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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11
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1answer
958 views

4 by 4 Matrix Puzzle

I was solving the puzzle for the Company interview exam. I found this puzzle, I cannot come up with the solution. How to solve it and what is the correct answer? Determine the number of $4\times ...
11
votes
1answer
89 views

Largest determinant of a real $3\times 3$-matrix

What is the largest determinant of a real $3\times 3$-matrix with entries from the interval $[-1,1]$ ? A result of John Williamson says that the largest value is equal to $4$, if the entries are just ...
11
votes
7answers
3k views

Sylvester rank inequality

If $A$ and $B$ are two matrices of the same order $n$, then $$ \operatorname{rank} A + \operatorname{rank}B \leq \operatorname{rank} AB + n. $$ I don't know how to start proving this ...
11
votes
1answer
2k views

Probability that a random binary matrix is invertible?

What is the probability that a random $\{0,1\}$, $n \times n$ matrix is invertible? Assume the 0 and 1 are each present in an entry with probability $\frac{1}{2}$. Is there an explicit formula as a ...
11
votes
2answers
3k views

Determine the matrix relative to a given basis

Question: (a) Let $f: V \rightarrow W$ with $ V,W \simeq \mathbb{R}^{3}$ given by: $$f(x_1, x_2, x_3) = (x_1 - x_3, 2x_1 -5x_2 -x_3, x_2 + x_3).$$ Determine the matrix of $f$ relative to the basis ...
11
votes
1answer
131 views

Prove a $n \times n $ matrix has rank 3

I have been examining a problem dealing with finding the rank of a $n \times n $ matrix $M$ as follows: \begin{bmatrix} 0&1&4&9&16&\cdots &(n-1)^2\\ ...
11
votes
1answer
2k views

The Determinant of a Sum of Matrices

Given $N$ $n \times n$ matrices $\mathsf{A}^{1}, \dots, \mathsf{A}^{N}$, \begin{align} \det \left( \sum_{i = 1}^{N} \mathsf{A}^{i} \right) = \sum_{\sigma \in S} \det \mathsf{A}^{\sigma}, \end{align} ...
11
votes
1answer
12k views

orthogonal eigenvectors

I have a very simple question that can be stated without proof. Are all eigenvectors, of any matrix, always orthogonal? I am trying to understand Principal components and it is cruucial for me to see ...
11
votes
3answers
227 views

Are matrices best understood as linear maps?

Any linear map between finite-dimensional vector spaces may be represented by a matrix, and conversely. Matrix-matrix multiplication corresponds to map composition, and matrix-vector multiplication ...
11
votes
2answers
373 views

Determinant of a finite-dimensional matrix in terms of trace

I have noticed that for the case of 1x1, 2x2 and 3x3 matrices $A$, $B$, I can write the determinant of their commutator $C=[A,B]$ in terms of traces: 1x1 matrices $A$, $B$: $$\det(C)=\text{tr}(C)$$ ...
11
votes
1answer
226 views

Prove that if $f$ is continuous at $0$, it is continuous on $\mathbb{R}$

Long story short, the question I'm stuck on is as follows: Let $f$ be a positive-definite function. Prove that if $f$ is continuous at $0$, then it is continuous everywhere. Here's the long ...
11
votes
2answers
395 views

Geometric interpretation of normal and anti-hermitian matrices

How do I interpret following types of matrices as special types of transformations? I mean what are the transformative properties of following types of matrices, from $\mathbb{R}^n $ to $ ...
11
votes
1answer
223 views

Proving a certain determinant $\left|\det A\right|$ is complete square

Consider the following matrix $$ A_{ij}= \begin{cases} 1\quad\text{ if }\space (i+j)\space\text{ is prime,}\\ 0\quad\text{ otherwise.} \end{cases} $$ How can one prove that $\left|\det A\right|$ is a ...
11
votes
3answers
183 views

Is there an easy way to find the sign of the determinant of an orthogonal matrix?

I just learned that if a matrix is orthogonal, its determinant can only be valued 1 or -1. Now, if I were presented with a large matrix where it would take a lot of effort to calculate its ...
11
votes
1answer
174 views

how prove the following statment for this matrix.

Let $A:=[a_{ij}]_{n×n}$ , $a_{ij}=0$ or $a_{ij}=1$ and $\exists m \in\mathbb N$ such that $A^m=J-I$, where $I$ is the identity matrix and $J=[1]_{n×n}$ (each entry is $1$). How to prove: $\exists ...
11
votes
2answers
434 views

Does this expression have a (“better”) determinant form?

