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
60 views

How would I find Matrix $B$ in the following equation?

Let $A$ be a $4\times 3$ matrix. Consider matrix $B$ which is a pre-multiplier of matrix $A$, that is, $BA$. Find matrix $B$ if it performs the following elementary row operation on $A$ Multiplies ...
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1answer
58 views

How to apply the pivot system in a matrix reduction

\begin{align} 4x_2 + 8x_3 &= 9 &(1)\\ 0.1x_1 + 2x_2 + 9x_3 &= 10 &(2)\\ 155x_1 + 2x_2 -7x_3 &= 0.001 &(3) \end{align} Which equations would I swap to use the pivot method when ...
0
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1answer
86 views

Are Singular Value Decompositions unique

I have a symmetric positive semi definite matrix K. The question is, is the decomposition U and D unique? Where $K=UDU^T$ and $D=diag(\lambda_1, \lambda_2,...,\lambda_n)$ where the lambdas are ...
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1answer
52 views

Projection self-adjoint

Let $A$ be a projection such that $A^2=A$ then I want to prove that $A=A^* \Leftrightarrow \ker(A) \perp \operatorname{im}(A)$. The implication $A=A^*\Rightarrow \ker(A) \perp ...
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1answer
2k views

Proof of matrix norm property: submultiplicativity

I've been searching for the definition of the submultiplicative (I think it has multiple names from what I've seen) property in proof form. Some books define it as part of the properties that define ...
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1answer
1k views

what are pivot numbers in LU decomposition? please explain me in an example

studying many pages like wikipedia, wolfram, Mathworks, Math Stack Exchange, lu-pivot, LU_Decomposition I couldn't find exactly whart are the pivot numbers in a specific example: for example what are ...
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1answer
646 views

Invertability of Singular 2x2 Matrix with all same real values.

Question: Let set G = { matrix [{a a},{a a}] such that a is real but not 0 } represent the set of 2x2 matrices with same elements of the reals excluding a = 0, show that G is a group under matrix ...
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0answers
70 views

Set of commuting Matrices $\Rightarrow $ Common Eigenvector [duplicate]

I am trying to prove that if we have an arbitrary set of commuting matrices in $M_n(\mathbb C)$ then they have a common eigenvector. Well, if we have only 2 matrices, the answer is easy and it has ...
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3answers
2k views

$I_m - AB$ is invertible if and only if $I_n - BA$ is invertible.

The problem is from Algebra by Artin, Chapter 1, Miscellaneous Problems, Exercise 8. I have been trying for a long time now to solve it but I am unsuccessful. Let $A,B$ be $m \times n$ and $n ...
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2answers
109 views

Change of basis (Gram-Schmidt)

I was wondering whether it is possible to write down explicitely the matrix that represents the change of basis from a basis $\{v_1,....,v_n\}$ to a basis $\{e_1,...,e_n\}$, where $e_i$ is the basis ...
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2answers
316 views

Is this relationship between spectral radius and singular values false?

I found the following relationship $\max_{i} \sigma_i \le \rho(A)$, where $A$ is a matrix. But somehow I do not trust this relation, I'd rather guess that the converse is true, but I do not know.
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1answer
113 views

Does $\exp(X)$ commute with $X$?

If I have a real invertible matrix $X$ and I take the matrix exponential of this matrix $Y = \exp(X)$, are there any reasons for $X$ and $\exp(X)$ to commute? Obviously if $A$ and $B$ commute then so ...
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1answer
72 views

Iteration to find squre root of positive semidefinite matrix

Suppose matrix $A$ is positive semidefinite and $I\succeq A$. Prove that the iteration $$Y_0=0,\hspace{3mm} Y_{n+1}=\frac{1}{2}(A+Y_n^2)$$ is nondecreasing (that is, $Y_{n+1}\succeq Y_n$ for all ...
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4answers
123 views

Book Searching in Stability Theory.

Can anyone recommend me a book on Stability Theory with an intuitive approach? I have some course notes on that subject, but it's really abstract and theoretical. I really want to understand it, ex: ...
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0answers
30 views

Is there an $M \in \mathbb{R}$ such that $\forall A \in \mathbb{M}^{n \times m},||A||_{\mathrm{frob}} \leq M ||A||_{\mathrm{op}}$?

Is there an $M \in \mathbb{R}$ such that $\forall A \in \mathbb{M}^{n \times m},||A||_{frob} \leq M||A||_{op}$? Research effort We can assume that $A \neq 0$. $$ ...
4
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1answer
355 views

The Principle of Mathematical Induction

The question is Let $( F_0, F_1, F_2,... )$ be the Fibonacci sequence defined by $F_0=0,\, F_1=1, and F_{n+1}=F_n+F_{n-1}$, n greater than or equal to 1. Prove the following identities. ...
4
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0answers
173 views

Proving the max of a quadratic form ${\mathbf x}^T\mathbf A \mathbf x$ can be attained when $x$ is from $n$-dimensional hypercube

updated: Maybe my original question is somewhat misleading. I rewrite some of the post. This is some research problem I'm working on. I have an $n\times n$ symmetric positive-definite matrix ...
0
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1answer
89 views

Perturbing/sensitivity of the eigenvalues of a block tridiagonal matrix

This question is related to my former question: Checking if one "special" kind of block matrix is Hurwitz Given the next matrix $$ J = \begin{bmatrix}-(B+B^T) & B \\ 0 ...
0
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1answer
195 views

Proving the following fact about the matrix exponential.

