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Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, ...

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1answer
19 views

Are there terminologies for ($A A^T$ or $A A^H$) and ($A^T A$ or $A^H A$)?

Are there terminologies for $A A^T$ and $A^T A$, respectively, where $A$ is a matrix? Like "$A A^T$ is the (something) of $A$." I know that if $A$ were a vector, we could use the terms inner product ...
0
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0answers
10 views

Decompose an equation into a matrix form

How can I decompose the following equation into a set of matrix multiplications? $f(x',\alpha,\beta) = \sum_{n=0}^{\infty}\sum_{n'=0}^{\infty}\sum_{l=0}^{n}\sum_{m=-l}^{l}A_{nn'l}(e^{(n-l)x'}-e^{(n'-...
1
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0answers
15 views

cost of gaussian elimination in numerical

In the following question I have solved the system $$2x_1-x_2=2$$ $$-x_1+3x_2 = 4$$ Using gaussian elimination. Below are the workings, however I am wonder what is the cost of this process? how do I ...
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0answers
8 views

Least squares solutions of matrices with redundant columns?

There was a similar question here, but I either did not understand the answers or the answers were too general. I am wondering specifically how to find the solutions. For example, what are the least ...
1
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2answers
41 views

Is there an elegant way to determine $Av$ given $Au_1, Au_2$, and $Au_3$ for a $3\times3$ matrix $A$?

Let A be a 3x3 matrix such that ${A} \begin{pmatrix} 3 \\ 4 \\ 5 \end{pmatrix} = \begin{pmatrix} 2 \\ 7 \\ -13 \end{pmatrix}, \quad \ {A} \begin{pmatrix} 4 \\ 5 \\ 6 \end{pmatrix} = \begin{pmatrix} -6 ...
1
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2answers
20 views

How many numerical multiplications for the product of three matrices?

For example you have the matrices $A, (5 \times 8), B, (8 \times 4), C (4 \times 10). $ The question wants you to find the number of multiplications if you were to multiply these matrices like $(A\...
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0answers
16 views

Multiplication of two matrices using convolution

Let's say I define my matrix $M$ such that $$m_{i,j}=m_{i-1,i-j}$$ Now, that means we only need the first column and row of this matrix as all the other elements in that matrix are just repetitions of ...
1
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1answer
13 views

Operator norm of a family of matrices

Let $c$ be a complex number. Consider the family of $n\times n$ matrices $M_n$ which have $c$'s on one off-diagonal, $\bar{c}$'s on the other off-diagonal, and zero everywhere else. So $M_4$ looks ...
0
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2answers
20 views

Name/notation for group of permutation matrices

$S_n$ is the group of permutations, and there is a bijection between each permutation and its permutation matrix. Is there a name/notation for the group of permutation matrices?
1
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2answers
30 views

Four Dimensional intersection point

How can I find the intersection point of two linearly independent 4D dimensional planes? I know that the first plane A goes through the four-dimensional points A1(1,2,3,4) A2(0,1,0,1) A3(0,1,1,0) and ...
0
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2answers
14 views

Matrix Representation of Linear Transformation from R2x2 to R3

We have a linear transformation T : $\mathbb R^{2\times2} \rightarrow \mathbb R^{3}$ defined by $$T\left(\begin{bmatrix}a&b\\c&d\end{bmatrix}\right)=(a+b,2d,b).$$ Let A and B be the ordered ...
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0answers
13 views

Question related to the trace and the commutator of matrices

Let $K$ be any field and $n\in \mathbb N$. For every $A\in M_n(K)$, define a linear form $\lambda_A: M_n(K) \rightarrow K$ by sending the matrix $M$ to $\lambda_A(M):= \operatorname{Tr}(AM)$. The map $...
2
votes
1answer
36 views

determinant of a complex matrix

Suppose we consider a complex $n\times n$ matrix $A$. It can be writen generally as $A=U+iV$, where $U$, $V$ are $n\times n$ real matrices. Now I hope to study the determinant of the following matrix ...
4
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0answers
17 views

If $M$ is symmetric posdef, then all diagonal band matrices derived from $M$ are also posdef?