[Edit I've found a determinant that satisfies the letter of the previous version of this question, but not its "Cayley-Menger" spirit.] A tetrahedron with face areas $w$, $x$, $y$, $z$ and ...
11
votes
1answer
557 views

spectral norm of random matrix

Suppose $A$ is a $n \times n$ random matrix with centered Gaussian (real) i.i.d entries with variance $\sigma^2/n$. What to we know about the spectral norm $s(A)$ of $A$, that is $\sqrt{\rho(A^t A)}$ ...
11
votes
1answer
667 views

Determinant of a symmetric matrix

Given an $n\times n$ matrix $C= [c_{ij}]$ which is symmetric (i.e. $c_{ij}=c_{ji}\ \forall i,j$) calculate the determinant of the following matrix (assume $c_{ij} \neq 0\ \forall i,j$): ...
11
votes
0answers
296 views

Determining the kernel of a Vandermonde-like matrix

The kernel of a Vandermonde matrix can be determined using this formula. The following type of matrix has a similar structure, and should also have a one-dimensional kernel. $V= \begin{bmatrix} 1 ...
11
votes
0answers
420 views

Inverse of Toeplitz Matrix Property

Sorry if this question has been asked already but I didn't find it. Given a symmetric Toeplitz matrix of the form $$\left[\begin{array}{llll} a_0 & a_1 & \dots & a_n\\ a_1 & a_0 ...
11
votes
1answer
251 views

Upper bound for the widest matrix with no two subsets of columns with the same vector sum

Over at PPCG there is an ongoing contest going on to find the largest matrix without a certain property, called property $X$. The description is as follows (copied from the question). A circulant ...
10
votes
4answers
657 views

If $A$ is singular, is $A^3+A^2+A$ singular?

Suppose that $A$ is singular, is $A^3 + A^2 + A$ singular as well?
10
votes
5answers
814 views

Matrix raised to 14th power

Calculate $\left(\begin{matrix} 6&1&0\\0&6&1\\0&0&6\end{matrix}\right)^{14}$ Whould I do it one by one, and then find a pattern? I sense $6^{14}$ on the diagonal, and zeroes ...
10
votes
2answers
1k views

Show that $ e^{A+B}=e^A e^B$

If $A$ and $B$ are $n\times n$ matrices such that $AB = BA$ (that is, $A$ and $B$ commute), show that $$ e^{A+B}=e^A e^B$$ Note that $A$ and $B$ do NOT have to be diagonalizable.
10
votes
7answers
726 views

Check if $\det(I + S) = 1 + \operatorname{trace}(S)$ holds ?

I saw the following statement in my homework and we are asked to make use of the statement: If $S$ is a symmetric matrix then $$\det(I + S ) = 1 + \operatorname{trace}(S).$$ However, I am not ...
10
votes
4answers
5k views

Prove that $A+I$ is invertible if $A$ is nilpotent [duplicate]

Possible Duplicate: Units and Nilpotents Given $A^{2012}=0$ prove that $A+I$ is invertible and find an expression for $(A+I)^{-1}$ in terms of $A$. ($I$ is the identity matrix).
10
votes
4answers
740 views

$AB \neq 0$ but $BA=0$

Do there exists to matrices or objects such that $AB \neq 0$ but $BA=0$? Another way to ask this question is if there exists objects or matrices $A$ and $B$ such that... $[A,B]=AB$ where $[ \, , \, ]$ ...
10
votes
7answers
9k views

practical uses of matrix multiplication

The use of matrix multiplication is usually given with graphics initially (scalings, translations, rotations, etc). Then there are more in-depth examples such as counting the number of walks between ...
10
votes
2answers
24k 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!
10
votes
5answers
432 views

Prove $BA - A^2B^2 = I_n$.

I have a problem with this. Actually, still don't have the right way to start :/ Problem : Let $A$ and $B$ be $n \times n$ complex matrices such that $AB - B^2A^2 = I_n$. Prove that if $A^3 + B^3 = ...
10
votes
1answer
2k views

Is the matrix $A$ diagonalizable if $A^2=I$

If $A$ is an involutory matrix, i.e. $A^2=I$, then is $A$ diagonalizable?
10
votes
3answers
2k 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 ...
10
votes
4answers
4k views

Solving very large matrices in “pieces”

Say you have a very dense matrix that is 30000x30000 elements. The very dense matrix comes from the radiosity equation, which I discussed here. Say you have Ax = B. You have B, and A is 30000x30000 ...
10
votes
2answers
168 views

Determinant of $4\times4$ Matrix

I tried to solve for a $4 \times 4$ matrix, but I'm unsure if I did this properly, can anyone tell me if I did this correct? Or if there were any mistakes where at? Also, I know this is an inefficient ...
10
votes
3answers
353 views

What is the largest determinant you can get by filling in 0,1 or 2 into a 4-by-4 matrix?

For example $$\left| \begin{array}{ccc} 2 & 0 & 0 & 2 \\ 2 & 0 & 2 & 0 \\ 0 & 2 & 1 & 2 \\ 2 & 2 & 0 & 0 \end{array} \right|=40$$ Can it get bigger ...
10
votes
2answers
806 views

Algebraic proof of a trig matrix identity?

