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
42 views

Find the matrix X such as A . X is close to B

Consider : A an m by n matrix B an ...
0
votes
0answers
16 views

Help understanding a homework problem (Preconditioning matrices, numerical methods)

Below is a link to the problem (because I didn't want to have to go through the pain of TeXing it all out myself), the basic idea is we are supposed to be first showing that a specific matrix has a ...
0
votes
1answer
18 views

symetric matrix inverse

Is there an easy way to invert a 3x3 symmetric matrix? for example A = $\begin{pmatrix} -1& 2& 0\\ 2& -5& 0\\ 0& 0& ...
0
votes
1answer
24 views

How to denote dimensions

I am struggling with nomenclature. If I have matrix $M \in \mathbb{R}^2 \times \mathbb{R}^4$ it would be considered an element of an 8-dimensonal vector space. If I index $M$ by two indices $i$ and ...
1
vote
1answer
24 views

Does $A \succeq B A^{-1} B$ imply that $A \succeq B$?

Let $A,B$ be two symmetric matrices with equal dimensions. Suppose $A \succeq 0 $ (ie, PSD), $B \succeq 0$ and $$ A - B A^{-1} B \succeq 0.$$ Then is it true that $A-B \succeq 0 $?
-1
votes
0answers
14 views

My professor asked me to link the Singular Values of a matrix to the Input matrix ? what does he mean by that [closed]

So i tried doing this solution Singular values = sqrt ( eigenvalues ( AA' ) ). But he said that it wasn't the correct answer . what does he mean by linking Singular Values to the Input matrix?
-1
votes
0answers
10 views

Exponential matrix using Laplace transform - reference request [closed]

I am looking for a book that covers the concept of exponential matrix in detail using the Laplace transform plz
0
votes
0answers
15 views

Question with Gauss-Jordan elimination produces the matrix

Please help me explain the problem below: How can I use Gauss-Jordan to get all bottom roll become 0? Thank you so much!
2
votes
1answer
41 views

Show that determinant is equal to determinant of each variable

Show that $$\begin{vmatrix} na_{1}+b_{1} & na_{2}+b_{2} & na_{3}+b_{3}\\ nb_{1}+c_{1} & nb_{2}+c_{2} & nb_{3}+c_{3}\\ nc_{1}+a_{1} & nc_{2}+a_{2} & nc_{3}+a_{3}\\ \notag ...
0
votes
0answers
17 views

As a square matrix's size increases from dimension 2 to say 50, how does the variance of the matrix's determinant change? [duplicate]

both for the situation where elements in the matrix are randomly assigned any number and where elements are assigned a value uniformly distributed between 0 and 1. Thanks so much! (This is not a ...
0
votes
1answer
26 views

How to find matrix $A$ from the relation: $A\times (A^TA)^{-1}\times A^T = B$

Kindly help me in the following: I have two Matrices, $A$ of size $(n\times m)$; and $B$ of size $(n\times n)$, where $n>m$. $A$ is unknown, but $B$ is known. $(A^TA)$ is invertible $B$ is ...
1
vote
1answer
28 views

Practical use of matrix right inverse

Consider a matrix $A \in \mathcal{R}^{m \times n}$ In the case when $rank(A) = n$ then this implies the existence of a left inverse: $A_l^{-1} = (A^\top A)^{-1}A^\top$ such that $A_l^{-1}A = ...
0
votes
0answers
10 views

Simultaneously Diagonalizable matrices using clustering

I'm interested in partitioning matrices into groups which are almost simultaneously diagonalizable. I'm aware that if matrices commute and one of them has no multiple eigenvalues then the matrices are ...
0
votes
0answers
18 views

Exponentiation of Gell-Mann Matrices

The Pauli Matrices satisfy the relation $e^{ia(n\cdot\sigma)}=I\cos(a)+ ia(n\cdot\sigma)\sin(a) $. Can a similar equation be derived for the Gell-Mann matrices? The main feature of the Pauli ...
0
votes
0answers
8 views

What are the properties of a very specific matrix built with a stochastic matrix and a matrix of weights.

Assume that you have an NxN stochastic matrix W (i.e. $\sum_j w_{i,j} = 1$, where $w_{i,j}$ is the element on row $i$ column $j$). Assume that you have an NxN matrix as follows: \begin{equation} ...
0
votes
0answers
13 views

HOW to converting matrix to normal form? [closed]

Now as the process goes ... converting the diagonal elements to 1 and then with the help of that 1, bringing zero in the first coloumn and row. Even after doing this process i cannot get the answer at ...
-2
votes
0answers
22 views

How to compute the gradient of the following matrix function? [closed]

$f(X)=\left\|XX^T-I \right\|_F^2$, where $\left\|\cdot \right\|_F^2$ is Frobenius matrix norm
0
votes
1answer
25 views

Positive definiteness of a bilinear form implies symmetry?

