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
votes
2answers
64 views

Two vector spaces with same dimension and same basis, are identical?

Let $V$ subspace of $W$ and both have same dimension and same basis. Then can we safely say that $V= W$ ? I believe yes. For example there may be an element $x \in V$ written as a linear combination ...
3
votes
2answers
45 views

Find m so that $(m+1,1,1)$ , $(1,-m,-1)$ , $(m,1-m,2)$ are linearly dependent

I formed an augmented matrix $$\left(\begin{array}{ccc|c}m+1&1&m&0\\1&-m&1-m&0\\1&-1&2&0\end{array}\right)$$ I now that we do reduced row echelon form for the ...
1
vote
3answers
24 views

Singular idempotent matrices

We know that if a square matrix $A$ is idempotent, then $$A^2 = A$$ If $A$ is non-singular, then the only possible matrix that is idempotent is $A=I$. But if $A$ is singular, then are there ...
1
vote
3answers
40 views

Invertibility of $BA$

I'm having trouble with the following question which may seem simple but to me it's not Let $A \in \mathbb{R}^{5 \times 7}, B\in \mathbb{R}^{7 \times 5}$. Prove that $BA$ is not invertible. ...
1
vote
1answer
36 views

How tight is this trace inequality?

I would like to know how tight the following trace inequalities are for real symmetric $A$ and real symmetric $B \succeq 0$ $$\mbox{trace} (AB) \leq \lambda_{\max} (A) \cdot \mbox{trace} (B) $$ or ...
0
votes
0answers
27 views

Minimum eigenvalue of a sum of symmetric matrices

Let $\{v_i\}$ be some orthonormal basis in $\mathbb{R}^n$, and let $\{w_i\}$ be a set of positive weights such that $\sum_{i=1}^n w_i = 1$. I am interested in bounding the smallest eigenvalue of the ...
0
votes
0answers
42 views

A kind of permutations and possible relation to cyclic groups.

Any permutation that moves $n$ elements in some fashion never revisiting the same until all others have been visited, in other words so that: $${\bf P}^n = {\bf I}, \text{ but no } 0<m<n \text{ ...
1
vote
1answer
54 views

Rank of orthogonal projection to prep of null-space, right-singular matrix and identity matrix

Let $\mathbf{A}$ be $m \times n$ ($m < n$) complex matrix and its SVD be $\mathbf{A}=\mathbf{U}\mathbf{\Sigma}\mathbf{V}^H$. Then we obtain an idempotent and Hermitian matrix, referred to ...
2
votes
1answer
47 views

Inverse of Hermitian matrix which is constant on the off-diagonal

For $N\in\mathbb N$, I am interested in the inverse of the matrix $$M=\left(\begin{matrix}N&a&\dots&\dots&a\\ \bar a&N & a&\dots&a\\ \vdots &\ddots & \ddots&...
1
vote
1answer
33 views

LASSO with equivalent quadratic costs

Is there any fundamental difference between the solutions obtained by minimizing following LASSO cost functions, if any? ( $A_{N \times n }$ and $ N >> n$) $ J=\Vert y-Ax \Vert_{2}^{2} + \...
1
vote
1answer
63 views

Eigenvalues and eigenvectors of $A^TA$ and $A$

For a square matrix $A$, I was wondering what the condition(s) are for the eigenvalues of $A^TA$ to be the same as the eigenvalues of $A$. Also what are the condition(s) for the eigenvectors of $A^TA$...
0
votes
1answer
34 views

Rewriting a complex matrix as a real matrix and changing basis to $(z,\bar z)$

If we have a complex $n\times n$ matrix $A$ and we rewrite it as a real $2n\times 2n$ matrix $B$ by using the identity $z=x+iy$. I don't get how if we change basis from $(x,y)$ to $(z,\bar z)$ that is ...
-1
votes
1answer
24 views

How to compute the inverse of a rank-$1$ matrix

I have a rank-1 matrix $R \in \mathcal{C}^{m \times m}$, how to compute another matrix X, such that $RX=I$, where $I$ is an identity matrix.
2
votes
2answers
58 views

