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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matrix powers problem

let $ A $ be the matrix :\begin{bmatrix}1 & 3 & 1\\4 & 2 & 3\\2 & 1 & 1 \end{bmatrix} Prove that $A$ verifies the expression : $ -A^{3}+4A^{2}+12A+5 I_{3} = O_{3}$ Deduct ...
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1answer
31 views

Prove that $BA^{-1} B \not=-B$ if $A + B$ is invertible for $A$ invertible and $B$ non-zero matrix

Let $A$ and $B$ be $n×n$ real square matrices. Matrix $A$ is an invertible and $B$ is a non-zero matrix. a)Prove that $BA^{-1} B \not=-B$ if $A + B$ is invertible b) Let $B= uv^T$ for $u,v \in \Bbb ...
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0answers
7 views

CDF of smallest eigenvalue of non-central Wishart matrix - how to evaluate the integral.

Does anybody know how to derive the distribution of the smallest root of a non-central Wishart matrix? I have got an integral expression that would give me the desired answer but cannot solve the ...
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1answer
21 views

Finding the Formula of the Product of $e_{i,j}$ and $e_{k,l}$ to Return Zero Matrix

My teacher for calculus this year gave a handout on the first day with an excerpt from Rings, Fields, and Vector Spaces by B.A. Sethuraman. The reason for this is in the beginning of Sethuraman's book ...
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0answers
23 views

Deriving equation in vector notation

I had some trouble deriving an equation from the book 'Elements of statistical Learning' p. 108 equation 4.9. This heavily relies on linear algebra, so I was wondering how the author came to his final ...
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2answers
26 views

Inverse of a Rotation matrix

If $R $ is a rotation matrix (determinant 1,orthonormal) can we say that $R^{-1}$ is also a rotation matrix? If yes how do we prove it?
2
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2answers
63 views

Binomial Theorem on a Matrix

Does the expression follow binomial theorem? $(A + I)^n$ where $A$ is matrix, $I$ is identity matrix. I know the binomial theorem but do not know whether it is applicable to matrices also.
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0answers
7 views

What is the closest self-adjoint (positive) operator to a given operator?

Given an operator $\rho$ on a Hilbert space $H$, is there a notion of nearest self-adjoint (positive) approximation of $\rho$ for a suitable norm? More specifically, does there exist an algebraic ...
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1answer
33 views

Permutation matrices

Let $\mathscr{M}$ be the set of all $n\times n$ matrices having entries $0$ and $I$ in such a way that there is one $I$ in each row and column. (a) If $M\in\mathscr{M}$, describe $AM$ in terms of ...
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0answers
38 views

How to find intelligently counterexamples for (dis)proofs about matrices?

Let's say I'm asked to give a counterexample for a claim about matrices, such as The elementwise product of two positive semi-definite matrices is positive semi-definite. It's easy enough to do ...
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0answers
13 views

Generalized Schur complement theorem

Let $M$ be an $(n+m)\times(n+m)$ real non-symmetric positive semidefinite (PSD) matrix partitioned as \begin{eqnarray*} M=\left(% \begin{array}{cc} A~~B\\ C~~D\\ \end{array}% \right), ...
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1answer
18 views

The nullity of a square matrix with linearly dependent rows is at least one. TRUE OR FALSE

Here is the answer my textbook gives. http://imgur.com/ycCRoWK I wonder: Why does the author ask this question specifically for square matrices? Is it different for other matrices.
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1answer
33 views

TRUE OR FALSE: Matrices with linearly independent row and column vectors are square.

Here is the answer of my textbook: http://imgur.com/vEoY31O Why must a matrice with linearly independent vectors have nullity(A)=0? That is where I lose track of the question. Are zero rows ...
2
votes
3answers
74 views

$A^2=cA$ for some $c \neq 0$

Let $A \in \mathbb{C}^{n \times n}$ and $0 \neq c \in \mathbb{C}$ a given constant. Suppose that $A$ has the following property: $$A^2 = cA.$$ Questions. 1) Is there a matrix class for matrices ...
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0answers
13 views

Leading eigenvalues of large sparse unsymmetric matrix

I have a matrix R which is sparse and all eigenvalues are -ve with a zero eigenvalue. Size of R is more than 1 million X 1 million. But I need to calculate only few large (by value not by magnitude) ...
0
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1answer
25 views

Prove that a normal matrix is unitary/Hermitian

I'm stuck with these two questions for while. I'd appreciate your help. ...
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0answers
16 views

Exponential and power of a special bidiagonal matrix

Given the bidiagonal matrix $$ \mathbf{A}=\begin{bmatrix} a_1 & b_1 & 0 & 0 & \dots & 0 & 0\\ 0 & a_2 & b_2 & 0 & \dots & 0 & 0 \\ 0 & 0 ...
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0answers
17 views

Immannt of a matrix.

