Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [matrix-analysis]

For question about matrices and their algebraic properties. Together with [tag:linear-algebra] if necessary.

In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties.

Some particular topics out of many include; operations defined on matrices (such as matrix addition, matrix multiplication and operations derived from these), functions of matrices (such as matrix exponentiation and matrix logarithm, and even sines and cosines etc. of matrices), and the eigenvalues of matrices (eigendecomposition of a matrix, eigenvalue perturbation theory).

See also: Wikipedia

297 questions
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Why is the matrix norm $||A||_1$ maximum absolute column sum of the matrix?

By definition, we have $$ \|V\|_p := \sqrt[p]{\displaystyle \sum_{i=1}^{n}|v_i|^p} \qquad \text{and} \qquad \|A\|_p := \sup_{x\not=0}\frac{||Ax||_p}{||x||_p} $$ and if $A$ is finite, we change sup to max. However I don't really get how we get to…
Cancan
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Does the exponential of a matrix commute with the matrix?

Can someone give me an idea for the proof that for every $t\in \mathbb{C}$ we have $e^{tA}\cdot A = A \cdot e^{tA} =$ ? I couldn't find a counterexample, so my gues is, that it would be true, but I'm not sure even how to begin the proof.
resu
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11
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Is the function $A \mapsto \sum\limits_{j=0}^{\infty} \langle A^j v, A^j v \rangle$ differentiable everywhere?

Suppose $v \in \mathbb R^n$ is a fixed vector. We define a scalar-valued function on $n \times n$ matrices $f: M_n(\mathbb R) \to \mathbb R$ by \begin{align*} A \mapsto \sum\limits_{j=0}^{\infty} \langle A^j v, A^j v \rangle. \end{align*} Let us…
MyCindy2012
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Operator norm (induced $2$-norm) of a Kronecker tensor

Let $A \in \mathcal M(n \times n; \mathbb R)$ with $\rho(A) < 1$. Then we know $I \otimes I - A^T \otimes A^T$ is invertible where $\otimes$ denotes kronecker product. Let $\text{vec}$ denote the vectorization operation and $\mathcal T = (I \otimes…
user1101010
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Is $l_2$ on $\mathbb{R}^n$ the only norm for which it is equal to its dual norm?

Given any norm $\|.\|$ on $\mathbb{R}^n$, its dual norm $\|.\|^D$ is defined as the following: $\|v\|^D = \sup_{\|x\|\leq 1} |(v,x)|$, where $(,)$ is the standard Euclidean Inner product. Under that definition, it turns out that the dual norm of the…
Rohan Didmishe
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Invertibility of infinite-dimensional matrix

I have a matrix $M \in \mathbb{R}^{n \times n}$ whose columns are linearly independent. Hence, $M$ is invertible. How to extend this conclusion to the case where $n$ is infinite? Specifically, given that $n\in\mathbb{N}$, let $X$ and $Y$ be Banach…
lulu
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Conjecture about $(0,1)$-matrices

Let $A$ be an $m$ by $n$ $(0,1)$-matrix. For $1\leq i \leq m$ and $1\leq j \leq n$, let $f(A,i,j)$ be the number of entries in $A$ not in row $i$, not in column $j$, and not equal to $a_{ij}$. I would like a proof or counterexample to the following…
Jeremy K
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7
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To find the inverse of a special kind of matrix.

In a matrix analysis problem, I encountered the following special kind of matrix $$ \begin{bmatrix} 0 & 1 & a & a & a & a \\ 1 & 0 & a& a& a& a \\ a& a &0 & 1& a& a \\ a& a &1 & 0 & a& a \\ a & a & a & a &0 & 1\\ a & a…
User8976
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How to prove that generalized Vandermonde matrix is invertible?

Given $$A = \left( z_i^{\lambda_k}\right)_{i,j = 1,\ldots, n} = \begin{pmatrix} z_1^{\lambda_1} & z_1^{\lambda_2} & \cdots & z_1^{\lambda_n} \\ z_2^{\lambda_1} & z_2^{\lambda_2} & \cdots & z_2^{\lambda_n} \\ \vdots & \vdots & \ddots & \vdots …
fixingmath
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Proving an inequality for operators.

Let $\mathbb P_n$ be the space of all $n \times n$ self-adjoint positive definite matrices. Consider the function $\varphi: \mathbb P_n \longrightarrow \mathbb R$ defined by $$\varphi (A) = -\text {tr}\ (A \log A).$$ Show that for all $t \in (0,1)$…
RKC
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Index of a Nilpotent matrix

I was wondering why there can't be a nilpotent matrix of index greater than its no. of rows. Like why there does not exist a nilpotent matrix of index 3 in $M_{2×2}(F)$ When I look up on the internet it says that it is related to ring theory and…
Ankush Kothiyal
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"Almost Normal" Matrix and Gap between Spectral Radius/Norm

Let's denote $$\Vert{A}\Vert := \max_{x\neq0}\frac{x^* Ax}{x^*x}$$ and let $\rho(A)$ denote the largest absolute value of the eigenvalues of matrix $A$. From basic linear algebra, one could characterize normal matrices as those unitarily…
Yu-Hsuan Huang
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Estimate parameter $a$ such that $tr \left[ A (B- (I-aC)B(I-aC) ) \right] > 0$.

Suppose $A, B, C$ are all real symmetric and positive definite matrices. Consider the function $f: \mathbb R \to \mathbb R$ given by $$ a \mapsto {\bf tr}\left[ A (B- (I-aC)B(I-aC) ) \right],$$ where $I$ is identity matrix. It is clear $f(0) = 0$…
MyCindy2012
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Is there a relation between the solutions to these two Lyapunov matrix equations?

Let $A \in \mathcal M(n \times n; \mathbb R)$ with $\rho(A) < 1$. Let $X, Y$ be the solutions to the following Lyapunov matrix equations \begin{align*} X &= A^T X A + Q \\ Y &= A Y A^T + Q \end{align*} where positive definite matrix $Q$ is given. By…
user1101010
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Help with Maximizing $\max_{U_i, V_j}\sum_{i,j} \frac{\operatorname{trace}^2(A_{ij}^\top U_i V_j)}{\|U_i V_j\|_F^2}$

I am trying to solve (approximately) the following maximization problem: $$ \max_{U_i, V_j} \sum_{i,j} \frac{\operatorname{trace}^2(A_{ij}^\top U_i V_j)}{\|U_i V_j\|_F^2}, $$ where: $ i, j \in \{1,2\} $, $ \|\cdot\|_F $ denotes the Frobenius…
Alex
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