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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0answers
33 views

show by using leibniz formula

There are given $ r, s,n \in\mathbb N$ and $r+s=n$. It also given $A \in M_{r,K} $, $B \in M_{r\times s,K} $ and $C \in M_{s,K} $. Let $M$ be the matrix $\begin{bmatrix}A & B\\0 & ...
8
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0answers
198 views

How prove this matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Question: let matrix $A,B,C\in M_{n}(C)$ is Hermitian matrix and is Positive definite matrices ,such $$A+B+C=I_{n}$$show that $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge ...
0
votes
1answer
10 views

invert lower triangular matrix

I am sorry if the question is simple,am trying to find the quicker method to invert a triangular matrix. Could you please provide some references where i could refer? Moreover,is there any known way ...
1
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1answer
34 views

Solving an augmented coefficient matrix so there are infinitely solutions

I am trying to figure out this math problem. For what values $a,b$ does the linear system have infinitely many solutions? This is the matrix $$ \left[ \begin{array}{ccc|c} ...
0
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1answer
22 views

what is the name of this matrix? does it have any special characteristics?

does anyone know the name of this matrix or if it has any special characteristics or how to calculate its inverse efficiently e.g. in a closed-form? [ \begin{array}{llllll} ...
0
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0answers
17 views

Correlating random numbers seems to skew the data

I am trying to generate a series of correlated random numbers that represent currency exchange rates for a Monte-Carlo simulation. I am attempting to do this via a Cholesky decomposition of the ...
2
votes
1answer
46 views

If direct limits of matrices are isomorphic, is the direct limit of the transpose matrices also isomorphic?

On the one hand, the following conjecture seems reasonable, but on the other hand it doesn't seem natural because some objects are being dualised while others are not. I would appreciate if anyone ...
1
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2answers
51 views

For which values of $a, b$ does the system of equations NOT have any solutions?

I am trying to solve this math problem: For which values of $a$ and $b$ does the linear system represented by the augmented matrix not have any solution? $$ \left[ \begin{array}{ccc|c} ...
1
vote
2answers
94 views

Square root of a $3\times3$ matrix

Here is $3\times3$ matrix$$\begin{pmatrix} 0& 0& 1\\ 0 & -1 & 0\\ 1& 0 & 0\end{pmatrix}$$ How can I solve this by using Cayley-Hamilton? I know how to ...
2
votes
1answer
25 views

Lower bound for the spectralradius of a matrix

Any submultiplicative norm (for example the row-sum-norm) is an upper bound for the spectralradius of a matrix A. But is there a way to get a suitable LOWER bound for the spectralradius ? ...
0
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1answer
18 views

Difference between matrices with altered eigenvalues

Given two p.s.d. matrices $X_1$ and $X_2$ with eigen decomposition $X_1 = U_1V_1U_1^T$ and $X_2 = U_2V_2U_2^T$ and a constant $\lambda > 0$ Now consider an altered version of the eigenvalue ...
3
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3answers
54 views

Cross product: matrix transformation identity

How can one prove the following identity of the cross product? $$(M a)\times (M b)=\det(M) (M^{\rm T})^{-1}(a\times b)$$ $a$ and $b$ are 3-vectors, and $M$ is an invertible real 3x3 matrix.
0
votes
1answer
48 views

Matrices and algebra

Given the matrix $A$ $=$ $$ \begin{pmatrix} -1 &3 & 5 \\ 1 & -3 & 5 \\ -1 & 3 & 5 \end{pmatrix} $$ and $X$ be the solution set of the equation ...
4
votes
2answers
97 views

Prove that $\det(M-I)=0$

$M$ is a $3 \times 3$ matrix such that $\det(M)=1$ and $MM^T $= I, where $I$ is the identity matrix. Prove that $\det(M-I)=0$ I tried to take $M$ $=$ $$ \begin{pmatrix} a &b & c \\ ...
1
vote
3answers
51 views

How find the matrix $K$ such $AKB=C$

Question: Find a matrix $K$ such that $$AKB=C$$ given that $$A=\begin{bmatrix} 1&4\\ -2&3\\ 1&-2 \end{bmatrix},B=\begin{bmatrix} 2&0&0\\ 0&1&-1 \end{bmatrix} ...
1
vote
0answers
30 views

Characterize matrices complying to certain constraints.

Characterize those matrices $ X $ (real symmetric), $ Y$ (real positive definite), $ R$ (real diagonal) and $ F $ (real diagonal) such that $ XRY + YRX = 0$ , (1) $ YRY - XRX = F$ . (2) ...
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votes
1answer
22 views

Cramer's rule and linear dependence/independence test

When you have the system of equations: $$ax + by = e\\cx + dy = f$$ And you do some row operations to eliminate $y$, we get: $$x = \frac{ed-bf}{ad-bc}\tag{1}$$ If we eliminate $x$ we get: $$y = ...
1
vote
3answers
43 views

Tridiagonal Symmetric Matrix

Could anyone help me to find the determinant of a $N\times N$ tri-diagonal symmetric matrix, named "$A[i,j]$" with $i,j \le N$, that has all the elements in the super-diagonal and sub-diagonal equal, ...
0
votes
1answer
41 views

