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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7answers
471 views

Is the square root of a triangular matrix necessarily triangular?

$X^2 = L$, with $L$ lower triangular, but $X$ is not lower triangular. Is it possible? I know that a lower triangular matrix $L$ (not a diagonal matrix for this question), $$L_{nm} \cases{=0 & ...
9
votes
4answers
1k views

What is the fastest way to find the characteristic polynomial of a matrix?

Finding the characteristic polynomial of a matrix of order $n$ is a tedious and boring task for $n > 2$. I know that: the coefficient of $\lambda^n$ is $(-1)^n$, the coefficient of ...
9
votes
2answers
17k views

Transpose of inverse vs inverse of transpose

I can't seem to find the answer to this using Google. Is the transpose of the inverse of a square matrix the same as the inverse of the transpose of that same matrix? Thanks!
9
votes
3answers
1k views

$I_m - AB$ is invertible if and only if $I_n - BA$ is invertible.

The problem is from Algebra by Artin, Chapter 1, Miscellaneous Problems, Exercise 8. I have been trying for a long time now to solve it but I am unsuccessful. Let $A,B$ be $m \times n$ and $n ...
9
votes
4answers
327 views

Example that the Jordan canonical form is not “robust.”

I'm working on this problem that asks to show that the Jordan canonical form is not robust in the sense that small changes in the entries of a matrix $A$ can cause large changes in the entries of its ...
9
votes
3answers
1k views

What's the meaning of the transpose?

I don't understand the motivation of the transpose (or better yet, I haven't even seen one). It feels like just something pulled out of a hat. Thinking about it makes it seem like a product of being ...
9
votes
1answer
881 views

How to understand spectral decomposition geometrically

Let $A$ be a $k\times k$ positive definite symmetric matrix. By spectral decomposition, we have $$A = \lambda_1e_1e_1'+ ... + \lambda_ke_ke_k'$$ and $$A^{-1} = ...
9
votes
2answers
643 views

Has there been a rigorous analysis of Strassen's algorithm?

According to Wikipedia, Strassen's Algorithm runs in $O(N^{2.807})$ time. Has anyone seen a more rigorous analysis displaying constants, possibly in a specific language such as C or Java? I ...
9
votes
1answer
916 views

Matrix raised to a matrix

Good evening, I was wondering if there is such a valid operation as raising a matrix to the power of a matrix, e.g. vaguely, if $M$ is a matrix, is $$ M^M $$ valid, or is there at least something ...
9
votes
1answer
207 views

Calculating the eigenvalues of a matrix

How to find the eigenvalues of $$\begin{bmatrix} 0 & 1 & & &\\ k & 0 & 2 & &\\ & k-1 & 0 & 3 &\\ & ...
9
votes
2answers
8k views

Prove that the eigenvalues of a block matrix are the combined eigenvalues of its blocks

Let $A$ be a block upper triangular matrix: $A = \left( \begin{matrix} A_{1,1}&A_{1,2}\\ 0&A_{2,2} \end{matrix} \right)$ where $A_{1,1} ∈ C^{p×p}$, $A_{2,2} ∈ C^{(n-p)×(n-p)}$ Show ...
9
votes
2answers
104 views

Subgroups of $\mathrm{GL}(n,\mathbb{Z})$ which are not finitely generated

The group $\mathrm{GL}(n,\mathbb{Z})$ is finitely generated: take for example diagonal matrices, permutations and one elementary matrix (upper triangular). Are there some simple / nice examples of ...
9
votes
2answers
544 views

Determine the winner of a tic tac toe board with a single matrix expression?

Assume a tic-tac-toe board's state is stored in a matrix. $$ S=\begin{bmatrix} -1 & 0 & 1 \\ 1 & -1 & 0 \\ 1 & 0 & -1 \\ \end{bmatrix} $$ Here, $X$ is mapped to $1$, $O$ is ...
9
votes
3answers
423 views

What are mandatory conditions for a family of matrices to commute?

Suppose that there are some matrices. Each matrix in the set must commute with another in the set. What are the mandatory conditions for this?
9
votes
1answer
200 views

Matrix algorithm convergence

Suppose I start with a $n \times n$ matrix of zeros and ones: $$ \begin{bmatrix} 0 & 0 & 0 & 1 & 1\\ 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1\\ 1 ...
9
votes
5answers
429 views

Smallest Non-negative number in a matrix

There is a question I encountered which said to fill an $N \times N$ matrix such that each entry in the matrix is the smallest non-negative number which does not appear either above the entry or to ...
9
votes
1answer
1k views

Does equality of characteristic polynomials guarantee equivalence of matrices?

I have a qualifying exam coming up in a couple days and I am just trying to understand some pathological examples I have in my notes. I will list a similar problem which I know the solution to and ...
9
votes
1answer
2k views

Under what circumstance will a covariance matrix be positive semi-definite rather than positive definite?

