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
1answer
26 views

Matrix representations of tensors

I've been trying to teach myself general relativity, and I always get stuck at the same point: I don't really understand what the metric tensor is. Unless I'm incorrect, and please correct me if I'm ...
1
vote
2answers
38 views

Explicit example of a basis of invertibles for $n\times n$ matrices

Using a topological (+linear algebra) argument, one can establish the existence of a basis spanning any square matrix using invertible matrices ( $span(GL_n (\Bbb{R}))=\mathcal{M}_n (\Bbb{R}) $). But ...
1
vote
0answers
32 views

Abuse of notation ? $(A\mid M_{n\times p})$ to denote a set of matrices…

Let $A\in M_{n\times m}$. Would it be considered an abuse of notation to write $$\left(A\mid M_{n\times p}\right)\subseteq M_{n\times (m+p)},\tag{1}$$ where $\mid$ denotes matrix augmentation ? By ...
0
votes
3answers
33 views

matrix times its transpose equals minus identity

What would be a good example for a $n\times n$ matrix such that $A^{T}A=-I$? It would be better if you can give a matrix which has a well-known name (like "rotation matrix" etc). Thanks!
0
votes
0answers
25 views

Find a relation between a,b and c

$ a,b,c\in \Bbb R$ $2x_1+2x_2+3x_3=a$ $3x_1-x_2+5x_3=b$ $x_1-3x_2+2x_3=c$ if a,b and c is a solution of this linear equation system find the relation between a,b and c I dont understand the ...
1
vote
1answer
22 views

proof of matrix positive semi definite

I have question about the proof about positive semidefinite (p.s.d) of a matrix. Let's say $M$ is a d by d p.s.d matrix, $H$ is any d by n matrix with n larger(or much larger) than d. Then how about ...
1
vote
1answer
8 views

Change of Basis in Canonical Correlation Analysis

I am studying canonical correlation analysis. And I'm completely stumped for the last few days at the following manipulation. How does the following change of basis works? The equation doesn't even ...
2
votes
2answers
44 views

Computing matrix exponential of non-diagonalizable 2x2 matrix

Compute $e^M$ where $M=\begin{bmatrix}8 & -1\\4 & 4\end{bmatrix}$ Because M is not diagonalizable i try to use Jordan decomposition so i find the Jordan matrix to be $J=\begin{bmatrix}6 & ...
2
votes
1answer
14 views

Computation of a matrix exponential for general dimensions

Originally I wanted to prove something else then I hit on this question that I find quite interesting but I don't know how to prove it elegantly. Let $$J=\begin{pmatrix} 0 & I \\ -I & 0 ...
1
vote
0answers
30 views

Eigenvalues of Moore–Penrose Pseudo-Inverse of a Symmetric Matrix

I was wondering if there is any bound or inequality for the eigenvalues of Moore–Penrose pseudo-inverse of a real $n\times n$ symmetric matrix $A$ in terms of eigenvalues of $A$, namely $\lambda_i$'s ...
0
votes
1answer
28 views

Union of subspace

Q. Say U and W are subspaces of a a finite dimensional vector space V (over the field of real numbers). Let S be the set-theoretical union of U and W. Which of the following statements is true: a) ...
0
votes
1answer
36 views

Is this “truncating” matrix function well known?

I'm working with a kind of "truncating" matrix function $\tau_r:M_{n\times n}\to M_{n\times r}$, where $r\leq n$, defined by $\tau_r(A)=B$, where $b_{ij}=a_{ij}$ for $j\leq r$. Is this a well known ...
1
vote
2answers
46 views

Eigenvectors and Kronecker product

Let us define $$ v:=v_A\otimes v_B\quad (*) $$ where $v_A$ is a fixed vector in $\mathbb{R}^{d_A}$, $v_B$ is any vector in $\mathbb{R}^{d_B}$ and $\otimes$ denotes the Kronecker product. To rule out ...
2
votes
0answers
44 views
+50

What do we call the result of wedging together the columns of a matrix?

