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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2
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1answer
202 views

Linear Algebra Question ( rank of matrix )

Let $\bf A$ be an $m \times n$ matrix. If $\bf P$ and $\bf Q$ are invertible $m \times m$ and $n \times n$ matrices, respectively prove $\operatorname{rank}(\mathbf{PA}) = ...
0
votes
1answer
93 views

real spectrum of an almost symmetric stochastic matrix

Let $M$ be a real nonnegative square matrix ''almost'' symmetric ($A_{ij}=0$ if and only if $A_{ji}=0$). In addition, $M$ is irreducible (not necessarily primitive) and row stochastic (the sum of each ...
0
votes
2answers
122 views

Adjacency matrix

Let $G$ be any simple and undirected graph. Let $A$ be the adjacency matrix of $G$. 1) Let $B$ be the number $\tfrac16\mathrm{tr}(A^3)$. What does $B$ count? That is $B$ counts the number of....? 2) ...
1
vote
2answers
635 views

Show that the identity matrix $I$ must have norm $1$.

I am trying to understand why the identity matrix $I$ must have a norm $1$, for any choice of matrix-norm $|\cdot|$? How would i show this?
31
votes
6answers
2k views

How many connected components does $\mathrm{GL}_n(\mathbb R)$ have?

I've noticed that $\mathrm{GL}_n(\mathbb R)$ is not a connected space, because if it were $\det(\mathrm{GL}_n(\mathbb R))$ (where $\det$ is the function ascribing to each $n\times n$ matrix its ...
1
vote
1answer
139 views

What happens to this infinite sum?

Assume $W$ is $n$ by $n$ matrix and $r<1$ is a real number. Let $$Q = \sum_{i=0}^{\infty} (rW)^i=[I_n-rW]^{-1}$$ Now assume that the $i$'th row of matrix $W$ is multiplied by a constant real ...
1
vote
1answer
42 views

Decomposition of a single 4D rotation

I have a $4\times 4$ matrix $M$ which represents a general 4-dimensional rotation. $$ M = \pmatrix{a_{11} &a_{12} &a_{13} &a_{14}\\a_{21} &a_{22} &a_{23} &a_{24}\\a_{31} ...
4
votes
3answers
922 views

Eigenvalues of matrix

I want to find all the roots of a polynomial and decided to compute the eigenvalues of its companion matrix. How do I do that? For example, if I have this polynomial: $4x^3 - 3x^2 + 9x - 1$, I ...
0
votes
1answer
69 views

matrix with distinct bounded eigen values is bounded?

I am looking from the numerical methods perspective. I have a mapping $G$ that maps points in the numerical iteration to the new level. I would like to show its stability. For that I need to show that ...
0
votes
2answers
604 views

Inverse of an matrix exponential

Consider the matrix exponential $$ e^{At} = \frac{1}{4} \begin{bmatrix} -e^{-t} + 5e^{3t} & e^{-t} - e^{3t} \\ -5e^{-t} + 5e^{3t} & 5e^{-t} - e^{3t} \end{bmatrix} $$ And $$ ...
5
votes
2answers
277 views

Determinant inequality about positive definite matrices.

Assume $A \in M_n(\Bbb{R})$ (not necessarily symmetric), and for $\forall \alpha\not=0$, $\alpha^TA\alpha>0$. Show that $$\det\left(\frac{A+A^T}{2}\right)\leq \det A.$$
1
vote
1answer
86 views

Adjugate matrix product

I have some problems understanding the proof of the Caley-Hamilton theorem (saying that a matrix the root of ith characteristic polynomial), namely: Why $A \cdot A^D = A^D \cdot A = \det A \cdot I$ ? ...
1
vote
1answer
99 views

What's this matrix called?

In an inner product space, $v_1,\dotsc,v_n$ are linear independent iff the matrix $A_{ij} := \langle x_i | x_j \rangle$ is invertible. What's the name of this matrix??
0
votes
1answer
50 views

About the spectral radius of a kind of matrices

Consider a square matrix $A=DS$ where $S$ is symmetric with diagonal entries being $0$ and $D$ is a diagonal matrix for normalizing $S$'s row sums so that $Ae=e$ where $e$ is a vector with all entries ...
0
votes
0answers
123 views

hermitian matrices, pauli matrices

These matrices are the Pauli matrices \begin{align} A_1 & = \left[\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right] \\ A_2 & = \left[\begin{array}{cc} 0 & -i \\ i & 0 ...
4
votes
4answers
627 views

Can you raise a Matrix to a non integer number? [duplicate]

So I heard you can take a matrix A to the power 2, take it to a -3th power and multiply it by an irrational number. You can also do some other non-intuitive things like taking e to the power of a ...
1
vote
1answer
66 views

Prove or disprove: the spectral radius of a matrix with negative entries and row sums as 1 is larger than 1

We all know that the spectral radius of a stochastic matrix is $1$. But how's the "negative" proposition: For a matrix $M=(m_{ij})_{n\times n}$, if there exists $m_{ij}<0$ for some $1\le i,j\le n$ ...
2
votes
1answer
98 views

How to Solve Bilinear Matrix Equation?