In the Wikipeda article about positive definite bilinear forms, there is the line It turns out that the matrix $M$ is positive definite if and only if it is symmetric and its quadratic form is a ...
1
vote
1answer
31 views

What is the determinant of cofactor matrix of a matrix? [duplicate]

For an $n \times n$ square matrix $A$, can determinant of its cofactor matrix (matrix consisting of cofactors of the elements of $A$) be expressed in terms of $\det(A)$ and $n$ ?
1
vote
1answer
28 views

$ [[[A,B],C],D] + [[[B,C],D],A] + [[[C,D],A],B] + [[[D,A],B],C] = 0 $

If A and B are $n \times n$ matrices, define the Lie product $[A,B] = AB-BA$. Exercise 1.37 of the book Basic Linear Algebra by T.S. Blyth and E.F. Robertson asks to prove that $$ (*) \ \ \ \ \ ...
1
vote
1answer
33 views

What are the Properties of a Matrix $X$ such that $X\times (X^tX)^{-1}\times X^t = I$

can you please help me in the following question: What are the Properties of a Matrix $X$ such that $X\times (X^tX)^{-1}\times X^t = I$ $X$ is not necessarily a square Matrix. I am interested in ...
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votes
3answers
30 views

How to find an unknown matrix which when multiplied with a vector gives the cross product of 2 vectors

Question Image I am trying to find the matrix $[u]_x$ as shown in the image. $u \times v$ is easy to calculate but how do I find the matrix $[u]_x$ such that $[u]_x v = u \times v$ ?
0
votes
1answer
30 views

Matrix Similarity for All Scalars

Show that if matrix $A$ is similar to $B$, then $A-\lambda I_n$ is similar to $B-\lambda I_n$, for all scalars $\lambda$.
0
votes
2answers
35 views

Pseudoinverse (Moore-Penrose) of rank 1 matrix is a scalar multiple of its transpose

Let $A$ be an $m \times n$ matrix of rank 1. Show that its pseudoinverse is $c^{-1}A^T$, where $c={\rm trace}(A^{T}A)$. I know that $A^{T}A$ is symmetric with rank 1, so it has exactly one non-zero ...
1
vote
1answer
51 views

Derivative of $\operatorname{tr}[(CC^{T})^{-1}]$?

What is the derivative of $$\operatorname{tr}[(CC^{T})^{-1}]$$ with respect to the matrix $C$ ? Thank you for your attention.
7
votes
1answer
68 views

Find the eigenvalues of a 3 x 3 matrix

I have a question on determining eigenvalues for a given matrix A: $$ A= \begin{bmatrix} 2 & 1 & 2 \\ 0 & 2 & -1 \\ 0 & 1 & 0 \\ \end{bmatrix} $$ Here's what I have so ...
1
vote
1answer
38 views

Matrix --> Scalar Valued Function: Differentiation

In class, we called a real-valued function from the space of matrices to the reals $f: \mathbb{R}^{m \times n} \rightarrow \mathbb{R}$ differentiable at $\mathbf{X}$ if: $$\lim_{\mathbf{H} \to ...
0
votes
2answers
24 views

solving an XOR matrix

I'm working on a somewhat-unique linear algebra problem arising from XORing files together in order to encode them, and then figuring out how to subsequently recreate the original files from the ...
0
votes
2answers
23 views

singular matrix statement [closed]

A and B are square matrices of the same order. for every singular A, there exists B that is different than zero that satisfies the equation $A^2B=A^5B$. prove or disprove the statement.
0
votes
1answer
18 views

Consistency of system of linear equations without taking it to echelon form

Establish the conditions under which the equations $$ax + by + cz = q-r;bx + cy + az = r-p;cx + ay + bz = p-q ,$$ are consistent. I am aware that by taking the system to echelon can get me the rank ...
2
votes
2answers
43 views

Show that the matrix $AA^T+\alpha I$ is positive definite, where $\alpha >0$ and $A$ is an $m\times n$ real matrix.

Show that the matrix $AA^T+\alpha I$ is positive definite, where $\alpha >0$ and $A$ is an $m\times n$ real matrix. So I need to show that $x^T(AA^T+\alpha I)x>0$ for all vectors $x$. I'm ...
0
votes
1answer
23 views

Which of the following statements about Matrix A are NOT TRUE

A = {(1 2 -6 0),(0 1 4 -1),(0 0 1 -3),(0 0 0 1)} 1) Rank of A is 4 2) Columns of A span R4 3) Rows of A are linearly independent 4) A is invertible my logic ...
0
votes
0answers
16 views

Prove that the eigenvalues of a skew-symmetric matrix are purely imaginary [duplicate]

Proof idea: $A$ is a skew symmetric matrix. $A$ is similar to $A^t$ because every matrix is similar to it's transpose. $$A^t = -A $$ $A$ is similar to $-A$. Let $P_{(\lambda)}$ be the characteristic ...
0
votes
0answers
40 views

Prove: If $\ker(A)\cap \mbox{Im}(A)\not = \{0\}$ then $A$ is a singular matrix

Prove: If $\ker(A)\cap \mbox{Im}(A)\not = \{0\}$ then $A$ is a singular matrix Here is my solution. Is it correct? So my thought was something along the lines of: $$\ker(A)\cap \mbox{Im}(A)\not ...
0
votes
1answer
24 views

Consistency of system of linear equations

Find when the equations $$\begin{cases}x + y - 2z = 0\\ax + by + cz = 0\\bx + cy + az = d\end{cases}$$ are consistent and solve them completely when they are consistent. I have tried the ...
1
vote
2answers
33 views

A matrix with the rank of 1 has only rational numbers. Are all the eigenvalues necessarily rational?