We can wedge together the columns of a square matrix to compute its determinant. More generally, the exterior product of the columns of a $b \times a$ matrix tells us the determinant of each $a \times ...
0
votes
1answer
46 views

Matrix derivative of a special function

I need some help for calculating the matrix derivative of a special function. I have checked Wikipedia and Matrix Cookbook, but could not get the answer or idea. Let us define $f(X)$ as ...
-4
votes
0answers
31 views

Compute the operator norm of the linear transformation defined by the following matrix. [closed]

Compute the operator norm of the linear transformation defined by the following matrix. \begin{bmatrix} 2 & 0 \\ 0 & -3 \end{bmatrix}
1
vote
2answers
40 views

How can you expand the adjoint of a matrix into a polynomial with matrix coefficients?

This book contains an algorithm which claims that a matrix $sI - A$, where $A$ is some $n \times n $ square matrix and $s$ a variable can be expanded into $$adj(sI - A) = K_0 s^{n-1} + K_1 s^{n-2} ...
2
votes
2answers
76 views

How to derive this matrix equation

$$ \left< Z,X-L-S \right> \quad +\quad \frac { r }{ 2 } \left\| X-L-S \right\|_F^2 \quad =\quad \frac { r }{ 2 } { \left\| L-\left( X-S+\frac {Z}{r} \right) \right\| }_F^2 $$ I think $ ...
-1
votes
2answers
46 views

If $A,H\in GL_{n}(\mathbb{R})$, What happen for $H A H^{-1}$, if $h(i,j)\longrightarrow‎ 0$?

If $A$ and $H$ is a two $n\times n$ matrixs, such that $\det(A)\neq 0$ and $ \det(H)\neq 0$, what happen for $H A H^{-1}$, if $h(i,j)‎\longrightarrow‎ 0 $? Is $H A H^{-1}‎\cong A$?$ \ \ \ (1\leq ...
0
votes
1answer
26 views

General Element of U(4)

Relating back to previous question about how to write a general element of $U(2)$, I am now wondering about how to write a general element of $U(4)$. Define $\Gamma_{(i,j)}:=\sigma_i\otimes\sigma_j$ ...
1
vote
1answer
28 views

Row equivalence implies independent columns?

I need to prove that "given" two matrices are row equivalent, a set of columns of the first matrix are linearly independent iff the corresponding columns of the second matrix are linearly independent. ...
1
vote
1answer
14 views

Linear transformation with matrices in base

Consider the vector space of real $2 x 2$ matrices and take as base $\{{E_{11},E_{12},E_{21},E_{22}}\}$. Where $E_{ij}$ represents the matrix with a $1$ in the $i$-th row and $j$-th column and the ...
3
votes
2answers
40 views

What is an Eigenbasis and how do I calculate it with the information below.

I have the matrix $$A = \begin{bmatrix} 4 & 2 & 2\\ 2 & 4 & 2\\ 2 & 2 & 4 \end{bmatrix}$$ I've calculated the Eigenvalues and Eigenvectors as follows with help in a previous ...
4
votes
1answer
39 views

Example of an nonidentity element in the kernel of the map.

This question is related to my previous question here. Let $n, m > 1$. The map $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/m\mathbb{Z}$, of reduction mod $m$, induces a group homomorphism $F: ...
1
vote
3answers
20 views

Finding the rank of an non-invertible matrix

I have a $3\times3$ matrix with three different eigenvalues $0,1, 2$. The question is: what is the rank of this matrix? If the matrix was invertible, I could say that the rank was equal to $n=3$. ...
0
votes
0answers
17 views

Why do the diagonals of a matrix have to be greater than 0 for the matrix to be positive definite? [duplicate]

Why do the diagonals of a matrix have to be greater than 0 for the matrix to be positive definite? Please provide an example (with numbers if possible).
9
votes
1answer
59 views

Induced group homomorphism $\text{SL}_n(\mathbb{Z}) \twoheadrightarrow \text{SL}_n(\mathbb{Z}/m\mathbb{Z})$ surjective?

