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
9 views

Matrices representing injective homomorphisms

Let $R$ be a ring and $M$, $N$ finitely genereated free modules modules over $R$. Let $A$ be a matrix representing a homomorphism $f: M \rightarrow N$. We know that the map $f$ is injective if and ...
2
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1answer
13 views

The lower bound of the smallest eigenvalue of a symmetric positive definite matrix

I encounter a symmetric positive definite matrix whose features are all diagonal entries are $1$. all the other entries are in $[0, 1)$, but the matrix is not diagonally dominant. Now I am ...
1
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1answer
17 views

Is orthogonality of column vectors preserved after right-multiplication by unitary matrix?

$\mathbf V$ is an $n \times (n-1)$ matrix with mutually orthogonal columns. $\mathbf Q$ is a unitary matrix of size $(n-1) \times (n-1)$. Is there a concise algebraic proof that the columns of $\...
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0answers
7 views

A matrix decomposition problem for row/column element order

Indeed, I don't know how to classify this problem, but I try to use matrix to describe it. The problem is that there exists a function $f(x, y)$ and its exact form remains unknown. But I have some ...
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1answer
19 views

Adjoint of a normal operator A is a polynomial in A

Is it true that adjoint of a normal operator A can be written as a polynomial in A?
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0answers
12 views

Calculating characteristic polynomial of linear transformation, cofactor expansion?

My question more is if cofactor expansion is really the best way to calculate the characteristic polynomial here: Linear transformation: $$T: P_2(R) -> P_2(R)$$ defined by $$T(a_o + a_1x + a_2x^2)...
1
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0answers
17 views

basic set of operation to transform square matrixes

There is any finite set of operations to transform any square matrix to any other square matrix (of the same order) ? If there isn't, can we construct a stable sub group with such property?
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0answers
10 views

Quantizing a matrix of reals while preserving row and column sums

Assume $E = [\epsilon_{i,j}]$, $i=1,2,\dotsc,m$, $j=1,2,\dotsc,n$, is an $m\times n$ matrix of reals. We know that $\forall i,j$, $\epsilon_{i,j}\in[-1/2,1/2]$. Moreover we know that both row-sums ...
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0answers
14 views

Optimize an Trace matrix form

In paper " Generalized Low Rank Approximations of Matrixces the Dimension of matrix are follow: $A_i$ is $r$ x $c$ L is $r$ x $l_1$ R is $c$ x $l2$ $D_i$ is $l_1$ x $l_2$ why it says ...
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0answers
15 views

Quadratic trace minimization with block diagonal constraints?

Given matrix $A$ and symmetric matrix $B$ with $0$'s on the diagonal and nonnegative values off-diagonal (imagine $B$ is a distance matrix), I have trouble solving $$\min_X\operatorname{tr}(A^TX^TAA^...
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2answers
39 views

Inverse of a matrix and a scalar

I'm asked o find $$\det((ad-bc)^{-1}\begin{bmatrix}a & b \\ c& d \end{bmatrix})$$ what I did was : This would equal to $$\det (\begin{bmatrix}\cfrac{1}{(ad-bc)} & 0 \\ 0 & \cfrac{1}...
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1answer
18 views

Singular Value Decomposition of Commuting Matrices

If two square matrices $M_1$ and $M_2$ commute, does it mean that the $U$ and $V^\dagger$ appearing in their singular value decompositions are the same? Specifically, does it imply that $$M_1 = U \...
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2answers
27 views

Find eigenvalues from characterestic equation

Equation of A : $λ^2 + 4λ - 12 = 0$ Find eigen values of $A$ and $A^3$ Find expression of $A^{-1}$ in terms of A I have no idea how to start solving it
2
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1answer
23 views

Trace minimization problem with “block diagonal diagonal” constraints?

I've reduced my optimization problem to the following trace minimization problem: $$\min_X\text{tr}(AXB),$$ subject to that $X$ is a block diagonal matrix whose blocks are all the same -- a diagonal ...
0
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6answers
128 views

Is it mathematically correct to replace 1 with I?

For example, $3=3I$ $=3\begin{bmatrix} 1 & 0 \\ 0 &1 \\ \end{bmatrix}$ $=\begin{bmatrix} 3 & 0 \\ 0 &3 \\ \end{bmatrix}$ ...
2
votes
1answer
45 views

Suppose that A satisfies $A^2 - 3A +2I = 0$. Find the eigenvalues of $A$ and $A^2$

knowing that A satisfies the equation $A^2 - 3A +2I = 0$ . I want to find the eigenvalues of $A$ and $A^2$. I don't know where to start. Can you explain how to solve such type of questions ?
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2answers
87 views

Is $A^T A$ similar to $AA^T$?

I saw in a proof somewhere that a square matrix $AA^T$ is similar to $A^T A$, so I thought about it and I don't know why (or whether) it's true. I tried using the fact that every matrix is similar to ...
1
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0answers
9 views

Orthogonally block diagonalize a special matrix

Consider a square matrix $A\in\mathbb{R}^{n\times n}$ of the form $$ A = S D $$ where $S\in\mathbb{R}^{n\times n}$ is symmetric and $D\in\mathbb{R}^{n\times n}$ diagonal with elements $\pm 1$ on the ...
2
votes
1answer
27 views

Skew-symmetric matrix and its eigenvalues

I checked some examples and I always received that skew-symmetric matrix of even dimension has only pure imaginary eigenvalues. For example: $\begin{bmatrix} 0 & 2 & 3 & 1 \\ -2 & ...
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votes
0answers
14 views

When pseudo inverse and general inverse of a invertible square matrix will be equal or not equal?

I calculated general inverse and pseudo inverse of a ivertible symmetrix matrix in MATLAB by using function inv and pinv respectively, but, I got different output. I didn't get the proper reason ...
0
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1answer
15 views

Calculate Real value Matrix from complex eigenvalues?

What I ask is exactly this Algorithm for real matrix given the complex eigenvalues But in my case, Im looking for 4*4 matrix which gives 4 pairs of complex eigenvalues. To be specific, I have ...
0
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1answer
28 views

Construct a defective matrix using scaling and rotation

Having read some material regarding the properties of defective matrix and methods to repair it (ie finding generalized eigenvalue).. I am wondering if there are ways to construct a defective matrix ...
2
votes
2answers
62 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 ...
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2answers
43 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 ...
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3answers
23 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 ...
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3answers
39 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
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1answer
26 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
35 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{ ...
2
votes
1answer
44 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
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1answer
30 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
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1answer
57 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
33 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 ...
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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
47 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
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1answer
52 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
21 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
69 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
37 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)...
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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.
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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
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2answers
25 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
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1answer
28 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 ...
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0answers
22 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
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1answer
10 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
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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
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2answers
51 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
16 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 ...