Let $M$ be a positive definite and symmetric matrix: $$M = \left(\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n}\\ a_{21} & a_{22} & \cdots & a_{2 n}\\ \vdots &...
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0answers
14 views

Show Stationary Distribution of a Markov Chain can be given by:

Let P be a one step matrix of a chain on S for which stationary distribution $\pi$ exists. Suppose $|S| = K$. Let I be the $K \times K$ identity matrix, and J be a $K \times K$ matrix of 1's. Also, ...
0
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0answers
13 views

Is it correct to perform operations on matrices inside of matrices rather than the whole matrix?

It seems to be a generally useful technique when proving theorems about matrices to consider smaller matrices inside of other matrices. For example, I might write $$M = \begin{pmatrix} I_{n \times n}...
0
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1answer
33 views

Write the derivative of the lower-triangular matrix $L(t)$ in terms of $L(t)$, $L^{-1}(t)$ and $\frac{d}{dt}A(t)$, where $A(t)=L(t)L^T(t)$

Let $A(t)$ be a symmetric positive definite matrix, thus by Cholesky decomposition, we have $A(t)=L(t)L^T(t)$ where $L(t)$ is lower triangular. Suppose $A(t)$ is differentiable. I want to write $\...
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0answers
25 views

Matrix notation bracket { [on hold]

An activity asks me to construct a $2 \times 4$ Matrix for $C_{ij} = \{ij\}$, I'm having trouble understanding this notation. Thanks.// The solution given was: \begin{matrix} 1 & 1 & ...
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0answers
9 views

Efficient Kronecker Product Formulation

Let $A$ and $B$ be two $p \times p$ matrices, where $p$ can be large. I am interested in finding $C$, where $$vec(C) = (I_{p^2} - A \otimes A)^{-1}vec(B)\,. $$ Here $\otimes$ denotes the Kronecker ...
3
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2answers
146 views

Eigenvalues of A are also eigenvalues of T

Let $V$ be the set of all $n\times n$ matrices over a field $F$. Let $A$ be a fixed element of $V$. Define a linear operator $T$ on $V$ by $T(B)=AB$. I am trying to show that if $\lambda$ is an ...
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0answers
14 views

2-norm, prove the properties for a symmetric matrix

Prove that in case A is a symmetric matrix the following properties $||A||_2=max_i\{|\lambda_i|;\lambda_i \ \ \text{is eigenvalue of A}\}$. $||A^{-1}||_2=\frac{1}{min_i\{|\lambda_i|;\lambda_i \ \ \...
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1answer
17 views

How to construct group Hermitian and unitary 3x3 matrices?

2x2 matrices of Pauli are $\sigma_x$, $\sigma_y$, and $\sigma_z$, with $$\sigma_x = \pmatrix{0 && 1 \\ 1 &&0},~\sigma_y = \pmatrix{0 && -i \\ i &&0}, \sigma_z = \...
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votes
1answer
21 views

Factorization - Quadratic terms

Is it possible to factorize the following matrix? \begin{equation} Q = \begin{bmatrix} q_1^2 \\ q_2^2 \\ q_3^2 \end{bmatrix} \end{equation} -- Edited -- I am wondering if I might write the matrixn ...
0
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3answers
23 views

Let $A \in M_{n\times n}(\mathbb{C})$. Which of the following statement(s) is/are true?

Let $A \in M_{n\times n}(\mathbb{C})$. Which of the following statement(s) is/are true? $(A)$ There exists $B \in M_{n\times n} (\mathbb{C})$ such that $B^2 = A.$ $(B)$ A is diagonalizable $(C)$ ...
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0answers
15 views

Finding a basis from given coordinate representation

Determine a basis in $\mathbb{R}^3$ for which the vector $x = [1, -1, 2]^T$, has the representation $[1, 1, 1]^T$. Is this basis unique? Explain.
2
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2answers
24 views

Calculating Null T and Range T for the linear transformation $T(x,y,z)=(x+2y-z,y+z,x+y-2z)$?