Let $n, m > 1$. The map $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/m\mathbb{Z}$, of reduction mod $m$, induces a group homomorphism $F: \text{SL}_n(\mathbb{Z}) \to ...
1
vote
2answers
50 views

What are the Eigenvectors in the following matrix?

I have the matrix A: \begin{bmatrix} 4 & 2 & 2\\ 2 & 4 & 2\\ 2 & 2 & 4\\ \end{bmatrix} I found $\lambda I_n - A$ to be: \begin{bmatrix} (\lambda -4) & -2 & -2\\ -2 ...
0
votes
2answers
25 views

Jordan Normal Form matrix decomposition into the sum of parts that commute.

I'm learning about Jordan Normal Form matrices, and I've read that we can decompose a Jordan Normal Form matrix into the sum of two parts $$J=N_J+D_J$$where $D_J$ is the diagonal part (i.e. the ...
0
votes
1answer
68 views

Find a matrix of the linear map in the given basis

Let $Y = \{y_1, y_2, y_3\}$ be a basis for $R^3$ where $y_1 = (1, 1, 1)$, $y_2 = (4, 1, 1)$ and $y_3 = (1, 1, 2)$. Let $W = \{w_1,w_2\}$, $w_1 = (1, 1)$ and $w_2 = (2, 4)$ be a basis in $R^2$. Need ...
-1
votes
0answers
16 views

Increasing a singular value [closed]

Can any one tell me the effect of increasing one singular value (say 10 times ) larger than others.Whether it has any importance in optimization Problems .
0
votes
1answer
12 views

If the Rref(A) of a 3x3 matrix is I(A), is this a valid eigenvector?

For the vector A: EDIT: I had originally multiplied the matrix by -1. Apologies. $$ \begin{bmatrix} 4 & 2 & 2 \\ 2 & 4 & 2 \\ 2 & 2 & 4 \\ ...
1
vote
1answer
61 views

Exponential of 4x4 matrix

It is asked to calculate $e^{tA}$, where $$A=\begin{pmatrix} 0&1 & 0&0 \\ 3\omega ^2&0 &0 &2 \omega \\ 0& 0 & 0 &1 \\ 0& -2 \omega &0 ...
0
votes
0answers
22 views

What does it mean to find the principal directions and radii, given a matrix?

Do I compute the SVD, getting $$U \Sigma V^*$$ and then read off the column vectors in U that correspond to the positive singular values? These column vectors span the range of the matrix A, and I ...
1
vote
0answers
36 views

minimize smallest eigenvalue

Assume $P_A,P_B$ are probability transition matrices (each element is nonnegative and row sum is 1) and $v$ is probability row vector (each element is nonnegative and sum of elements is 1). How to ...
0
votes
0answers
31 views

Help explain “3d algebra”

The following is part of my lecture note, but I get lost after the first paragraph. I know what is "even" permutation and "odd permutation" which I learned from my abstract algebra course, and figured ...
-1
votes
2answers
59 views

A square matrix is called skew-symmetric if $A^T=-A$. Prove that if $A$ and $B$ are skew-symmetric matrices, then $A+B$ is skew symmetric.

A square matrix is called skew-symmetric if $A^T=-A$. Prove that if $A$ and $B$ are skew-symmetric matrices, then $A+B$ is skew symmetric.
2
votes
1answer
31 views

$f(x)$ be the characteristic polynomial of a matrix $A \in M_n(\mathbb R)$ ; then is it true that $f(1)=1+\operatorname{trace}(A)+O(\|A\|^2)$?

Let $f(x)$ be the characteristic polynomial of a matrix $A \in M_n(\mathbb R)$; then is it true that $f(1)=1+\operatorname{trace}(A)+O(\|A\|^2)$ ? I need a proof if it is true ; or any modification ...
1
vote
1answer
24 views

How to show that $D \det_A (H)$ exists and equals $\det( adj(A)H)$?

