Articles of matrices

How to construct magic squares of even order

Could someone kindly point me to references on constructing magic squares of even order? Does a compact formula/algorithm exist?

rank of block triangular matrix and its relation to the rank of its diagonal blocks

Prove that $$rank\begin{pmatrix}A & X \\ 0 & B \\ \end{pmatrix}\ge rank(A) + rank(B)$$ where $$A,B\in \mathbb C^{m \times m}$$. I know the intuition behind it (i.e. maximal independent rows, etc.), but I am looking for a formal proof. I have tried QR decomposition of A and B, then broke the block triangular matrix up […]

Proving that the algebraic multiplicity of an eigenvalue $\lambda$ is equal to $\dim(\text{null}(T-\lambda I)^{\dim V})$

Without using induction, prove that the the algebraic multiplicity of an eigenvalue $\lambda$ is $$\dim (\text{null} (T-\lambda I)^{\dim V});$$ here, the algebraic multiplicity of an eigenvalue $\lambda$ refers to the number of times it appears on the diagonal of an upper triangular matrix. Attempt: Please note that the book this is taken from, Linear Algebra […]

Partial Derivative v/s Total Derivative

I am bit confused regarding the geometrical/logical meaning of partial and total derivative. I have given my confusion with examples as follows Question Suppose we have a function $f(x,y)$ , then how do we write the limit method of representing $ \frac{\partial f(x,y)}{\partial x} \text{and} \frac{\mathrm{d} f(x,y)}{\mathrm{d} x}$ at (a,b)? What is the difference? Imagine […]

Derivative of a Matrix with respect to a vector

I know that for two k-vectors, say $A$ and $B$, $\partial A/\partial B$ would be a square $k \times k$ matrix whose $(i,j)$-th element would be $\partial A_i/\partial B_j$. But could someone please explain how the partial derivative look like if we were differentiating $k \times k$ matrix instead? That is, $M$ is a $k […]

How adjacency matrix shows that the graph have no cycles?

Let $G$ a directed graph and $A$ the corresponding adjacency matrix. Let denote the identity matrix with $I$. I’ve read in a wikipedia article, that the following statement is true. Question. Is it true, that $I-A$ matrix is invertible if and only if there is no directed cycle in $G$?

Second derivative of $\det\sqrt{F^TF}$ with respect to $F$

I need to evaluate $$ \frac{\partial^2W}{\partial F^2}(I)(F,F)=\sum_{i,j,k,l=1}^3\frac{\partial^2W}{\partial F_{ij}F_{kl}}(I)F_{ij}F_{kl} $$ for $W=\det\sqrt{F^TF}$. Here $F\in\mathbb{R}^{3\times3}$. From First derivative we know that $$ \frac{\partial W}{\partial F}=WF^{-T}. $$ To evaluate the sum above, I write $$ \frac{\partial^2W}{\partial F^2}=W\left[\left(F^{-T}\right)^2+\frac{\partial F^{-T}}{\partial F}\right]․ $$ Here I got stack.

An annoying Pell-like equation related to a binary quadratic form problem

Let $A,B,C,D$ be integers such that $AD-BC= 1 $ and $ A+D = -1 $. Show by elementary means that the Diophantine equation $$\bigl[2Bx + (D-A) y\bigr] ^ 2 + 3y^2 = 4|B|$$ has an integer solution (that is, a solution $(x,y)\in\mathbb Z^2$). If possible, find an explicit solution (involving $A,B,C,D$, of course). Motivation: I […]

$A \in {M_n}$ is normal.why the range of $A$ and ${A^*}$ are the same?.

Let $A\in {M_n}$ be normal. Why the range of $A$ and ${A^*}$ are the same?

Decompose rotation matrix to plane of rotation and angle

I would like to decompose an $n$-dimensional orthogonal rotation matrix (restricting to simple rotation with a single plane of rotation) to the two basis vectors of the plane of rotation, and an angle of rotation. The common method is decomposing the rotation matrix to an axis and angle, but this doesn’t work in higher dimensions. […]