How should I solve this matrix equation? What is the solution for $X$? \begin{equation} BXC +B^TXC^T=D \end{equation}
3
votes
3answers
105 views

Can every symmetric regular complex matrix be decomposed into a product of a matrix times its transpose?

Let $A\in GL(n,\mathbb{C})$ be symmetric. Is there a always a $B\in GL(n,\mathbb{C})$, such that $A=B^TB$?
1
vote
0answers
28 views

Delocalization of eigenvectors in Expanding Graphs

Given an adjacency matrix A, can we say something about whether the eigenvectors corresponding to its highest(or second-highest) eigenvalues are delocalized ? By delocalization I mean that every ...
1
vote
2answers
90 views

Prove that $A^*A + I$ is invertible

Let $A$ be an $m\times n$ matrix. Prove that $A^*A + I$ is invertible. I'm not sure what to do because everything I try ends up at $A^*A$ again and not $AA^*$.
0
votes
0answers
76 views

Matrix norm applications?

There'are many different ways to calculate Matrix norm. But once calculated, what is the practical use/application of it (e.g. in computer programming)? Or does it let define something that can be ...
1
vote
1answer
72 views

Diagrammatic Representations: $\dim(Skew_{n\times n}(\mathbb{R}))+\dim(Sym_{n\times n}(\mathbb{R})) = \dim(M_{n\times n}(\mathbb{R}))$

SEE AUTHOR'S ANSWER BELOW So I'm trying to derive the dimensions of both $Skew_{n\times n}(\mathbb{R})$ and $Sym_{n\times n}(\mathbb{R})$. I know that $\dim(M_{n\times n}(\mathbb{R}))=n^2$, but I ...
1
vote
1answer
103 views

Trace Constraints and rank-one positive semi-definite matrices.

Let $C_1$,$C_2$...$C_N$ be $M\times M$ hermitian matrices and $c$ be a given positive constant. Let $W$ be a positive (possibly semi) definite matrix such that \begin{align} \text{trace}\{WC_1\}\geq ...
4
votes
1answer
6k views

Understanding rotation matrices

How does $ {\sqrt 2 \over 2} = \cos (45 \text{ degrees })$? Is my graph (the one underneath the original) accurate with how I've depicted the representation of the triangle that the trig function ...
2
votes
0answers
211 views

Proving invertibility of matrices using banachs lemma

I'm studying for finals and trying to understand how you can possibly use banach's lemma for anything worthwhile, more particularly we have a bunch of sample questions that say it can be used to prove ...
0
votes
1answer
55 views

Rank of the entry-wise-product of an orthogonal matrix and its transpose

Let $P$ be an $n \times n$ real-valued orthogonal matrix, ie, such that $ P P^\top = I_n$. Consider the matrix $M$: $$ M_{i,j} = P_{i,j} P_{j,i} \ , \ \ i,j = 1,..,n. $$ Is it possible that rank$(M) ...
1
vote
2answers
123 views

Property of Hermitian matrix

How to prove that the sum of diagonal elements of Hermitian matrix equals to the sum of its eigenvalues? Thanks much!
0
votes
1answer
61 views

Range and null space

For a given $N \times 1$ vector $x$, $x \neq 0$, consider the matrix $R = xx^H$. Identify the range space and null space of $R$, and thus determine its rank. Here $H$ is a Hermitian operator.
2
votes
1answer
357 views

How to prove unitary matrices require orthonormal basis

How to prove unitary matrices require orthonormal basis thanks much!
2
votes
1answer
78 views

Hermitian Matrix

Can you show that the range space and the null space of a Hermitian matrix are mutually orthogonal. I was reading a conference paper and they used it too easily I suppose. Can you please prove.
1
vote
1answer
43 views

How to prove that the corresponding matrix is unitary

Let's say we are given hermitian matrix $H$. How to prove that the matrix $M$, formed from eigenvectors of $H$ is unitary? Thanks
1
vote
2answers
51 views

Logarithm of a Gram matrix

Given a Gram matrix $K$, we are interested in calculating its matrix logarithm $\log(K)$, and in particular, to relate minus this logarithm to the Laplacian of a graph. We have noticed that ...
0
votes
2answers
92 views

Finding eigenvalues of sparse integer matrix

I need to find eigenvalues of a sparse matrix with integer coefficients. I understand in general this is not done by explicitly computing the characteristic polynomial due to numerical instability, ...
0
votes
4answers
189 views

Given two column vectors $a$ and $b$, what is the determinant of $A$ if $A=ab^T$

Given two column vectors $a$ and $b$ in $\mathbb R^n$ , $n \ge 2$, form the $n×n$ matrix $A = ab^T$. What is the determinant of $A$? (Hint: Examine linear dependence).
2
votes
2answers
3k views

Dimension of the corresponding eigenspace?