Are circulant matrices open

Are the set of positive definite symmetric circulant matrices open in the set of positive definite symmetric matrices?
1
vote
0answers
29 views

Matrix with trace zero [duplicate]

Question is that : Suppose a matrix $A\in M_n(\mathbb{C})$ is a commutator by which i mean $A=BC-CB$ for some $B,C\in M_n(\mathbb{C})$ then we see that Trace of $A$ is $0$ But Suppose a matrix is of ...
1
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0answers
21 views

Matrix that shows how close indices are to each other

I have a vector of n words, not all distinct. [the, quick, brown, fox, jumps, over, the, lazy, dog] From this, I want to make an n×n matrix that shows how ...
1
vote
4answers
55 views

TT* + I is invertible

I've the following exercise which I can't solve: Prove that: $$ AA^* + I $$ is invertible for all Matrix $ A $ in finite-dimensional field $V$ with inner product. $ A^* $ is the adjoint operator. ...
9
votes
2answers
96 views

Subgroups of $\mathrm{GL}(n,\mathbb{Z})$ which are not finitely generated

The group $\mathrm{GL}(n,\mathbb{Z})$ is finitely generated: take for example diagonal matrices, permutations and one elementary matrix (upper triangular). Are there some simple / nice examples of ...
2
votes
2answers
184 views

Matrix Manipulation Question

Given that A $\begin{bmatrix}1\\-2\\1\end{bmatrix}=\begin{bmatrix}2\\-3\\-5\end{bmatrix}$ and A $\begin{bmatrix}1\\4\\-2\end{bmatrix}=\begin{bmatrix}0\\2\\-1\end{bmatrix}$ find A $\begin{bmatrix}1 ...
0
votes
2answers
17 views

Solving a system of equations using the inverse of the coefficient matrix

Let A be 2*2 matrix with A={{1 1}, {2 -2}} use A^-1 to solve the matrix equation A{{x_1 x_2}} ={{3 5}} I got A^-1 ={{1/2 1/4 },{1/2 -1/2}} just need to know how to solve equation .
1
vote
1answer
27 views

find an elementary matrix $E$ such that $EA=B$

Let matrix $$A = \begin{pmatrix} 1 & 2 & 0 \\ -3 & 1 & 1 \\ 0 & 4 & 2 \end{pmatrix} $$ and $$B = \begin{pmatrix} 1 & 2 & 0 \\ 0 & 7 & 1 \\ 0 & 4 ...
0
votes
0answers
31 views

Can I solve this problem with matrices?

So I have some two dimensional data sets thats I want to analyse. They can be viewed in 2D form as below: $M1$: $$\begin{matrix}00 & 00 & 00 & 00 & 00 & 00 & 00 & 00 ...
1
vote
1answer
22 views

$A$ is hermitian if and only if $\langle A\alpha,\beta\rangle= \langle\alpha ,A\beta\rangle$ for $\alpha$ and $\beta \in \mathbb{C}^n$

How can i prove that $A$ is hermitian if and only if $\langle A\alpha,\beta\rangle= \langle\alpha ,A\beta\rangle$ for $\alpha$ and $\beta \in \mathbb{C}^n$ i stuck in this problem i know that if $A$ ...
0
votes
2answers
27 views

Symmetric Matrix Quadratic Form

Let $A,B\in\mathbb{M}_{n\times n}(\mathbb{R})$ and $A,B$ are symmetric matrics. Prove that if $\vec{x}^TA\vec{x} = \vec{x}^TB\vec{x}$ $\forall\vec{x}$, then $A=B$. Since $A,B$ are symmetric, they are ...
10
votes
4answers
331 views

Converting recursive equations into matrices

How do we convert recursive equations into matrix forms? For instance, consider this recursive equation(Fibonacci Series): $$F_n = F_{n-1} + F_{n-2}$$ And it comes out to be that the following that ...
7
votes
5answers
114 views

Positive semi-definite of a matrix composed of semi-definite blocks

Say a matrix A is positive semi-definite. Let B be a square matrix composed of replicas of A as sub-blocks, s.t. $$B=\begin{pmatrix} A & A \\ A & A \\ \end{pmatrix},$$ or $$\begin{pmatrix} A ...
1
vote
1answer
37 views

Constructing regular integer matrices with distinct integer eigenvalues

How can I construct matrices with positive integer values and distinct integer eigenvalues (not necessarily positive, but 0 should not be an eigenvalue). The standard-method to construct matrices ...
1
vote
1answer
25 views

Triangularisation of a linear transformation

I understand that Upper triangular matrices must have at least one eigenvector, but why does this mean that the basis of $[T]_B$ must contain an eigenvector for $[T]_B$ to be upper triangular?
7
votes
2answers
98 views

Proof of the inequality $\sqrt{\det X} \leq \frac{\operatorname{tr}X}{2}$

Let $A, B \in M_2(\mathbb{R})$ be symmetric and positive definite. Put $X:=AB$. then, we have the following inequality: $$\sqrt{\det X}\leq \dfrac{1}{2}\operatorname{trace}X.$$ and the equality ...
1
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0answers
33 views