I want to know in details about immanant of matrix. I have come to know about it in here but it could not give me sufficient knowledge. Kindly provide me some link where I can get introductory ...
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1answer
31 views

Normal Matrix Having all real eigen values is Hermitian

$A$ is a normal matrix (i.e. $AA^*=A^*A$, where * denotes the hermitian conjugate). If all its eigenvalues are real, prove that it is Hermitian (i.e. $A^*=A$). I have tried many things but could not ...
3
votes
1answer
37 views

Matrix inequalities question

Let $A, B \in \mathbb{R}^{n \times n}$. Assume that: $$ 0 \preccurlyeq 2 A^\top A \preccurlyeq A^\top + A $$ $$ B^\top + B \preccurlyeq 0 $$ Is the following inequality true? $$ A B + B^\top ...
5
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2answers
91 views

An interesting linear algebra question

Let $A$ and $u$ be $n\times n$ matrix and $n\times 1$ vector of $\mathbb{C}$. Denote $\overline{A}$ is the matrix $(\overline{A})_{ij}=A_{ij}^*$, the conjugate number; ($\overline{A}$ is not the ...
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0answers
8 views

What are the upper bound and stability conditions for the following simple linear system

Consider the following linear system $$\dot{x}=\sum\limits_{i=1}^{m}{{{\alpha }_{i}}}\left( t \right)\cdot {{A}_{i}}\cdot x \quad (1) $$ where, $x\in {{\mathbb{R}}^{n}}$ represents the state vector, ...
0
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1answer
29 views

When does a matrix fail to be positive definite?

I am wondering how to think about a matrix being "bigger" than another. If I have the inequality X - Omega Sigma^-1 > 0 where all matrices are quadratic and X = Z'Z with Z positive definite and Omega ...
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0answers
9 views

Exponentially weighted rank ordered correlation matrix

Is there any well-known method to apply exponential weighting (similar to EWMA) to rank ordered correlation matrices such as Kendall tau or Spearman?
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0answers
12 views

Kronecker product and the vec operator: confusion on proof of vec(AXB) = (transpose(B) ⊗ A) vec(X)

I was reading up on Kronecker products and vec operator from couple of sources and landed on the equation: vec(AXB) = (transpose(B) ⊗ A) vec(X) suppose ...
3
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0answers
25 views

Analytical Matrix Inversion

I have a matrix of the form $A = bI - J$ where 1. $b$ is a large positive constant so that $A$ is positive definite 2. $J_{ij} = 0$ for $i=j$ and follows a power law off-diagnol. In index ...
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0answers
23 views

Trick for Jordan-Matrix and transformation of basis

some time ago I found a 'trick' for getting a basis-transformation-matrix for jordan. I'd like to understand it, but at a certain point I stuck. Maybe you can help me? Given is a matrix A: ...
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0answers
7 views

What is pseudodiagonality in matrix/tensor?

What is the difference between diagonality and pseudodiagonality? Does this apply to tensor too? https://www.math.uzh.ch/fileadmin/math/preprints/06_11.pdf
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1answer
20 views

Canonical form for orthogonal similarity classes

Could someone point me to a reference re canonical forms for classes of matrices in $M_n(\mathbb{C})$ which are unitarily similar? That is, canonical representatives for the equivalence class defined ...
0
votes
2answers
27 views

Show that every row of matrix $S$ is a linear combination of its bottom row and the row (1 1 1 1 1 1 )

Couldn't solve the following three questions. $$S=\begin{pmatrix} 36 & 35 & 34 &33&32&31 \\ 25 & 26 & 27&28&29&30 \\ ...
0
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2answers
41 views

How do you calculate this third eigenvector in this 3x3 matrix?

Scroll down to the bottom if you don't want to read how I arrived at my original two answers. My question is how are all the online calculators I check coming up with this third eigenvector (1, 1, ...
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0answers
47 views

Find the fundamental matrix of a system of ODEs?