Decomposition to rotation around arbitrary axis

In 3d, I have a $4\times4$ matrix $M$, which has only a rotation part and a translation part. In other words, I can compute $X'=RX+T$ ( with $R$ a $3\times3$ rotation matrix, $T$ a vector for the ...
1
vote
1answer
60 views

Feasibility of Matrix Inequality

I need to show if the following inequality is true $$ (A + B)^{-1}M (A + B)^{-1} - A^{-1} M A^{-1} \preceq 0$$ given that $(A,B)=(A^T,B^T) \succ 0$ and $M = M^T \succeq 0$ also we have that $A + B \...
2
votes
1answer
22 views

Factoring difference of products of orthogonal matrices

I am working through some problems in Golub and van Loan's Matrix Computations and have come across the following problem that has me stumped. The question involves producing an upper bound on $||Q'...
7
votes
1answer
70 views

For which dimensions is it possible to have $A \succeq B \succeq 0$ with $A^2 - B^2$ having $n-1$ negative eigenvalues?

For any dimension $n$, can we write down two symmetric, positive semi-definite matrices $A,B$ with $A \succeq B$ in the sense of the usual ordering (i.e., $A-B$ is positive semidefinite) such that $A^...
0
votes
1answer
38 views

Using Cayley-Hamilton theorem to get a formula for $A^{-1}$ from $\chi_A$

I'm trying to prove that if an invertible n-by-n matrix $A$ has characteristic polynomial $$\chi_A(t)=(-1)^nt^n+a_{n-1}t^{n-1}+\ldots+a_2t^2+a_1t+a_0$$ with $a_0\not=0$then $$A^{-1}=\frac{-1}{a_0}((-1)...
-1
votes
2answers
33 views

For which values of t does a matrix not have eigenvalues

I need help solving this problem "For which values of real parameter t does the matrix: \begin{bmatrix} π^2t^2 & 36\\ -36 & 0 \\ \end{bmatrix} NOT have real eigenvalues. Thank you.
0
votes
0answers
18 views

Matrix Calculation Significance and Multivariate Bayesian Methods

Suppose I have the matrix given by: $$X = \begin{bmatrix}1 & 0 & 0\\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{bmatrix}$$ This matrix actually represents whether a user interacted with a ...
2
votes
1answer
26 views

Non-strict column diagonally dominant matrix inner product

Let $A \in \mathbb{R}^{n \times n}$ be a normalized non-strict column diagonally dominant matrix, that is: $$a_{j,j} = \sum_{i \ne j} \left|a_{i,j}\right|$$ where $$0 \le a_{j,j} \le 1$$ and $$-...
1
vote
1answer
29 views

Operatornorm of $(\mathbb{R}^d, \|\cdot\|_1) \to (\mathbb{R}^d, \|\cdot\|_{\infty})$

Determine the operatornorm of the mapping $I:(\mathbb{R}^d, \|\cdot\|_1) \to (\mathbb{R}^d, \|\cdot\|_{\infty})$! Unfortunately I haven't many ideas for this task. I know that the definition of the ...
0
votes
0answers
23 views

commutativity of log(I + A) and log( A−1) (matrix function)

I'm self-(re)learning linear algebra since the beginning of the summer, and i have a problem with the following exercice entitled additive logarithmic. If i'm right, we need to prove the ...
0
votes
1answer
13 views

Convex set or not?

This question is related to my previous question Set of all positive definite matrices with off diagonal elements negative I know that the set of all positive definite matrices form a convex set. ...
1
vote
0answers
14 views

nth product of sequential matrices

$\forall n \in \mathbb{N}$, let: $$P_n = \left( \begin{matrix} a & 1-a \\ b_n & 1-b_n \end{matrix} \right). $$ Whereby $\{b_n\}_{n \in \mathbb{N}}$ is a monotonically increasing sequence of ...
1
vote
2answers
54 views

Order of $\mathrm{SL}(n,\mathbb{F}_p)$ (Constructive proof)

Most proofs of $$ \vert ~\mathrm{GL}(n,\mathbb{F}_p) ~\vert = \prod_{k=0}^{n-1} (p^n-p^k) $$ I have seen so far, are done by counting the possibilities to build up invertible matrices i.e. counting ...
0
votes
1answer
52 views

Find $B(B^{T}B)^{-1}B^{T}$.

