J_k(-\lambda)$ also appears $m$ times in $B$. This is my ...
0
votes
1answer
38 views

Finding a sixth degree polynomial that goes through 8 points

For a summative math research assignment, I will have to find a sixth degree polynomial that would ideally go through the following points: (0, 20.5625) (10, 27.5625) (30, 14.5625) (50, 14.6875) (60, ...
5
votes
3answers
121 views

nilpotent endomorphism on finitely generated modules over a domain

If $R$ is a domain and $f: R^n \to R^n$ is an $R$-module endomorphism. Suppose $f^m = 0$ for some $m> 0$. Show that $f^n = 0$. The cases $ m \le n$ is trivial. When $m>n$, I don't have much ...
0
votes
1answer
33 views

Is matrix $A^i A^j = A^j A^i$

I want to know if $$A^i A^j = A^j A^i$$ holds or not. It seems like an obvious, but I am wondering if there is a more formal proof
0
votes
2answers
45 views

How can I show that for matrix $A$ , $A^t A $ is not equal to $ A A^t $ in general?

How can I show that for matrix $A$ , $A^t A \neq A A^t $ $A^t$ means the transpose of $A$. That is the entire question and I have no idea how to begin... please help!
0
votes
0answers
24 views

Looking for mathematical/combinatorial and computational explanation regarding adding values in a $5 \times 4$ (matrix?) with a constraint.

Given the following matrix (not sure if I should call it that): Matrix $5 \times 4$ I want to add all possible combinations of values such that each Horse gets but one value from each Bookie. What I ...
0
votes
0answers
34 views

maximum frequencies of numbers in a matrix

I have a matrix A of size n*n.Consider a new matric M : M[i][j]=max of frequencies of numbers occuring in ith row and jth column(A[i][j]) counted once. I have a ...
2
votes
0answers
22 views

A variation on the $AB$ vs $BA$ nonzero eigenvalues question.

Let $A\in\mathbb{R}^{m\times n}$ and $B\in\mathbb{R}^{n\times m}$, so that $AB\in\mathbb{R}^{m\times m}$ and $BA\in\mathbb{R}^{n\times n}$ both exist. Thanks to Sylvester's determinant identity, we ...
0
votes
2answers
34 views

How to find $\dim W_1$, $\dim W_2$, $\dim W_1+W_2$, $\dim W_1\cap W_2$ for the following spans?

Let $W_1=\{(1,1,2,1), (3,1,0,0)\}$ and $W_2=\{(-1,-2,0,1), (-4,-2,-2,-1)\}$ Apparently $\dim W_1=\dim W_2=2$. For $\dim W_1\cap W_2$, since $(-4,-2,-2,-1)$ can be expressed as ...
1
vote
1answer
22 views

choosing a square matrix to have a product with one 1 und other 0's

Let $A$ be a $m\times n$ real matrix with maximal rank. Let $i\in\{1,\dots,m\}$, $j\in\{1,\dots,n\}$. I'm curious if it is possible (for any choice of $i,j$) to find a square matrix $B$ such that ...
0
votes
1answer
28 views

Dimension of the image of a matrix

So the question asks: Verify if the image of the linear map $T : \mathbb{R}^6 \to \mathbb{R}^3$ given by left multiplication by A= $$\begin{bmatrix}6 & 0 &2 & 2& 3& 4\\0 & -1 ...
1
vote
2answers
37 views

Unit vectors with imaginary numbers

I'm trying to determine if the matrix: \begin{bmatrix} 0 & i \\ 1 & 0 \end{bmatrix} is a unitary matrix. Therefore, the first step I'm taking is to figure out if both $\langle 0, 1\rangle$ ...
1
vote
0answers
23 views

Matrix Inverse as Series

I am looking for different representations of the inverse of a matrix as a power series. One obvious candidate is the Von Neumann series which is given $$A^{-1} = \sum_{k=0}^{\infty} (I-A)^k$$ ...
0
votes
0answers
27 views

Linear transforms and their corresponding invertible matrix.

Let $(1,x,x^2,x^3)$ be a basis for $\mathscr{P_3}(\mathbb{R})$ and let $(1,x,x^2,x^3,x^4)$ be a basis for $\mathscr{P_4}(\mathbb{R})$. Suppose $R \in ...
0
votes
0answers
11 views

Is it always possible to find the Reduced Row Echelon form of a matrix, given the basis of its null space? [on hold]

I tried starting with multiple bases of the null space and each time I was able to write the RREF form of the matrix. However, I have not been able to prove that this is true for all possible bases.
1
vote
1answer
34 views

When is $\mathbf{X}^{T}\mathbf{X}+\lambda\mathbf{I}$ invertible?