Assume the formula $\det(e^A)=e^{\operatorname{tr}(A)}$ for all matrices $A \in \mathbb{C}_{n\times n}$. Show why this implies that the exponential always yields a regular matrix.
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1answer
31 views

ODE with solution in a subspace

The task is to show that if $A$ is a Markov matrix, as to say the sum of all the entries in $A$ for each column equal $0$ and all the entries $a_{ij} ≥ 0$ for $i\neq$j then the solution to the ODE ...
4
votes
3answers
125 views

Calculating determinant with real number on diagonal and units everywhere else

I'm solving a problem and I'm having difficulties in calculation of the determinants of two matrices. There is two $N\times N$ matrices: $$\left( \begin{array}{cccc} a & 1 & \ldots ...
1
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1answer
42 views

The relation between Schimidt and Cross Product

The relation between the two kind process of Orthogoniz:Schimidt and Cross Product method1 Schimidt When orthoganize three $3$ 3-tuple vectors, we have $\beta _1=\alpha _1$ $\beta _2=\alpha ...
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2answers
102 views

Prove the set of all $m \times n$ matrices over $R$ is a free $R$-module

How to prove the set of all $m \times n$ matrices over $R$ is a free $R$*-module* with a basis of $mn$ elements? For math advanced please guide me..thanks!
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0answers
51 views

Can we perform this operation on block matrices?

We have a block matrix: $$ \left[\begin{array}{c|c|c} A & 0 & 0 \\ \hline 0 & B & 0 \\ \hline 0 & 0 & C \end{array}\right] $$ Here $A$, $B$ and $C$ are all permutation ...
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0answers
78 views

Closed form solution for ${d \operatorname{Tr}(A\log(X))\over dX}$

I need a closed form solution for ${d \operatorname{Tr}(A\log(X))\over dX}$. Here $A, X$ are $n\times n$ matrices, $\log$ is the matrix logarithm. ${d \operatorname{Tr}(\log(X))\over dX}=X^{-1}$, ...
6
votes
4answers
953 views

How to prove that the normalizer of diagonal matrices in $GL_n$ is the subgroup of generalized permutation matrices?

I'm trying to prove that de normalizer $N(T)$ of the subgroup $T\subset GL_n$ of diagonal matrices is the subgroup $P\in GL_n$ of generalized permutation matrices. I guess my biggest problem is that I ...
1
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1answer
199 views

Transvection matrices generate $SL_n(\mathbb{R})$

I need to prove that the transvection matrices generate the special linear group. I want to proceed using induction on n. I was able to prove the 2x2 case, but I am having difficulty with the n+1 ...
6
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3answers
251 views

If $A^3=I$ for a real matrix, is $A$ normal/orthogonal?

I read earlier that if $A$ is a real $3\times 3$ matrix satsifying $A^3=I$, then $A$ is similar to a matrix of form $$ \begin{bmatrix} 1 & 0 & 0\\ 0 & \cos\theta & -\sin\theta\\ 0 ...
1
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1answer
134 views

simpler way to calculate a determinant?

Simpler way to calculate this? $$A = \begin{bmatrix}\lambda -2 & 2 & 0 \\ 2 & \lambda -1 & 2 \\ 0 & 2 & \lambda \end{bmatrix}$$ my method: \begin{align*} \det A &= \det ...
2
votes
2answers
80 views

Does log of a matrix factor through similarity? Is it a bijection up to branch choice?

When taking the log of a matrix we have various choices, but fixing a particular choice, we should have $$P^{-1}\log{(A)} P = \log(P^{-1}AP),$$ right? (Here $P \in GL$.) It is supported by the ...
1
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4answers
255 views

Matrix Differentiation of $x^TAx$

I know the matrix differentiation of $x^TAx$ is if A is symmetric is 2Ax. I saw that some people write as $2x^TA$. Are these two results the same? I am not sure how they are same. Can anyone explain ...
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0answers
157 views

Calculating Gimbal Angles by converting Euler Angles to Matrix

I have a Gimbal (a movable platform) which can move in 3 axis (roll, pitch and heading). I have a Device which is connected directly to the Gimbal which reads back current roll, pitch and heading ...
3
votes
3answers
382 views

Given a vector equation with $n$ vectors, how can we determine if the span of the vectors is equal to $\mathbb{R}^n$

