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
8 views

Multiplying inverse matrices easily.

Okay, this is more of a confirmation question than anything: I have been given two matrices $A^{-1}$ and $B^{-1}$. Then the inverses of these are: $A$ and $B$. I need to calculate $(AB^{T})^{-1}$. ...
0
votes
1answer
12 views

Alternative methods to solve DLP for $GL_{3}(\mathbb{F}_2)$

Is there (or rather what is) a more elegant/efficient way to solve the DLP for $g^x=h$ in $GL_3(\mathbb{F}_2)$ where $$g=\begin{pmatrix}0 &1 & 1 \\ 1 &1 &1 \\ 1&0&1 ...
1
vote
1answer
11 views

Row Equivalent Matrices

If I have a matrix $A$, where there are zeros everywhere apart from the first row, what are the matrices that are not row equivalent to $A$. I know that if two matrices are row equivalent, we can ...
0
votes
1answer
16 views

Can square matrices be represented as the union of vectors and some other set?

I believe all invertible matrices can be representable as $A = |A| \, \mathrm{adj}\left(A\right)^{-1}$ (a rotation part times a scaling part.) All invertible matrices can then be mapped to vectors ...
1
vote
3answers
26 views

Matrices and diagonalization.

I could verify that $P$ statement is false by just calculating the determinant but couldn't answer $Q$ statement. Any clue about $Q$??
0
votes
2answers
32 views

It is true that $rank(xy^T)=1$? [on hold]

Let $x,y\in \mathbb{C}^n$. It is true that $rank(xy^T)=1$?
-2
votes
0answers
21 views

Gaussian Elimination vs matrix inversion [on hold]

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
8 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
14 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
1answer
32 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
23 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
22 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
40 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
126 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
42 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
2answers
34 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
29 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
39 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
29 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
12 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
36 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
48 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
18 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
45 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
16 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
31 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
37 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
40 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
29 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
27 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
21 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 ...