The question is quite simple: for a $N \times p$ matrix $\mathbf{X}$ with real entries, when is $\mathbf{X}^{T}\mathbf{X}+\lambda\mathbf{I}$ invertible (where $\mathbf{I}$ is the $p \times p$ identity ...
3
votes
2answers
40 views

Testing the diagonalizability of matrix $B= \left(\begin{array}(\lambda_1 & a & b \\ 0 & \lambda_1 & c\\ 0 & 0 & \lambda_2\end{array}\right)$

How to show that the matrix $$B= \left(\begin{array}(\lambda_1 & a & b \\ 0 & \lambda_1 & c\\ 0 & 0 & \lambda_2\end{array}\right)$$ is diagonalizable when $a\neq0$, when ...
0
votes
0answers
15 views

Matrix transformation into block off-diagonal form

Consider the 4-by-4 matrix $\boldsymbol M = \boldsymbol M_0 + \boldsymbol M_1$, where $\boldsymbol M_0 = \alpha \left( \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 ...
-1
votes
2answers
44 views

Find an upper triangular matrix $A$ such that $A^3=\begin{pmatrix}8&-57\\0&27\end{pmatrix}$

Find an upper triangular matrix $A$ such that $A^3=\begin{pmatrix}8&-57\\0&27\end{pmatrix}$. I tried to solve this problem using Cayley–Hamilton Theorem, but I am unable to solve that.
0
votes
2answers
44 views

What is a basis and dimension of $span\{I,M,M^2,…\}$ where $I$ is the identity matrix and $M$ is invertible squared matrix?

Putting all vectors (matrices) in one gives $$ \begin{bmatrix} 1 & 0 & 0 & m_1 & \cdots\\ 0 & 1 & 0 & m_2 & \cdots\\ 0 & 0 & 1 ...
2
votes
1answer
58 views

Show that $ 4\times4$ matrix has real eigenvalues

I have a real $ 4\times4$ matrix of the form $$ C = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ c_{31} & c_{32} & 0 & c_{34} \\ c_{41} & c_{42} & ...
-1
votes
0answers
54 views

Why this is not always true $\det(A +B ) = \det(A) + \det(B)$ for square matrices [duplicate]

Why $\det(A +B ) = \det(A) + \det(B)$ is not always true for square matrices $A$ and $B$? Note that $A$ and $B$ are square matrices.
0
votes
1answer
25 views

If $A$ and $B$ are $n\times n$ matrices, then $((AB)^{-1})^T=(A^{-1})^T (B^{-1})^T$

Please help me to solve this. Prove that if $A$ and $B$ are $n\times n$ matrices, then $((AB)^{-1})^T=(A^{-1})^T (B^{-1})^T$. a problem involve transpose and inverse of matrices. check the ...
-1
votes
2answers
15 views

Finding the values of the rank and nullity

$Q$ is a $3 \times 3$ matrix that is not invertible. What are all the possible values of the rank and nullity of $Q$ ?
0
votes
1answer
29 views

Centralizer of $A$ is equal to $\langle A \rangle$

Let$$A=\begin{pmatrix} 0 & a \\ 1 & b \end{pmatrix}.$$ How to prove or disprove that the centralizer of $A$ is equal to $\langle A \rangle$ (matrices generated by A)? For a matrix to be in ...
0
votes
0answers
11 views

Translate and Rotate mesh

I have a mesh constituted of some vertices in 3d space, let's call them $(x_1,y_1,z_1),(x_2,y_2,z_2),\cdots,(x_n,y_n,z_n)$. The mesh's central point is $(0,0,0)$. How to find out the new coordinates ...
2
votes
3answers
35 views

Scaling a matrix to make its eigenvalues fall within a certain interval

Suppose I have a diagonalizable matrix $M$ which has all its eigenvalues between $a$ and $b$. Is it possible to scale $M$ to $M_S$ such that all the eigenvalues of $M_s$ lie in the interval $[-1,1]$? ...
3
votes
1answer
39 views