Consider the function $\det : M_n(\mathbb R) \to \mathbb R$ ; how to show that for any $A , H \in M_n(\mathbb R)$ , the derivative operator of determinat of $A$ evaluated at $H$ i.e. $D \det_A (H)$ ...
1
vote
1answer
25 views

Condition for guaranteed minimum-rank solution

Consider the following rank minimization problem of a positive semi-definite matrix $X$: \begin{equation*} \begin{aligned} & \underset{X}{\text{minimize }} & & rank(X) \\ ...
1
vote
0answers
35 views

Inverse of $A^\top BA+C^\top DC$?

I'm numerically solving a system of equations of the form: $$Mx = b$$ where: $$M = A^\top BA+C^\top DC,$$ $B$ and $D$ are block-diagonal, $A$ and $C$ are $n\times m$ matrices with $m \leq n+3$. ...
0
votes
0answers
32 views

What kind of transformation does each matrix determine?

I have been given 5 matrices, been asked 'What kind of transformation does each matrix determine?', to multiply them and to 'Explain how the transformations determined by each matrix in the pair are ...
0
votes
1answer
27 views

A complex matrix with real eigenvalues

Let $A$ be a $10\times 10$ matrix with complex entries and all eigenvalues non-negative real numbers and at least one eigenvalue strictly positive . Then there exist a matrix $B$ ...
1
vote
1answer
43 views

If $V$ is a vector space and $U$ & $W$ are subspaces of $V$, such that $U \oplus W = V$! Need help with proofs!

Consider the map $\rho : V \to V$, defined by $\rho(v) = u − w$, where $v = u + w$, $u \in U$, $w \in W$. Show that: i. $\rho$ is well defined and it is linear; ii. $\rho(u) = u$, $\forall u ∈ U$; ...
0
votes
0answers
12 views

Conversion of network-like matrix

I have given a network in the following form (Example): x1 + x2 - x3 = 0 x3 + x4 - x5 = 0 x5 + x6 - x7 = 0 where = is something like a node, where flow needs to ...
2
votes
0answers
25 views

Prove that this matrix is total unimodular

Is there an easy way to prove that this matrix is total unimodular ? $$ \begin{bmatrix} 1 & F_1 & 0\\ 1 & 0 & F^T_1 \\ 0 & F_2 \end{bmatrix} $$ $1$ is the identity matrix, ...
0
votes
0answers
26 views

For any $n>1$ , $ Aut(\mathbb Z^n) \cong GL(n , \mathbb Z)$ ? And for any $m,n >1$ , $Aut(\mathbb Z_m ^n) \cong GL(n , \mathbb Z_m)$ ?

Is it true that for any $n>1$ , $ Aut(\mathbb Z^n) \cong GL(n , \mathbb Z)$ ? And is it true that for any $m,n >1$ , $Aut(\mathbb Z_m ^n) \cong GL(n , \mathbb Z_m)$ ? ( One problem I'm ...
2
votes
3answers
39 views

Is $A(x, y, z) = (xy + z, yz - x)$ a linear function?

Is this a Linear function? If yes, what is its matrix? $$A:\mathbb{R}^3\to\mathbb{R}^2, A(x, y, z) = (xy + z, yz - x);$$ Since they are all first degree variables, i think that this is a linear ...
-4
votes
2answers
27 views

Question about basis of Vector Space [closed]

Show that $B$ is a basis for $\mathbb{R}^2$:Where $$B=\{(-1,1) , (2,3)\}$$
1
vote
0answers
34 views

Generalized inverse of matrix product involving a positive semi-definite matrix

I have the following: A real square positive definite matrix $A$, and a real square conformable positive semi-definite matrix $B$. I form the product $$C = A^{-1}BA^{-1}$$ and I wonder, is it true ...