I have a covariance matrix: $\operatorname{cov}(\mathbf{X}, \mathbf{X}) = \operatorname{E}[(\mathbf{X} - \operatorname{E}[\mathbf{X}])(\mathbf{X} - \operatorname{E}[\mathbf{X}])^T]$ According to ...
9
votes
3answers
492 views

Is there a general formula for the derivative of $\exp(A(x))$ when $A(x)$ is a matrix?

It's easy for scalars, $(\exp(a(x)))' = a' e^a$. But can anything be said about matrices? Do $A(x)$ and $A'(x)$ commute such that $(\exp(A(x)))' = A' e^A = e^A A'$ or is this only a special case?
9
votes
1answer
765 views

Do these matrix rings have non-zero elements that are neither units nor zero divisors?

Let $R$ be a commutative ring (with $1$) and $R^{n \times n}$ be the ring of $n \times n$ matrices with entries in $R$. In addition, suppose that $R$ is a ring in which every non-zero element is ...
9
votes
2answers
131 views

$AB=z \mathrm{Id}_n$ implies $z^m BA = z^{m+1} \mathrm{Id}_n$ for what $m$?

This question builds on a series of questions looking for elementary proofs that $AB=\mathrm{Id}$ implies $BA=\mathrm{Id}$, for $A$ and $B$ both $n \times n$ matrices over a commutative ring. First ...
9
votes
3answers
4k views

Get Transformation Matrix from Points

I have built a little C# application that allows visualization of perpective transformations with a matrix, in 2D XYW space. Now I would like to be able to calculate the matrix from the four corners ...
9
votes
1answer
517 views

Detecting symmetric matrices of the form (low-rank + diagonal matrix)

Let $\Sigma$ be a symmetric positive definite matrix of dimensions $n \times n$. Is there a numerically robust way of checking whether it can be decomposed as $\Sigma = \mathcal{D} + v^t.v$ where $v$ ...
9
votes
1answer
100 views

Does the Trace product in a semigroup have any relation with Trace of a matrix / matrix product

I recently read an article on generalized inverses and Green's relations (by X.Mary). The framework is semigroups, but obviously it has a lot of application within matrix theory. In the article ...
9
votes
0answers
121 views

How many arrays with crossed cells, order of rows/columns irrelevant

I've been struggling with this simple problem for months though as I am a newbie to… well, maths, there's high chance someone more educated than myself may get it right! Let's consider an array or a ...
9
votes
1answer
104 views

Intuition & Proof of rank(AB) $\le$ min{rank(A), rank(B)} (without inverses or maps) [Poole P217 3.6.59, 60]

I'm aware of analogous threads; I hope that mine is specific enough not to be esteemed one. $\mathbf{a^i}$ is a row vector. $A, B$ are matrices. Prove: $1$. $\mathbf{a^i}B$ is a linear ...
8
votes
6answers
2k views

If $A^2$ is invertible, then $A$ is also invertible?

True or False: If $A^2$ is invertible, then $A$ is also invertible. ($A$ is a matrix here.) The answer is true. I was trying to come up with an example that makes this false. But I couldn't. ...
8
votes
6answers
510 views

$\left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array} \right)$ not diagonalizable

I would like to ask you about this problem, that I encountered: Show that there exists no matrix T such that $$T^{-1}\cdot \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array} ...
8
votes
3answers
684 views

Trace of powers of a nilpotent matrix

Let $A$ be an $n\times n$ complex nilpotent matrix. Then we know that because all eigenvalues of $A$ must be 0, it follows that $\text{tr}(A^n)=0$ for all positive integers $n$. What I would like to ...
8
votes
3answers
293 views

$A \in M_{n}(\mathbb{R})$ satisfies $A+A^{t}=I$, then does it imply $\text{det}(A)>0$

I am stuck on this simple question for a long time. If $A \in M_{n}(\mathbb{R})$ satisfies $A+A^{t}=I$, then does it imply $\text{det}(A)>0$? I tried finding a counter-example as well as tried ...
8
votes
5answers
481 views

What is the motivation defining Matrix Similarity?

I'm taking the course Linear Algebra 1, and recently we've learned about matrix similarity. What is the motivation defining it? or, What are the uses/applications for this definition? Thanks
8
votes
4answers
534 views

“weird” ring with 4 elements - how does it arise?

For no real reason, I did a classification of (associative, with a 1) rings with less than 8 elements (they all happen to be commutative). Most of the rings I got were of a type I knew - namely: ...
8
votes
2answers
2k views

Matrix of Infinite Dimension

Any linear map between two finite-dimensional vector spaces can be represented as a matrix under the bases of the two spaces. But if one or all of the vector spaces is infinite dimensional, is the ...
8
votes
4answers
3k views

How do I calculate the $p$-norm of a matrix?