I'm studying for my linear exam and would appreciate any help for this practise question: You are given that λ = 1 is an eigenvalue of A. What is the dimension of the corresponding eignspace? A = ...
2
votes
2answers
238 views

Determining eigenvalues, eigenvectors of $A\in \mathbb{R}^{n\times n}(n\geq 2)$.

Let $a$ and $b$ be distinct nonzero real numbers and let $A\in \mathbb{R}^{n\times n}(n\geq 2)$ with each diagonal entry equal to $a$ and each off-diagonal entry equal to $b$. Determine all ...
1
vote
1answer
24 views

A Problem with a Proof Concerning the Properties of Orthogonal Matricies

OK, I'm working on the following problem and don't understand how my Linear Algebra text has made it to a certain conclusion here is the problem: Let u be a unit vector in $R^{n}$ and let $H = I - ...
0
votes
1answer
174 views

Cholesky update for an added diagonal

I have to (numerically) compute $$\int_0^\infty f(z)(z\mathbf{K}+\sigma^2I)^{-1}\mathbf{u}\,dz$$. Where u is a constant vector, z is a scalar and f(.) is a known function. The problem that I have is ...
1
vote
0answers
118 views

Matrix of Discrete fourier transform $F^4$ is identity

I already showed that Discrete fourier transform matrix is unitarian matrix. Now I would like to show that $F^4$ is identity. On wikipedia is written: "This can be seen from the inverse properties ...
1
vote
1answer
376 views

Show that every ideal of the matrix ring $M_n(R)$ is of the form $M_n(I)$ where $I$ is an ideal of $R$ [duplicate]

Suppose $R$ is a commutative ring. Show that every ideal of $M_n(R)$ is of the form $M_n(I)$ where $I$ is an ideal of $R$. I have spent 30 minutes on this question and I still got nowhere. Can ...
5
votes
1answer
181 views

Condition of the eigenvalue problem

[Ciarlet, 2.3-1] I know this result: Let $A$ a diagonalisable matrix, $P$ a matrix such that $$P^{-1}AP\ =\ \mbox{diag}(\lambda_i)\ =\ D,$$ and $\|\cdot\|$ a matrix norm satisfying ...
5
votes
2answers
269 views

Error in argument regarding the Cayley Hamilton theorem

I cannot spot the mistake in the following argument regarding the Cayley Hamilton theorem: Let $A\in M_n$, then, $$\begin{align*} P_A(t)&=\det(tI-A)\\ &\implies P_A(A)=\det(AI-A)\\ ...
2
votes
1answer
168 views

Proving that the sum of the errors of a least square linear approximation is $0$

Let $(x_1,y_1),\dots,(x_n,y_n)$ be points in $\mathbb{R^2}$ and $e=[\epsilon_1,\dots,\epsilon_n]^T$ the error vectors belonging to the least square solution of the linear approximation. Prove that ...
10
votes
1answer
1k views

In-place inversion of large matrices

In Solving very large matrices in "pieces" there is a way shown to solve matrix inversion in pieces. Is it possible to apply the method in-place? I am refering to the answer in the ...
-1
votes
2answers
219 views

Linear combination problem from linear algebra

Express the column matrix $b$ as a linear combination of the columns of $A$. (Use $A_1$, $A_2$, and $A_3$ respectively for the columns of A.) $$\begin{align} A&=\begin{pmatrix}-2 &-5 ...
1
vote
1answer
35 views

Show that $ PH^T(HPH^T + N)^{-1} = (P^{-1} + H^TN^{-1}H)^{-1}H^TN^{-1} $

Show that: $$ PH^T(N + HPH^T)^{-1} = (P^{-1} + H^TN^{-1}H)^{-1}H^TN^{-1} $$ Given: P and N are symmetric matrices. All matrices are square matrices in the same size. I tried using Woodbury Matrix ...
1
vote
3answers
304 views

Looking for a quick way to prove a matrix identity

It's quite simple, it takes two seconds to understand it, but I can't find a quick demonstration. Let's say I have three matrices $A$, $B$ and $C$. The matrices $A$ and $B$ do not commute with one ...
5
votes
5answers
379 views

How to prove that $\det(M) = (-1)^k \det(A) \det(B)?$

Let $\mathbf{A}$ and $\mathbf{B}$ be $k \times k$ matrices and $\mathbf{M}$ is the block matrix $$\mathbf{M} = \begin{pmatrix}0 & \mathbf{B} \\ \mathbf{A} & 0\end{pmatrix}.$$ How to prove that ...
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vote
0answers
39 views

What is the matrix norm in defining the generator of a continuous time Markov chain?

For a continuous time Markov chain with finite state space and Markov transition function $p(t)$, its generator $G$ can be defined entry-wise as $$ G_{i,j}:=\lim_{t\to 0^+} \frac{p_{i,j}(t) - ...