Matrix multiplication in quaternions is not necessarily linear

I tried to show by example that matrix multiplication for quaternionic matrices is is not necessarily $\mathbb H$-linear. If $A \in M_n(\mathbb H)$ is a quaternionic matrix and $x$ is a vector in ...
1
vote
0answers
34 views

Schur decomposition of real-eigenvalue matrix

Is Schur decomposition of real-eigenvalue matrix a real orthogonal decomposition? If yes, why is it? Is it because all the eigenvectors are real? If I have $$ A^T+A^2=I $$ then I deduced ...
1
vote
1answer
43 views

$A,B$ matrices , prove $Bv = \Lambda v$

$A,B$ are $n \times n$ matrices and $AB = BA$ Also, there is an eigenvalue $\Lambda$ in $A$ which its geometric complexity is $1$. Also there is $ v \ne 0 $ that $v$ is an eigenvector of $A$. ...
-1
votes
0answers
28 views

Order of eigenvectors in jblas?

I am using jblas to compute eigenvectors of a double symmetric matrix. Using symmetricEigenvectors(myMatrix)[0], I can get a matrix which columns are the eigenvectors of my matrix. However I need them ...
3
votes
2answers
115 views

What is the intutive explanation of why the notation of matrices is as it is?

If I want to solve a system of linear equations, like 2x-y=1 x+2y=4 Then the matrix notation for the same would be: $$ \begin{bmatrix} 2 & -1 \\ 1 & 2 \\ \end{bmatrix} \begin{bmatrix} X\\ ...
0
votes
1answer
36 views

Show that $\lambda_1 = \min \{ Q(u) \mid \|u\| = 1 \}$ and $\lambda_m = \max \{ Q(u) \mid \|u\| = 1 \}$

Let $V$ a vector space over $K$ and $Q(u) = \langle u, Tu \rangle$ a quadratic form. $T$ is a symmetric operator. The eigenvalues of $T$ are sorted by size $\lambda_1 < \dots < \lambda_m$. How ...
3
votes
1answer
44 views

If matrix $A$ is invertible, then there is a permutation of its rows leaving no-zeros on the diagonal

I need to prove this statement: "If $A$ invertible, then exist a permutation of its rows leaving no-zeros on the diagonal" and I tried using the definitos of invertible matrices and $LU$ ...
2
votes
1answer
50 views

When does $x^TAx + c^Tx$ have a global minimum?

This question is closely related to my last question about extended quadratic forms. I figured out a nice criterion, when $$f : \mathbb R^n \rightarrow \mathbb R$$ $$f(x) = x^TAx + c^Tx$$ has a ...
2
votes
3answers
49 views

Proving linear independence of matrices

Let $A = \textrm{diag}(a_{1},a_{2},a_{3})$ where $a_{1},a_{2},a_{3}$ are distinct. I am trying to show that every diagonal $3\times3$ matrix cane be made up of linear combinations of $I$, $A$ and ...
0
votes
1answer
18 views

Does cofactor expansion generalize to complex matrices?

When finding the determinant of some $n * n$ matrix $A$ when $$\forall i,j\in\mathbb{N} ,i\leq n\land j\leq n\implies A_{ij} \in \mathbb{C}$$ Can cofactor expansion be used under the normal definition ...
0
votes
1answer
15 views

Proof of product of symmetric matrices

Use the inverse of a 2x2 matrix formula to confirm that: "The product of two symmetric matrices is symmetric if and only if the matrices commute." IN THE CASE where the symmetric matrix ...
0
votes
1answer
13 views

covariance matrix in multivariate Gaussian distribution (semi-positive or positive definite)

The book that I'm reading states clearly that the covariance matrix $\Sigma$ in the following equation is a positive semidefinite and symmetric matrix. $$ p(x) = det(2\pi\Sigma)^{-1/2} exp\{ ...
1
vote
2answers
191 views

Newly Developed With Details - Describing orthographic projection using simple 2D transformations

Thanks to Pedro for helping me further develop my question into something tangible. His (most recent) answer below clearly and formally outlines what I am asking. This is similar to this question, ...
1
vote
2answers
107 views

$A^2$ is diagonalizable leads to $A$ diagonalizable?

If $A^2$ is diagonalizable, is it necessary true that $A$ is diagonalizable? Also, the opposite: If $A$ is diagonalizable, is it necessary true that $A^2$ is diagonalizable? I'm not sure yet, tried ...
-1
votes
1answer
29 views

Prove the sum of symmetric and non-symmetric matrix is $M^R_{2\times2}$

Prove that: $M^R_{2\times2} = \{A \in M^R_{2\times2} | A^t = A\} \oplus \{A \in M^R_{2\times2} | A^t = -A\}$ Well, it's pretty obvious that I need to show that the sum of symmetric and non-symmetric ...
12
votes
2answers
472 views
+200

Properties of 4 by 4 Matrices

Define $ A=\begin{pmatrix} x_1 & x_2 & 0 & 0\\ 0 &1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{pmatrix}, B=\begin{pmatrix} 1 & 0 & 0 & 0\\ x_3 ...