I know that all of the eigenvalues except for one are $0$. Can the last eigenvalue be non-rational?
3
votes
1answer
61 views

How to prove this result about the interlacing of eigenvalues.

Let $A$ be a real symmetric matrix of order $n$ with eigenvalues $\mu_1\ge \mu_2,\ldots, \mu_{n}$. $B=\begin{bmatrix} x&x&x&x&x \end{bmatrix}$, where $x$ is non zero column vector in ...
0
votes
3answers
47 views

Show that any conjugate pair of complex numbers (with non-zero imaginary part) cannot be the spectrum of any 2x2 matrix with real, nonnegative entries [duplicate]

My professor showed me this in her office today but I didn't like her method and wanted to use another method. So, I computed the characteristic polynomial of some arbitrary $2 \times 2$ matrix ...
0
votes
1answer
26 views

Determinant of the matrix representation of an isomorphic linear transformation

Are there any theorems or special properties about the determinant of a matrix representation of an isomorphic linear transformation?
0
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0answers
23 views

Linear Algebra Base Change Matrix

I'm currently learning eigenvectors and Linear Transformations. And this basis change part is particularly confusing for me.. I feel like there are two types of questions, First type, you have a ...
1
vote
1answer
31 views

Form of pure states on $M_n(\mathbb{C})$

Related question: The form of the states on an algebra of $n\times n$ matrices with complex entries I have tried to show that pure states on $M_n(\Bbb{C})$ are of the form $\phi(A)=Tr(\rho A)$, just ...
0
votes
2answers
26 views

Show Matrix relationship

Say $A$ and $B$ are 2x2 matrices with integer entries in the group of matrices with determinant 1, under matrix multiplication. Let $A$ and $B$ have the same first column, show there is exists an ...
2
votes
1answer
43 views
+50

Block matrix of order $m$ with three block matrices

How to find eigenvalues of following block matrices? $M=\begin{bmatrix} A & B & O & O & O & O & O & \cdots & O & O\\ B & A & B & O & O & O ...
2
votes
1answer
36 views

Find $A^{20}x$ using eigenvectors and eigenvalues.

Find $A^{20}x$ A is a 3X3 matrix with the following eigenvectors and eigenvalues: $V_1 = [1, 0, 0]... V_2 = [1, 1, 0]... V_3 = [1, 1, 1]$ corresponding to Eigenvalues.. $\lambda1 = -1/3, \lambda2 = ...
0
votes
1answer
12 views

Find invariant factors of a power of a matrix, given a matrix in Jordan canonical form

Suppose \begin{align} A&=(I_4+(N_1 \oplus N_3))\oplus(N_2\oplus N_4 \oplus N_5)\oplus(-I_3+N_3) \\ &= \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 ...
-1
votes
0answers
29 views

Gradient of a vector [closed]

Matrix $V$ is 200 by 785. Matrix $X$ is 785 by 1. Matrix $W$ is 10 by 201. Matrix $y$ is 10 by 1. First, I do: $ V * X$ Then, I apply $tanh()$ to every element of that resulting matrix. The result ...
2
votes
0answers
33 views

3D shaped matrices - how would multiplication work? [duplicate]

I've been thinking about vectors and matrices lately, and I got a little curious. Why don't we have cubic shaped matrices? After all, vectors are 1-dimensional matrices, so it follows that there ought ...
0
votes
3answers
44 views

finding an orthogonal basis for $\mathbb{R}^3$

I have a vector $[-1/3, 1/3, 4/3]$ and another vector $[1, 1, 0]$ and I need to find an another orthogonal vector that are orthogonal to both vectors but could not find a method. Any help would be ...
0
votes
1answer
22 views

on Matrix Inequality

Let $A=(a_{ij})$, and $B=(b_{ij})$ be two $n$ by $n$ real symmetric matrices such that $$ a_{ij}\leq b_{ij}+\alpha, \quad \alpha>0. $$ Can we conclude that $A\leq B +\textbf{1}\alpha$? Note ...
0
votes
1answer
25 views

Relationship between the ranks of matrices A and B , where B is obtained by changing one element of A.

Let $A$ be a $5 \times 5$ matrix and let $B$ be obtained by changing one element of $A$. Let $r$ and $s$ be the ranks of $A$ and $B$ respectively. Which of the following statement is/are correct? $s ...