To find: $$B(B^{T}B)^{-1}B^{T}$$ for $B=[0,1,-1]^T$ I have $$\begin{bmatrix} 0\\ 1\\ -1 \end{bmatrix} \left ([0,1,-1]\begin{bmatrix} 0\\ 1\\ -1 \end{bmatrix} \right )^{-1}[0,1,-1]$$ but ...
2
votes
0answers
44 views

Taylor series of a square matrix

Let $A$ be a constant square matrix, $\Delta t$ is a scalar. I would imagine a taylor series of its exponential like this: $$e ^{A\,\Delta t} = I + \sum_{n=1}^\infty \frac{(A\,\Delta t)^n}{n!} $$ ...
0
votes
1answer
18 views

Write summation of vector outer products into matrix form

My question is as follows: Given the weighted summation of vector outer products $\sum_i\sum_jh_{ij}{\bf v_i}{\bf u_j}^T$, where $h_{ij}$ is the weight, and ${\bf v_i,u_j}$ are column vectors, I was ...
0
votes
2answers
34 views

Mathematical calculation

I encountered during my reading to ridge regression that $$(X^TX+\lambda I)^{-1}X^TX = I-\lambda(X^TX+\lambda I)^{-1}$$ What mathematical manipulation has been done here? Thanks in advance
1
vote
2answers
39 views

Getting characteristic polynomial from a small matrix

Sorry I don't know how to format matrices, but if I have this matrix $\pmatrix{1& 1& 0\\ 0& 0& 1\\ 1 &0& 1\\}$ How is the characteristic polynomial $λ^3 − 2λ^2 + λ − ...
0
votes
1answer
15 views

Matrix for a recurrence

The matrix for a recurrence of the form $a_{k+2} = ka_{k+1}+a_{k}$ where $a_0 = 0$ and $a_1 = 1$ is given by $$\begin{bmatrix}k & 1\\ 1 & 0 \end{bmatrix}^n = \begin{bmatrix} a_{k+1} & a_k \...
1
vote
2answers
47 views

Find non-diagonal matrices $A$ and $B$ such that $B^TAB$ is diagonal

Here $B^T$ denotes the transpose of $B$. $A$ and $B$ are invertible $3\times 3$ matrices with integer entries. $A$ is symmetric positive definite with at most two zero entries. We want the ...
0
votes
0answers
22 views

sending basissen

Lets say we have this $3\times3$ matrix: $$ \begin{bmatrix} 4 &−4 &12\\ 1& -1& 3\\ −1& 1 &−1 \end{bmatrix} $$ What is the algorithm to find a basis of $\Bbb R^3$ for which ...
2
votes
1answer
34 views

Row sum of inverse of a matrix

Let's say I have a matrix A, $$A= \begin{bmatrix} a_{11}& a_{12} & a_{13} \\ a_{21}& a_{22} & a_{23} \\ a_{31}& a_{32} & a_{33} \end{bmatrix} $$ All the elements of A are ...
1
vote
2answers
64 views

Compute $R^{2016}$ of a given counterclockwise rotation.

Write out the matrix $R$ of counterclockwise rotation by 30$^{\circ}$ in $\mathbb{R}^2$. Compute ${R}^{2016}$. Now this is an easy question to answer overall; 30 goes into 360 12 times and one twelfth ...
1
vote
1answer
40 views

Element-wise derivative of matrix logarithm

$E = ln(C) = -\sum_{a=1}^{\infty}\frac{1}{a}(I-C)^a$ I want to find a simple formula for $\frac{\partial E_{ij}}{\partial C_{pq}}$ $\frac{\partial C_{ij}}{\partial C_{pq}} = \delta_{ip}\delta_{...
0
votes
1answer
49 views

$A^2$ is bounded $\implies$ $A$ is bounded?