So I've recently started taking Linear Algebra, and I've been thinking about how to determine if the linear combinations of any n vectors can represent any vector in $\mathbb{R^n}$. More formally put, ...
0
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0answers
96 views

Dimension of a subspace (Hermitian matrices)

I have the following exercice from T. Tao's blog which I want to solve: Suppose that $n \geq2$. Show that the space of Hermitian matrices with at least one repeated eigenvalue has codimension 3 in ...
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2answers
258 views

find values of a, b and c when $y=ax^2 +bx + c$ is a curve?

find values of $a, b$ and $c$ when $y=ax^2 +bx + c$ is a curve and the passes through the points $(x_1, y_1), (x_2, y_2), (x_3, y_3)$. The question is needed to be solved using matrices that involves ...
1
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3answers
646 views

How prove this matrix have a positive eigenvalue

let $$A=\begin{bmatrix} a_{1}&a_{2}&a_{3}\\ a_{4}&a_{5}&a_{6}\\ a_{7}&a_{8}&a_{9} \end{bmatrix}$$ where $a_{i}>0$, show that the matrix $A$ At least one positive ...
2
votes
1answer
260 views

Matrix Norm Bounds

A natural consideration of matrix norms is to compare them, and one of the many standard results on the induced 1, 2, and $\infty$-norms indicate that $$\frac{1}{\sqrt{n}}\|A\|_\infty\leq \|A\|_2\leq ...
0
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1answer
83 views

Permutation matrix notation

What is the official notation (and the source of the notation) of the following permutations matrices (one shift left of identity matrix): For $n=2$ $P = \left(\begin{matrix}0 & 1\\ ...
3
votes
2answers
96 views

What's this theorem?

While reading an old book, I came across this theorem: Neither name nor proof was given, can somebody provide some further information about this throrem? Thanks.
2
votes
1answer
67 views

Matrix generating $\operatorname{SL}_n(\mathbb{R})$

How do I show that the following matrices generate $\operatorname{SL}_2(\mathbb{R})$ $\begin{pmatrix} 1 & a \\ 0 & 1 \\ \end{pmatrix}$ or $\begin{pmatrix} 1 & 0 \\ a & 1 \\ ...
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3answers
91 views

Matrix equation $X + X^T = \operatorname{Tr}(X)A$

Let $A \in M_n(\mathbb{R})$. Solve $X+X^T= (\operatorname{Tr}(X))A $ where the unknown $X$ is in $M_n(\mathbb{R})$. $X^T$ is the transpose of $X$ and $\operatorname{Tr}(X)$ is the trace of $X$.
3
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1answer
208 views

Applications/Motivations of matrix decomposition techniques

Matrix decomposition is one area of matrices that has always intrigued me. Every time I open a matrix book, I can interestingly follow it till Eigen values and Eigen vectors because they are well ...
1
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1answer
669 views

Fast way to solve a system of linear equations from Givens QR decomposition

I have this system of linear equations: $$ A= \begin{bmatrix} 2 & 0 & 1 \\ 0 & 1 & -1 \\ 1 & 1 & 1 \end{bmatrix} $$ $$ b= \begin{bmatrix} 3 \\ 0 \\ 3 \end{bmatrix} $$ I ...
2
votes
1answer
9k views

Finding an orthogonal basis from a column space

I'm having issues with understanding one of the exercises I'm making. I have to find an orthogonal basis for the column space of $A$, where: $$A = \begin{bmatrix} 0 & 2 & 3 & -4 & ...
6
votes
3answers
165 views

Eigenvalues of block matricies

If the eigenvalues of a matrix $A$ are $\lambda_1,\lambda_2,\dots,\lambda_n$, what are the eigenvalues of the matrix? $\begin{bmatrix}0 &A\\A&0\end{bmatrix}$ From some numerical examples I ...
1
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0answers
108 views

Proving that an matrix have integers entries

Considering the Matrix below with $n \in \mathbb{N}$ $$ A = \left( \begin{array}{ccc} 1 & \frac{1}{2} & \cdots & \frac{1}{n} \\ \frac{1}{2} & \frac{1}{3} & \cdots & ...
1
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1answer
42 views

How can we transform matrices into scalars?

If we have the three matrices: $$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} , \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 ...
5
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1answer
139 views

Inner Product on Division Algebras

Here, Wikipedia gives a proof that the only finite dimensional associative division algebras over $\mathbb{R}$ are $\mathbb{R}, \mathbb{C}, \mathbb{H}$. The proof proceeds by taking such a division ...
1
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4answers
741 views

Example of matrices that do not satisfy the submultiplicative property

In rare cases there are some matrices that do not satisfy the submultiplicative property: $$\|AB\| \leq \|A\| \|B\|$$ Is there an example of such a matrix $A$ and $B$ such that $$\|AB\| > \|A\| ...
2
votes
1answer
114 views

If one of the sub-matrices is singular, the whole matrix will be singular?

A larger matrix is formed by four sub-matrices A,B,C,D. If one of the sub-matrices is singular, the whole matrix will be singular?