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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1answer
31 views

A is a square matrix where $3A^9- 7A^4 + 4A = I$. Prove that A is invertible by finding $A^{-1}$

The question is: A is a square matrix where $3A^9- 7A^4 + 4A = I$. Prove that A is invertible by finding A^-1. I have looked at other similar questions on this site: 1. Here 2. and Here But they use ...
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0answers
15 views

How can I backward substitute this 4x4 guassian elimination

I was trying to solve this guassian elimination and I think i have it into the required 'staircase' pattern needed for guassian elimination. And here it is after forward elimination: 1 2 0.5 | 3.5 ...
1
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1answer
32 views

Coding a Message using Matrices

Ok, so this problem I've been working on for the past hour, with no answer. In coding a message, a blank space was represented by 0, an $A$ by 1, a $B$ by 2, a $C$ by 3, and so on. The message was ...
3
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0answers
59 views

Statement about $(I-A)^{-1}$ matrices

Let $A \in \mathbb{R}^{n \times n}$ and let denote $I$ the $n \times n$ identitiy matrix. Theorem. If $(I-A)$ is invertible and $(I-A)^{-1}$ is a nonnegative matrix and there is such a diagonal ...
0
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1answer
46 views

If the inner product of Ax with x is 0 for all x, then A=0. [duplicate]

Given matrix $A\in M_{n}(\mathbb{C})$, if $\left<Ax,x\right>=0$ for all $x\in \mathbb{C^n}$, then $A=0_{n}$. (Here $\left<a,b\right> = b^{\ast}a$ where "*" is the conjugate transpose.) ...
2
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0answers
15 views

Clifford algebra - Gamma matrices

Let's say we have $\gamma^{a}$ matrices $(a=1,2,...,D)$. They satisfy the following condition $$\gamma^{a}\gamma^{b}+\gamma^{b}\gamma^{a}=2\delta^{ab}I^{N\times N}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$ ...
0
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3answers
17 views

If $A$ and $B$ are real matrices and $X,Y$ are are non-singular square matrices such that $XA=BY$

If $A$ and $B$ are real matrices and $X,Y$ are are non-singular square matrices with real entries such that $XA=BY$ then which of the following is true? $1. \dim(X)=\dim(Y)$ $2. \dim(A)=\dim(B)$ ...
1
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0answers
8 views

A matrix being “diagonal with a $c$-column border”, what does it mean?

The matrix in the middle of the beginning part, i.e., $$ Q=\begin{bmatrix} \operatorname{diag}(s) & L \\ 0 & K \end{bmatrix} $$ In the context, $L$ is a $r$ by $c$ matrix. What is the ...
3
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1answer
18 views

Estimate the Cauchy integral for matrix-valued functions

Recently, I became familiar with the concept of the "matrix function via Cauchy integral", i.e., $$f(A):=\frac{1}{2\pi i}\int_\varGamma f(z)(zI-A)^{-1} \mathrm{d}z$$ Furthermore, it can be shown that ...
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2answers
14 views

Inspecting vector linear dependence, one line in matrix all zeros

Suppose we have vectors $v1, v2,v3$ and we want to inspect their linear dependence. They are linearly dependent when the only solution for the equation $\alpha * v1 + \beta * v2 + \gamma * v3 = 0$ is ...
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1answer
23 views

Matrix equations?

I came across this problem: I have successfully found the bases of the null space but I can't seem to understand the second part. I looked around online and found nothing useful. I would appreciate ...
1
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1answer
16 views

Minimal polynomial of the operator $T:V\oplus W\to V\oplus W$

Let $V$ & $W$ be two finite dimensional vector spaces over $R$ and let $T_{1}:V\to V$ & $T_{2}:W\to W$ be two linear transformations whose minimal polynomials are given by ...
0
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1answer
14 views

For what values of K is the matrix diagonalizable?

Can someone please help with this problem. I have tried it several times but can't get the answer as k*0 = 0 makes it hard to work with k.
3
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0answers
64 views

What do these symbols mean? One looks like an inverse “T”, another looks like ^

I saw an equation as the following one. What is the meaning of the symbol looking like an inverse T? How about the symbol "^"? //////////////////////////////////////////////////////// The ...
1
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1answer
27 views

show that $A(t)\exp(\int_{t_0}^t A(s)\,ds )=\left(\exp(\int_{t_0}^t A(s)\,ds )\right)A(t)$, when $A(t)$ is symetric.

$A(t)$ is a symetric matrix for $t\in [t_0,a]$. show that $$A(t)\cdot \exp\left(\int_{t_0}^t A(s)ds \right)=\exp\left(\int_{t_0}^t A(s)ds \right)\cdot A(t)$$ it is easy but exhausting to show for ...
1
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1answer
26 views

Prove that a matrix is invertible?