I have a doubt regarding the calculation of range of a linear transformation. I will explain my doubt with an example. Suppose, $T:R^3 \to R^3 \ni$ $T(x,y,z)=(x+2y-z,y+z,x+y-2z)$ is a Linear ...
0
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1answer
16 views

How to determine basis and dimensions of subspaces?

I'm new into linear algebra and I encountered the following problem: In $\mathbb{R}^4$ we conside the following subspaces: $U=\{(x.y.z.w)\in \mathbb{R}^4 | x+y+z=0\}$, $V=\mathscr{L}((1,1,0,0), (2,-...
0
votes
1answer
32 views

Prove that 2-by-2 identity matrix $I_2$ has infinitely many distinct square root matrices [on hold]

Consider a square matrix $A$. Matrix $B$ is called as the square root of matrix $A$ if it satisfies $B^2 = A$. Prove that $I_2$ has infinitely many distinct square root matrices, where $I_2$ is the ...
0
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1answer
27 views

If a non zero vector multiplied by a matrix is 0, then the determinant of the matrix is 0

I have a question with regards to a 3x3 matrix, in a proof, this claim is made: If, $$\begin{bmatrix} a & b & c \end{bmatrix} \begin{bmatrix} x_{11} & x_{12} & x_{13} \\ ...
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0answers
12 views

Choice of representatives to partition by a subgroup of $GL_2(\mathbb R)$ of another subgroup of $GL_2(\mathbb R)$

My question is: Is it that it doesn't matter in 2.12.5 if we use $\begin{bmatrix} 1 & 0\\ 0 & d \end{bmatrix}$ or $\begin{bmatrix} a & b\\ 0 & d \end{bmatrix}$ for the same reason ...
2
votes
1answer
48 views

An inequality about positive matrices

Suppose that $Q_1,\ldots,Q_m\in M_n(\mathbb R)$ are positive definite, $v_1,\cdots,v_m\in \mathbb{R}^n$ are $m$ given vectors and $\alpha_1,\ldots,\alpha_m$ are $m$ nonnegative real numbers that sum ...
0
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0answers
14 views

Internal stability of a discrete-time system

These are two parts of a much larger proof I'm working on, can't figure how ii implies iii though. $x(k+1)=Ax_k,x(0)=x_{0}$ Where $A∈\mathbb{R}^{n×n}$ is a real constant matrix. i) All the ...
0
votes
0answers
28 views

$k$th power of $n \times n$ matrix

I am given an $n \times n$ matrix ${\bf A}$, with the following properties: It's diagonals elements are all $0$'s, it is symmetric, and any element of ${\bf A}$ have value either $0$ or $1$. How do I ...
1
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0answers
18 views

Discrete-Time External Stability

Consider a discrete-time system $\sum_{L}^{}$ of the form $x(k+1) = Ax(k) + Bu(k)$ $y(k) = Cx(k)$ Show that if all the eigenvalues of A are on the open unit disc, show that $\sum_{L}^{}$ is BIBO ...
0
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0answers
24 views

Find the matrix A that transforms a non invertible 3x3 matrix to 2x2 matrix

I'm trying to figure out an elegant way (or any way that doesn't involve multiplying everything out) to do this matrix algebra but am stuck. The equation is $A \begin{bmatrix} 1 & 2 & 2\\ 2 &...
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votes
3answers
32 views

Cofactor matrix and trace [on hold]

Let $A$ be a square matrix. Is there any relation between $\operatorname{cofactor}(A)$ matrix and $\operatorname{trace}(A)$? Thanks!
0
votes
1answer
20 views

Find the condition on x, y and z for $span{(2, 1, 1), (1, -1, 1), (x, y, z)} = R^3$