I'll put the question first, and then the background, because I'm not sure that the background is necessary to answer the question: I have a geometric proof, but is there an elegant algebraic proof ...
10
votes
2answers
249 views

Let the matrix $A=[a_{ij}]_{n×n}$ be defined by $a_{ij}=\gcd(i,j )$. How prove that $A$ is invertible, and compute $\det(A)$?

Let $A=[a_{ij}]_{n×n}$ be the matrix defined by letting $a_{ij}$ be the rational number such that $$a_{ij}=\gcd(i,j ).$$ How prove that $A$ is invertible, and compute $\det(A)$? thanks in advance
10
votes
2answers
407 views

Variety of Nilpotent Matrices

Let $k$ be an algebraically closed field and view $M_n(k)$ as $\mathbb{A}^{n^2}$. $A\in M_n(k)$ is nilpotent if and only if $A^n=0$. Since the equation $A^n=0$ is given by $n^2$ polynomial ...
10
votes
1answer
401 views

Probability of a random $n \times n$ matrix over $\mathbb F_2$ being nonsingular

Given a random square matrix of size $n\times n$ in the field $\mathbb F_2$, what is the probability that its determinant is $1$? (This is also the probability that the matrix is non-singular, since ...
10
votes
2answers
2k views

Calculating RGB plus Amber

I'm currently working on a wide gamut light source using red, green and blue LED emitters. From an internal xyY (or CIE XYZ) representation, I can reach any color or color temperature via a 3x3 ...
10
votes
1answer
477 views

Example of similar matrices $A$ and $B$ such that products $AB$ and $BA$ are not similar

I'm looking for the simplest possible example of square matrices $A$ and $B$ such that $A$ is similar to $B$, $AB$ is not similar to $BA$. Such an example should exist, but I would like to find ...
10
votes
1answer
122 views

How many matrices in $M_n(\mathbb{F}_q)$ are nilpotent?

I have strong computational evidence to think that the answer is $q^{n(n-1)}$, although a proof eludes me. Any ideas?
10
votes
3answers
148 views

Determinant of $a_{i,j}=(x_i+y_j)^k$

How can I find the determinant of the matrix $A\in\mathcal{M}_n(\mathbb{R})$ with coefficients $a_{i,j}=(x_i+y_j)^k,k<n$ ? All the $x_u,y_u$ are real numbers. Derivating won't help, and I didn't ...
10
votes
3answers
222 views

Show that $A$ is symmetric, with $A \in M_n(\mathbb R)$

Let $A \in M_n(\mathbb R)$. Show that if $A(\,{}^t\!A)A$ is is symmetric, then $A$ is also symmetric. My attempt: If $A \in Gl_n(\mathbb{R})$, We have : ${}^t(A^{-1})=(\,{}^t\!A)^{-1}$ ...
10
votes
3answers
221 views

Existence of some type matrix

Is there square matrix $A$ of size $3$ with real entries such that $$ \operatorname{tr}(A)=0\text{ and }A^2+A^T=I. $$ I have proved that there is not with size $2$ using definition of "trace", but ...
10
votes
2answers
403 views

Characteristic polynomial of a matrix with zeros on its diagonal

Let $p(x)=x^n+a_{n-2}x^{n-2}+a_{n-3}x^{n-3}+\cdots+a_1x+a_0=(x-\lambda_1)\cdots(x-\lambda_n)$ be a polynomial with real coefficients such that every $\lambda_i$ is real. Is there always a symmetric ...
10
votes
2answers
155 views

Prove or disprove : $\det(A^k + B^k) \geq 0$

This question came from here. As the OP hasn't edited his question and I really want the answer, I'm adding my thoughts. Let $A, B$ be two real $n\times n$ matrices that commute and $\det(A + ...
10
votes
2answers
264 views

Given $BA$, find $AB$.

If $A, B$ are $4\times 3, 3\times 4$ real matrices, respectively, $$BA=\begin{bmatrix} -9 & -20 & -35 \\ 2 & 5 & 7 \\ 2 & 4 &8 \end{bmatrix}$$ $$AB=\begin{bmatrix} -14 & ...
10
votes
1answer
287 views

What can we say about $(I-AD)^{-1}$ if $D$ is a diagonal matrix?

Assume we know that square matrices $A$ and $(I-AD)^{-1}$ are invertible and also $D$ is a diagonal matrix. Also assume that $A$ is a symmetric matrix. My question is when we can express ...
10
votes
3answers
416 views

Generation of unimodular matrices with bounded elements

Does anybody know what is the algorithm for generating random unimodular matrices (integer matrices with determinant $\pm 1$) whose elements do not exceed a given bound? Such an algorithm is mentioned ...