I know that the $p$-norm for a matrix is: $$\|A\| = \max_{x\neq 0} \frac{\|Ax\|_p}{\|x\|_p}$$ but I don't know what this really means. So how would I compute the $2$-norm, $3$-norm, etc for the ...
8
votes
5answers
474 views

Matrices with eigenvalues 0 and 1

How can you describe all $2\times 2$ matrices whose eigenvalues are 0 and 1? My attempt: I know that 0 and 1 has to be solutions of the characteristic polynomial. And I've considered some examples ...
8
votes
2answers
3k views

Looking for insightful explanation as to why right inverse equals left inverse for square invertible matrices

The simple proof goes: Let B be the left inverse of A, C the right inverse. C = (BA)C = B(AC) = B This proof relies on associativity yet does not give any insight as to why this surprising fact ...
8
votes
3answers
87 views

Find the expansion for $\det(I+\epsilon A)$ where $\epsilon$ is small without using eigenvalue.

I'm taking a linear algebra course and the professor included the problem that prove $$ \rm{det}(I+\epsilon A) = 1 + \epsilon\,\rm{tr}\,A + o(\epsilon) $$ Since the professor hasn't covered the ...
8
votes
3answers
279 views

Relationship between Nilpotent Matrix and Matrix with all zero diagonal factors.

solving Linear Algebra HW, I suddenly became curious about the relationship between Nilpotent Matrix and matrix with all zero diagonal factors such that $A_{11} = A_{22} = \cdots = A_{nn} = 0$ Does ...
8
votes
1answer
282 views

Why is the permanent of interest for complexity theorists?

Studying a bit about the determinant and the permanent, I'm told that although both concepts have very similar formulas, the permanent was of not much interest historically - it was until later that ...
8
votes
3answers
4k views

Derivative of determinant of a matrix

Good morning everyone, I would like to know how to calculate: $\frac{d}{dt}\det \big(A_1(t), A_2(t), \ldots, A_n (t) \big)$ Help me please. Thank you
8
votes
3answers
264 views

Intuitively (concretely actually) what happens when you multiply a matrix by its transpose?

The construct $A^TA$ for $A$ any $m \times n$ matrix seems to appear often in formulae and results. For example I was reading that square root of eigenvalues of $A^TA$ (an $n \times n$ matrix) are ...
8
votes
2answers
6k views

What is the proof that covariance matrices are always semi-definite?

Suppose that we have two different discreet signal vectors of $N^{th}$ dimension, namely $\textbf{x}[i]$ and $\textbf{y}[i]$, each one having a total of $M$ set of samples/vectors. $\textbf{x}[m] = ...
8
votes
4answers
309 views

What does it mean if $\det(A)$ equals $1$?

What does it mean if $\det(A)$ equals $1$? Does it mean that the identity matrix can be obtained from $A$ by only adding multiples of rows onto others?
8
votes
3answers
343 views

When will $AB=BA$?

Given two square matrices $A,B$ with same dimension, what conditions will lead to this result? Or what result will this condition lead to? I thought this is a quite simple question, but I can find ...
8
votes
2answers
5k views

How to calculate the gradient of log det matrix inverse?

How to calculate the gradient with respect to $X$ of: $$ \log \mathrm{det}\, X^{-1} $$ here $X$ is a positive definite matrix, and det is the determinant of a matrix. How to calculate this? Or ...
8
votes
4answers
22k views

Diagonalizable Matrices: How to determine?

I am trying to figure out how to determine the diagonalizability of the following two matrices. For the first matrix $$\left[\begin{matrix} 0 & 1 & 0\\0 & 0 & 1\\2 & -5 & ...
8
votes
3answers
331 views

Two introductory linear algebra problems

I remember when I was in Moscow one of my homework questions was: Is there a $2\times 4$ matrix whose $2\times 2$ minors are: a) $(2,3,4,5,6,7)$ b) $(3,4,5,6,7,8)$ c) ...
8
votes
3answers
225 views

Matrix Equation $A^3-3A=\begin{pmatrix}-7 & -9\\ 3 & 2\end{pmatrix}$

How can I solve in $\mathcal{M}_{2}(\mathbb{Z})$ the equation $$A^3-3A=\begin{pmatrix}-7 & -9\\ 3 & 2\end{pmatrix}?$$ I try to use $$A^2-Tr(A)A+detA\cdot I_2=O_2$$ but I don't still obtain ...
8
votes
1answer
343 views

Is the zero matrix the only symmetric, nilpotent matrix with real values?

My intuition tells me that the zero matrix is the only matrix that is symmetric and nilpotent with real values, but I'm having trouble proving it (or finding a counterexample.) I have searched for ...
8
votes
3answers
223 views

Minimize $||Ax-b||$ but for $A$, not $x$

I have a machine learning regression problem. I need to minimize $$ \sum_i||Ax_i-b_i||_2^2 $$ However I am trying to find matrix $A$, not the usual $x$, and I have lots of example data for $x_i$ and ...