Prove a vector in $\ell^2(\mathbb{Z})$ is zero

Suupose we take a vector $\vec{c}\in\ell^2(\mathbb{Z})$ where $$c(i)=\sum_{k=1}^\infty\frac{c(-k+i)+c(k+i)}{k+1}$$ That is, every elements of the vector is a series with the other terms in $\vec{c}$. ...
0
votes
1answer
18 views

Let $A$ be a $2\times2$ matrix with real entries such that $A$ is invertible. If $Det(A)=k$,and $Det(A+kadj(A))=0$

Let $A$ be a $2\times2$ matrix with real entries such that $A$ is invertible. If $Det(A)=k$,and $Det(A+kadj(A))=0$, then find the value of $Det(A-kadj(A))$ My attempt: $Det(A+kadj(A))=0$ ...
0
votes
1answer
18 views

Simplification of a product of three matrices

Define $$\mathbf{c}_t = \begin{bmatrix} x_{1t} \\ x_{2t} \\ \vdots \\ x_{Nt} \end{bmatrix}\in \mathbb{R}^N$$ where all entries are in $\mathbb{R}$, $t = 1, 2, \dots, p+1$. I am trying to simplify ...
1
vote
0answers
32 views

Matrices representing a map between free modules of infinite rank and Fitting's Lemma (Eisenbud)

p.497 of Commutative Algebra with a View Toward Algebraic Geometry, Eisenbud: If $\phi: F \rightarrow G$ is a map of free modules, then $I_j\phi$ is the image of the map $$\Lambda^j F ...
0
votes
1answer
28 views

Find the dimension and a basis of a subspace

Let $U$ is the set of all commuting matrices with matrix $A= \begin{bmatrix} 2 & 0 & 1 \\ 0 & 1 & 1 \\ 3 & 0 & 4 \\ \end{bmatrix}$. Prove ...
0
votes
2answers
25 views

System of linear equations - Resolution

$$ \left( \begin{matrix} \pi_1 & \pi_2 & \pi_3 \end{matrix} \right) = \left( \begin{matrix} \pi_1 & \pi_2 & \pi_3 \end{matrix} \right) \begin{bmatrix} 0.6 & 0.3 & ...
0
votes
0answers
24 views

If $ 0<A <B$, is it true that $||A||_p < ||B||_p$ for all $p$ positive integer?

I have got some questions regarding matrix norms and inequalities. We only consider square, nonsingular matrices in the following. If $ 0<A <B$, is it true that $||A||_p < ||B||_p$ for all ...
0
votes
2answers
19 views

Finding the general solution of a system of linear equations

so I've come across this question in preparation for an exam: Let $A$ be a $4\times 4$ matrix where $rank(A)=3$. The vectors $(1,2,0,-1),(0,2,1,1)$ are solutions to the system ...
0
votes
1answer
23 views

Multiplication of block matrices

Let $J_{m \times n}$ be an $m \times n$ matrix of $1$'s (and to abbreviate we write $J_m=J_{m \times m}$) and let $M=\begin{pmatrix} 0_n & J_{n \times m} \\ J_{m \times n} & 0_m ...
1
vote
2answers
36 views

Linear maps of polynomials, the bases of the space and their corresponding matrix.

Suppose $T \in \mathrm{Hom}(\mathscr{P}_3(\mathbb{R}),\mathscr{P}_4(\mathbb{R}))$ is defined by: $$Tp(x)=(x^2p(x))',$$ for all $x \in \mathbb{R}$ and $S \in\mathrm{Hom} ...
0
votes
1answer
54 views

Theorems restricting the eigenvalue of a matrix

I have a square matrix $C$, whose entries I will denote by $c_{ij}$, and I would like to bound the magnitude of its eigenvalues. Each $c_{ij}$ is defined in terms of $s_{ij}$ and $S_j$ as follows: ...
0
votes
1answer
20 views

How to calculate the variation of a matrix?

Suppose we have two diagonal matrices $$ A_{\mu \nu}=\left(\begin{array}{cccc} \rho(t) & 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0 ...
2
votes
1answer
39 views

Group isomorphism between two groups .

Let $SO_2(\mathbb{R})$ be the group of rotations of the circle under the operation of composition. And consider $\mathbb{R}$ as an additive group. Prove that $SO_2(\mathbb{R}) \cong \begin{Bmatrix} ...
2
votes
1answer
25 views

Check if two square matrices are similar.

Check if matrices $A= \begin{bmatrix} 1 & 1 & 5 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \\ \end{bmatrix}$ and $B=\begin{bmatrix} 1 & 7 ...