Given that $$span{(2, 1, 1), (1, -1, 1), (x, y, z)} = R^3$$ find the condition of x, y and z that satisfies the above. After converting it to matrix form, we get $$ \left[ \begin{array}{ccc|c} 2&...
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0answers
17 views

Nunber of invertible matrices [duplicate]

I got this problem: find the number of invertible $n×n$ matrices with elements from $Z_p$ with $p$ a prime integer.
0
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0answers
17 views

Iterative Vector Matrix Product

Consider the following iterative matrix product: $x_{i+1} = A_{i} \cdot x_{i}$. The matrices $A_i$ are defined as follows: All rows except of the first and last row are stochastic. The first and the ...
2
votes
1answer
25 views

Distance between subalgebras and commutants in matrix algebras.

This is a question on how to relate two different distances in the matrix setting. Everywhere below, $M_n$ denotes the square matrices $n\times n$ whose entries are in $\mathbb C$. We consider the ...
0
votes
1answer
33 views

How to determine basis and dimension of subspaces?

I'm new into linear algebra and I encountered the following problem: In $\mathbb{R}^4$ we conside the following subspaces: $U=\{(x.y.z.w)\in \mathbb{R}^4 | x+y+z=0\}$, $V=\mathscr{L}((1,1,0,0), (2,-...
1
vote
1answer
30 views

Find a constant so the matrix is negative definite

$A=\begin{bmatrix} -7 &2 &-4 \\ 2& c &1 \\ -4& 1 &-6 \end{bmatrix}$. Find $c$ such that $A$ is negative definite. What I tried so far is to find minors and that's how I ...
0
votes
1answer
39 views

If $A$ is negative definite, can $A$ squared be negative definite?

If I have a symmetric matrix $A$ that is negative definite, then spectral decomposition theorem says that there is an orthogonal matrix $Q$(matrix of eigenvectors) such that $A=Q \Lambda Q^T$. Does ...
1
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2answers
68 views

Show that $\{ v_1, v_2, v_3 \}$ is linearly independent if and only if $A$ is invertible.

Let $\{u_1, u_2, u_3\}$ be a set of linearly independent vectors in $V$. Assume that (1): $$v_1 = a_1u_1 + a_2u_2 + a_3u_3$$ (2):$$v_2 = a_4u_1 + a_5u_2 + a_6u_3$$ (3): $$v_3 = a_7u_1 + a_8u_2 + ...
5
votes
1answer
24 views

Deducing the null space from column relations

Let $A=\begin{bmatrix} a_{1} & a_{2} & a_{3} \end{bmatrix}$ and $3a_{1}+2a_{2}+a_{3}=0$ where $a_{i}$ are matrix columns. Find the nullspace of $A$. So, my initial thought was that the ...
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votes
2answers
41 views

Matrix/vector proof

Let $C$ be an $m\times n$ matrix with real entries. For vectors $u,v\in\mathbb R^m$, we define $u\sim v$ if there exists a vector $x\in\mathbb R^n$ such that $Cx = u−v$. Prove that for all $u, v,w \in\...
3
votes
2answers
50 views

If S is a normal subgroup, identify the quotient group G/S. What are the $\varphi(G)$'s?

The following is an exercise from Artin's Algebra: (Kiefer Sutherland's voice) Let G be the group of upper triangular real matrices $\begin{bmatrix} a & b\\ 0 & d \end{bmatrix}$ with a and ...
0
votes
1answer
31 views

When and when isn't it permissible for matrix multiplication to a replacement for a linear transformation?

I've been told that what a matrix of $T$ actually represents$^*$ can be a bit complicated, and that, when a mapping is from $F^m \to F^n$, the matrix of $T$, when multiplied by an arbitrary vector in ...
0
votes
0answers
31 views

Operator norm bounds

For $A \in \mathbb{R}^{m \times n}$m need to prove the following bounds: \begin{align*} ||A|| &\leq \sqrt{m} \max_{i \in \{1, 2, \dots, m\}} \big( \sum_{j=1}^n A_{ij}^2 \big)^{1/2}\\ ||A|| &\...