Let $A_{20 \times 20}$ be a real matrix such that: $\ \ \ \bullet$ $a_{ii}=0$ for $1 \le i \le 20$ $\ \ \ \bullet$ $a_{ij} \in \{-1;1\}$ for $1 \le i,j \le 20$ and $ i \neq j$ Prove that $A$ is ...
2
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3answers
37 views

Inverse Matrices and Infinite Series

Given that $C=I+A+A^2+A^3+ \ldots$ Prove that I-A is the inverse of $C$ Hint: Use the infinite series technique for finding inverse of a matrix. Now I know with an infinite geometric series with a ...
1
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1answer
28 views

How to Change Summation Expression $\sum_{i=1}^N \mathbf{X}_i^{\top}\mathbf{\Omega}^{-1}\mathbf{X}_i$ into Matrix Expression

Let $\mathbf{X}_i$ be a $G \times K$ matrix, and suppose are $i=1,...,N$ of these matrices. Note that \begin{align} \sum_{i=1}^N \mathbf{X}_i^{\top}\mathbf{X}_i &= \begin{bmatrix} ...
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0answers
25 views

Does this guassian elimination have a solution?

I was asked to find the following solutions using guassian elimination, but I was unsure of my answers since it became quite messy but the variables still somehow fit: $$\left[\begin{array}{ccc|c} ...
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0answers
23 views
0
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1answer
29 views

Prove True or false : If A and B are nxn invertible matrices and (AB)^2=A^2B^2, then AB=BA

This looks like it is false but the thing is I can't find a counter example for it.
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1answer
23 views

Number of positive, negative eigenvalues and the number of sign changes in the determinants of the upper left submatrices of a symmetric matrix.

How do we prove that the number of sign changes in the sequence of the determinants of the upper-left matrices of a symmetric matrix $A$ corresponds to the number of positive and negative eigenvalues ...
1
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1answer
19 views

How to prove that an M-matrix is inverse-positive?

Wikipedia says that The inverse of any non-singular M-matrix is a non-negative matrix." To be more precise, if $A$ is an M-matrix, then the entries of the inverse of $A$ are all non-negative, ...
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2answers
19 views

Orthogonal Matrices and Similarity Transforms

Sorry I can't be more specific with the title. I really don't know what to call this and about 2 hours of Googling has yielded no results. All we are given: $U$ is $n\times n$ and orthogonal $Ax = ...
0
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1answer
12 views

Covariance matrix of Y when we have the covariance matrix of X

If the random vector $\mathbf{X}$ is transformed according to \begin{align*} Y_1 &= X_1\\ Y_2 &= X_1 + X_2 \end{align*} and has a covariance matrix $$ \mathbf{C}_X = ...
3
votes
1answer
28 views

For what kind of matrix does it hold $\|XA\|_{1} \leq 1$ for a given $\|A\|_{1} \leq 1$.

All matrices are real. By $\| \cdot \|_1$ I mean a matrix norm induced by the vector norm $L_1$, i.e. the max of the column sums of absolute values. The matrix $A$ is given and we have $\|A\|_{1} ...
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0answers
25 views

Prove that theorems about trace of matrix: [on hold]

$ 1-) $    If A is an nxn symmetric matrix with r nonzero characteristic roots $ \lambda_1,\lambda_2,...,\lambda_r $, then $ tr(A^-)=\sum_{i=1}^r \lambda_i^{-1}$ $ 2-) ...
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1answer
26 views

Prove that theorem about trace of non-negative matrix: [on hold]

If A is a non-negative nxn matrix , then $\ tr(A) = 0 $ if and only if $ A =0$
1
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2answers
26 views

How to show that $\| QA\|_2=\| A \|_2$ where $Q$ is unitary (for a matrix A)

I want to show that for a unitary matrix $Q$ and a matrix $A$ that $$ \|QA\|_2=\|A\|_2$$ I start with the definition of matrix induced norms: $$\| QA \|_2 = \sup_{x \neq ...
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1answer
12 views

Are the following vectors in the range of A

Let $A$ be the following matrix: $$ \left( \begin{array}{cccccc} 1 & 2 & 1 & 3 & 2 & 1\\ 2 & 0 & 3 & 2 & 3 & 0 \\ 4 & 2 & 1 & 1 & 2 & 1 \\ ...
0
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2answers
23 views

Number of Jordan canonical form of a matrix

Let, $A\in M(3,C)$. Assume that the characteristic & minimal polynomial of $A$ are known. Then what is the number of possible Jordan form of $A$ and how? What changes if we replace $C$ by $R$ or ...
5
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2answers
41 views

About integral binary quadratic forms fixed by $\operatorname{GL_2(\mathbb Z)}$ matrices of order $3$

I am reading this paper of Manjul Bhargava and Ariel Shnidman, and I want to prove this claim, which appear at the first paragraph of Theorem $14$: Up to $\operatorname{SL_2}(\mathbb Z)$ ...
0
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2answers
18 views

'A' transpose inverse equals to 'B' transpose

I searched everywhere but I could not find a solution to this problem. Let $A$ and $B$ be invertible matrices with $AB = I$. Show that
0
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1answer
24 views

Build a rotation matrix that rotates 30 degrees along the axis (1,1,1)?