To linearize a system, in one of the steps I am required to find the fundamental matrix $\Phi$(t) of a system such that $\Phi$(0)=I. The example system my professor used: $\dot{x} = x - y - x^3 - ...
0
votes
1answer
19 views

Prove that Frobenius matrix norm is compatible with the vector norm

Show that, the Frobenius matrix norm $||.||_F$ is compatible or consistent with a vector norm $||.||_2$ , that is, $||Ax||_2 \leq ||A||_F ||x||_2, \forall x \in \mathbb{R}^n$. Where $||A||_F = \sqrt{ ...
0
votes
1answer
14 views

exponential of a product of any two matrices commuting with one of the matrices

I'm trying to show that for any arbitrary matrices A and B, $e^{AB}A = Ae^{BA}$ I checked this other answered question, but this case differs as I have a product of matrices as opposed to a ...
0
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0answers
30 views

About lemma $\rho(A) \leq \|A^k\|^{1/k}$

In the Spectral radius wikipedia article in section Matrices there is a lemma, what states that: Lemma. Let $A \in \mathbb{C}^{n \times n}$ be a complex-valued matrix, $\rho(A)$ its spectral ...
0
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1answer
34 views

If $u$ and $v$ are vectors in $3$-space, then $u\cdot v$ is a scalar

My understanding is that B is definitely true because of the below picture but I cannot understand A. Please would someone point me to the right direction! Thanks!
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2answers
28 views

General Solution Of Linear Equations

$x_1+x_2-6x_3+4x_4=6$ $3x_1-x_2-6x_3-4x_4=2$ $2x_1+3x_2+9x_3+2x_4=6$ I have row reduced the matrix and got $$\left(\begin{array}{cccc|c} 1 & 1 & -6 & 4 &6\\ 0 & 1 & -3 ...
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0answers
45 views

Is there a closed-form expression for this matrix power series?

I encountered a matrix power series: $$ X = M + PMP^{t} + P^{2}M(P^{t})^{2} + \cdots, $$ where $M$ is a real symmetric matrix, and $P$ is a real square matrix. Assuming that this series converges, ...
1
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1answer
59 views

Minimum eigenvalue of product of two matrices

Abstract description: Let $\mathbf{A}$ and $\mathbf{B}$ be two $n \times n$ real matrices. Let $\sigma( \mathbf{A B} )$ denote the spectrum of $\mathbf{A B}$. Assume that (A1) $\mathbf{A}$ is ...
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+50

Different method for QR decomposition - is it possible

This method could also possibly be applicable to matrices of higher dimension, but for the simplicity of my question i will only ask it for $2$x$2$matrices. Suppose $A=\begin{pmatrix} a_{11} & ...
1
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3answers
44 views

Span of columns (or rows) of a given matrix?

Consider the following matrix: $$A = \begin{pmatrix} 1 & 0 & 2 \\ 2 & 1 & 3 \\ \end{pmatrix}$$ The columns of $A$ span $\mathbb{R}^2$. The columns of $A$ span $\mathbb{R}^3$ ...
1
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1answer
54 views

Determine cycle from adjacency matrix

Is there a way/algorithm to determine if there is a cycle in a graph if I only have the adjacency matrix and can not visualize the graph?
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2answers
50 views

$A^{T}A$ is diagonal. What can I say about $A$?

Is there any special property about the elements of $A$ if $A^{T}A$ is diagonal? I imagine you need some sort of symmetry but I can't see what it should be. Edit: Sorry, maybe it's better phrased ...
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0answers
3 views

How to prove any pinhole camera matrix can be factorized into KRt factors?

This is a problem in computer vision. The definition of $KRt$ decomposition/factorization of pinhole cameras can be found here: equation $(10)$ in Elements of Geometric Computer Vision or see ...
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1answer
48 views
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2answers
34 views

Why can I not write every $N \times M$ matrix as multiplication of an $N \times 1$ and a $1 \times M$ matrix?

My intuition says I simply can't express $N \times M$ independent variables in terms of $N+M$ variables but how can I show that?
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2answers
88 views

Determinants of Matrices det(4A) equals?

Suppose A is a 4 x 4 matrix such that det(A) = 1/64. What will det(4A^-1)^T be equal to? Here's my thinking, det(A^T) = det(A) I has no effect on the determinant. And det(A^-1) = 1/det(A) so ...
0
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0answers
17 views

Which matrix norm gives the minimal variation of eigenvalues?

This is a follow-up of this question. The original question is intentionally as general as possible, because I was interested in the most general possible answer. I am now trying to understand its ...
1
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1answer
14 views

How to compute the group inverse of $M+aI_n-\frac{a}{b}J_{n\times n}$?

For a square matrix $M$, the group inverse of $M$, denoted by $M^\#$, is the unique matrix $X$ such that $MXM=M$, $XMX=X$ and $MX=XM$. Given an $n \times n$ matrix $M$, let $I_n$ denote the identity ...
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2answers
48 views

Proving that $A$ is diagonalizable

Let $A\in\mathbb{C}^{n\times n}$ be a $n$ by $n$ matrix such that $A^k = I$ for some natural number $k$. Find a nonzero annihilating polynomial of A and prove that A is diagonalizable. I will say ...