# Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

42,680 questions
Filter by
Sorted by
Tagged with
7 views

40 views

### Does there exist a non-symmetric involutory matrix?

Let $A$ be a real involutory matrix i.e. $$A^2 = I.$$ Is it necessarily symmetric? Any help will be highly appreciated. Thank you very much.
12 views

### Equivalence of semidefiniteness and quadratic inequality

My question is about a logical equivalence in Robert Freund's lecture notes "An Introduction to Semidefinite Programming (SDP)" from MIT's OCW. The claim is that for $Q$ an $n\times n$ real ...
14 views

### Bounding the norm of the inverse matrix

Very strangely, I do struggle with answering the following, seemingly elementary, question. For a square matrix $A$, is there an upper bound for the norm of its inverse? In terms of e.g. the norm of ...
15 views

### I have a quadratic minimization problem such that, $e=x^TAx+x^Tb+k$, I wish to find its minima using recursive least square.

I have arrived at a generic quadratic minimization problem, such that, \begin{equation} e=x^TAx+x^Tb+k \end{equation} where,$x=[x_1,x_2,x_3]^T \in \mathbb{R}^3$, $b=[\gamma,0,0]$ and $k$ is a constant....
16 views

### Commutator of 2 Rotational Matrix in x and z axis

How do i find the commutator of rotational matrix in x and z axis? Dx = \begin{pmatrix}1&0&0\\ 0&cos\gamma &-sin\gamma \\ 0&sin\gamma &cos\gamma \end{pmatrix} Dz = \begin{...
53 views

39 views

26 views

### Are all the conditions of the Moore-Penrose inverse definitions necessary?

The Moore-Penrose inverse of a real or complex matrix $M$ is the unique matrix defined by four conditions. Can any of these four conditions be relaxed with no loss of uniqueness? I noticed there was a ...
60 views

### How to find the determinant A from an equation having A as variable?

I'm currently struggling because I can't find the answer do this. If anyone can help me, it would be great. A is a $5\times 5$ non scalar matrix, $(A+2)(A+A^3+1)^2 (A^2+A^3+1)^3 =0$ a) ...
16 views

### Matrix Operations: Name of function to rotate each column such that the first element is on the main diagonal

Suppose I have a matrix, A = [ 1 1 1 2 2 2 3 3 3 ] And I would like to create or find some function g such ...
28 views

### Matrice Algebra

I am reading a paper and stumbled upon this piece: $$(2u-1)^TQ(2u-1) = 4u^TQu - 4(1^TQ)u + 1^TQ1$$ I have two conflicting results but both results are different $4u^TQu-2u1^TQ-2u^TQ1+1^TQ1$ another ...
24 views

### When is a series of matrices divergent. How to define divergence in this case?

In Quantum Mechanics we deal with series of operators represented as matrices like $$e^A = 1+ A + \frac{A^2}{2} + \dots$$ and similarly for $\sin(A)$, etc., where $A$ is a matrix. Now my question ...
35 views

### Approximation of a matrix multiplication

Let $A_{n \times n}$ be a positive-definite matrix. There is a way to approximate the following product? $$\left(I-\frac{1}{n}11^{T}\right) A \left(I-\frac{1}{n}11^{T}\right)$$ where $1$ is a column ...
25 views

### If $B$ is positive definite, then $(g^TBg) (g^TB^{-1}g) \ge (g^Tg)^2$

I am trying to prove something in Optimization, and it comes down to proving the inequality $$(g^TBg) (g^TB^{-1}g) \ge (g^Tg)^2$$ where $B$ is positive definite and $g$ is any vector. I am able to ...
38 views

### Expected value of $1$'s in a matrix product defined over $\mathbb{Z}_2$

Let $\mathbf{A}$, $\mathbf{B}$ be random boolean matrices of $n \times n$ size, such that the matrix entry is $1$ with probability $p$ and $0$ otherwise. All entries are independent. How many $1$'s on ...
21 views

### Applying Isometries to Matrix Inequalities [closed]

If $A$, $B$ are two matrices such that $A-B\geq 0$, i.e., the difference is positive semidefinite. Let $V$ be an isometry. Is it true that $V (A-B) V^\dagger \geq 0$? Or under what conditions does ...
17 views

23 views

### Demonstrate this matrix derivative expression with the formulas of this table.

I'm doing this problem: Calculate $\frac{∂||Ax-b||^2}{∂x} = 2A^T (A-b)$. Knowing that $\frac{∂||x||^2}{∂x} = 2x$ so far I have this: $$\frac{∂||Ax-b||^2}{∂(Ax-b)} \frac{∂(Ax-b)}{∂x} = 2(Ax-b)A$$ ...
46 views

### Doubt regarding the proof of row rank = column rank

Wikipedia provides two methods to prove row rank of a matrix is equal to its column rank. My doubt is regarding the second method. But the wikipedia page mentions that this proof is valid only for ...
38 views

### $A^{-1}XB = I$ Solve for X matrix equation

$A^{-1}XB = I$, $A$ and $B$ are given and they are square matrixes. If I want to solve this matrix equation for $X$, I need to change it to the form like this $X = A×B×I$?
16 views

### Eigenvector of a complete graph Laplacian

Can somebody help me prove why $v=\begin{bmatrix} 1 \dots 1\end{bmatrix}^T$ is the eigenvector of every complete graph Laplacian matrix? Thanks!
20 views

### product of matrices, and its norm

I have a product of matrices $\prod\limits_{i=1}^{n} a_i$, If $b$ is an eigenvalue of $a_i$ for any $i$, then $|b|<1$. (1) Under what norm or condition, $\|\prod\limits_{i=1}^{n} a_i\|<r<1$ ...
71 views

### to reduce an equation into the simplest form using translation and rotation [closed]

Given the equation $3x^2+10xy+3y^2-2x-14y-5=0$, I want to reduce it into the simplest form using translation and rotation of the coordinate axes by $2 \times 2$ orthogonal matrix. How can I proceed ...
20 views

17 views

### Determine the orthogonal complement of x in span [p1, p2, p3]. [duplicate]

Let vectors P1, P2, and P3 be defined as follows: P1 = [1 2 3 4] $^{T}$, P2 = [4 -2 -6 -7]$^{T}$, and P3 = [3 4 -2 1]$^{T}$ Let $x = [1 \, 2\, 3\, 7]^{T}$ This question is compromised into two ...
Is it possible to find an analytical expression for the innverse of the matrix $A$ defined by : $$i\neq j \;:\; A_{ij}=x_ix_j$$ $$A_{ii}=\alpha x_i$$ With $\alpha$ a constant and $x$ a vector. If ...