Why does the following image equal what it equals? Why does x,y,z equal that? 1/sqrt(3),1/sqrt(3),1/sqrt(3)
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0answers
21 views

Which curve we will get under $\mathcal{A} \in M_2(\Bbb{R})$ from a unit circle

If I have a circle: $x^2+y^2=1$, It's parametric equation is : $$\begin{cases} x = \cos\theta \\ y = \sin\theta \end{cases}$$ under some transform: $A=\begin{pmatrix} a & b\\ c & d ...
1
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2answers
53 views

determine signature of matrix

what is the signature of this matrix: $\begin{pmatrix} -3&0&-1 \\0&-3&0 \\ -1&0&-1 \end{pmatrix}$ ? I tried calculating them without eigenvalues; this should be done via ...
0
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1answer
21 views

what do these odds ratios represent?

I am reading this article in which is given the matrix of the joint probabilities of two random variables, X=$(x_1,x_2)$ and Y=$(y_1,y_2)$. Let's say they are $(p_{1,1},p_{1,2},p_{2,1},p_{2,2})$. ...
1
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0answers
30 views

Matrix Decompositions: Difference between Cholesky Decomposition, Eigendecomposition and Jordan Normal Form Decomposition

I recently created a related topic about the square root matrix, in case you'd like to refer to that one. Here's what we want: Consider the matrix $\Omega=E(\mathbf{u}^{\top}\mathbf{u})$, where ...
1
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1answer
21 views

Multivariate Calculus - Partial Derivatives - Implicit Differentiation - Chain Rule

Let $z = z(x,y)$ be defined implicitly by $F(x, y, z(x,y)) = 0$, where $F$ is a given function of three variables. Prove that if $z(x,y)$ and $F$ are differentiable, then $$\frac{dz}{dx} = - ...
3
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0answers
33 views

Comparing/contrasting hyperbolic and Euclidean geometry - or, on how ${\rm PSO}_2(\Bbb R)$ sits inside ${\rm PSL}_2(\Bbb R)$

I am studying hyperbolic geometry, in particular comparing and contrasting it with familiar Euclidean geometry. Let $\Bbb E$ be the Euclidean plane, and $G={\rm Iso}^+(\Bbb E)$ be the group of ...
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0answers
13 views

Finding equation of plane in 3D

I was given 3 points on a plane: (5, 4, −8),(1, 6, −3) and (7, −2, 5) I was trying to find the equation of the plane and did the following: I chose two vectors to cross multiply to find the normal ...
1
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1answer
20 views

If $A_{n\times n}$ and $B_{n\times n}$ are both nonsingular real matrices, where $n$ is odd, show that $AB + BA \neq0$.

I have been puzzling over this for a while now. I tried to find something in the properties of nonsingular matrices as well as the properties of determinants that might relate, but so far I've found ...
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0answers
23 views

Linear systems, eigenvectors

For each of the following linear systems of differential equations, (i) find the general real solution (ii) show that the solutions are linearly independent (iii) draw the phase portrait a. $$\dot ...
0
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1answer
13 views

Finding hermitian conjugate and inverse of a complex matrix

I have the following matrix: $$ F = [e^{i\frac{2\pi kl}{n}}]^{n-1}_{k,l=0} \in \mathbb{C}^{n,n} $$ for $n = 1,2,3,...,i$ I need to find $F^HF$ and $F^{-1}$ where $F^H$ is a hermitian conjugate ...
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0answers
9 views

Copositive matrices. [on hold]

Copositive matrices. A matrix X^2∈Sn is called copositive if zTXz≥0 for all z≥0. Verify that the set of copositive matrices is a proper cone. Find its dual cone
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0answers
33 views

Solving matrix equation of the form $(AX)^2+(BY)^2=D$

Is there any method that can solve the matrix equation of the form $(AX)^2+(BY)^2=D$? $A$ and $B$ are matrices, $X$, $Y$ and $D$ are column vectors. (Solve for $X$ and $Y$) I originally have two ...
0
votes
1answer
28 views

About kernel space

Both the square and symmetric matrices $A$ and $B$ are positive semidefinite. Moreover, $A-B$ is positive semidefinite and $\text{rank}(A)=\text{rank}(B)$. Based on these conditions, can we have ...
1
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1answer
49 views

Given the matrix $A^k$, how to get $A^{k+1}$?

Given: $$A^k = \left(\begin{array}{rr} \cos kx & \sin kx \\ -\sin kx & \cos kx\end{array}\right)$$ $$A^{k+1} \overbrace{=}^? \left(\begin{array}{rr} \cos kx & \sin kx \\ -\sin kx & ...
0
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0answers
26 views

Finding a linear transformation with respect to different bases

Let $f: \Bbb R^2 \rightarrow \Bbb R^2$ be the linear transformation which rotates objects in the plane around the origin by 30 degrees counterclockwise. Find a matrix F for $f$ with respect to the ...
1
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0answers
21 views

How to find the basis of a matrix by using Gauss-Elliminaton?

I confuse that, This is my calculating process, Where i do the mistake in this process? I hope to understand this error.