Let $A_n$ be a sequence of $k \times k$ real matrices. Assume $A_n^2$ is bounded w.r.t some norm. Is $A_n$ also bounded? I was able to show this is true if $A_n$ are symmetric matrices (using SVD). ...
0
votes
1answer
53 views

Is there an effect for the eigenvalues on vectors other than the Eigenvectors?

Does having an eigenvalue greater than one mean that the magnitude of any vector multiplied by the matrix will be increased?
1
vote
1answer
64 views

Linear independence of standard basis vectors from Vandermonde style vectors

Is it true a statement that all $n$ dimensional vectors of the standard basis (e.g. $[1 \ 0 \ 0 \ ...]^T$, $[0 \ 1 \ 0 \ ...]^T$ etc ..) are linearly independent from the set of the $n-1$ vectors $...
0
votes
1answer
25 views

QR decomposition subcases

Is the full QR decomposition the most general, which includes the reduced QR, i.e, is it alright to always compute the full QR Decomposition for a given matrix blindly? What's the point of having two ...
0
votes
0answers
30 views

Linearize Matrix Equation

I want to find a linearized formula for G in terms of A. $G = B^TC^{-1}T(I+BA)$ $G$ is 4x2 $B$ is a constant matrix 2x4 $A$ is a variable matrix 4x2 $C = I + A^TB^T + BA + BAA^TB^T$, so $C$ is ...
2
votes
1answer
29 views

Characterization of a square matrix.

I would like to see a proof to this fact. For a square matrix the following are equivalent: $A$ has a right inverse. $A$ has rank $n$, where $A$ is $n \times n$. $A$ is invertible.
5
votes
1answer
57 views

Rank of a lower triangular block matrix

For $$A= \begin{bmatrix}B&0\\C&D\end{bmatrix}$$ where $B, C, D$ are matrices that may be rectangular, is it true or false that $$\text{rank}(A)=\text{rank}(B)+\text{rank}(D)$$ I think that if ...
1
vote
1answer
31 views

Right inverse matrix

I know that if $A, B$ and $C$ are square matrices such that $$ AC=I \quad \mbox{and} \quad BA=I, $$ then \begin{eqnarray*} AC=I & \Rightarrow & BAC=B\\ & \Rightarrow &IC=B\\ & \...
1
vote
2answers
47 views

A proof of the Continuity of the inverse matrix function

I would like to see a proof to this fact. If $A$ is an invertible matrix and $B \in \mathcal{L}(\mathbb{R}^n,\mathbb{R}^n)$, that is an bounded linear opertor in $\mathbb{R}^n$. Then, if there ...
1
vote
1answer
64 views

How to prove that the vectors of the Krylov space of A are linearly independent if A is nonsingular.

$\mathbf{K}$ is a Krylov matrix. \begin{align} \mathbf{K}&= \left[ \begin{array}{ccccc} \mathbf{b} & \mathbf{A}\mathbf{b} & \mathbf{A}^2\mathbf{b} & \cdots & \mathbf{A}^{N-1}\...
4
votes
4answers
130 views

Not understanding derivative of a matrix-matrix product.

I am trying to figure out a the derivative of a matrix-matrix multiplication, but to no avail. This document seems to show me the answer, but I am having a hard time parsing it and understanding it. ...
2
votes
2answers
52 views

Given a symmetric matrix $A$, find $P$ such that $P^T A P$ is a diagonal matrix

Given $$A =\begin{pmatrix} 0 & 3 & 0 \\ 3 & 0 & 4 \\ 0 & 4 & 0\end{pmatrix}$$ find a matrix $P$ such that $P^T A P$ orthogonally diagonalizes $A$. Verify that $P^TAP$ is ...
3
votes
1answer
95 views

Is it possible to endow $\text{GL}_2(\Bbb R)$ with a ring structure?

My question is the following: Is it possible to find a binary operation $*$, seen as an addition, such that $(\text{GL}_2(\Bbb R),*,\cdot)$ has a ring structure (not necessarily with a unit)? [We ...