algorithms – Given $n$ sets of matrices, find $n$ matrices that have the least number of LCDs among their entries

Let’s say I have $n$ sets of matrices

$$
A = left{begin{pmatrix}
2 & 4 & 17\
5 & 6 & 9\
end{pmatrix}
begin{pmatrix}
2 & 4 & 18\
5 & 6 & 9\
end{pmatrix}
right}
$$

$$
B = left{begin{pmatrix}
13 & 20\
3 & 16\
end{pmatrix}
begin{pmatrix}
14 & 20\
3 & 16\
end{pmatrix}
begin{pmatrix}
13 & 21\
3 & 16\
end{pmatrix}
begin{pmatrix}
14 & 21\
3 & 16\
end{pmatrix}
begin{pmatrix}
13 & 20\
3 & 17\
end{pmatrix}
begin{pmatrix}
14 & 20\
3 & 17\
end{pmatrix}

right}
$$

Let’s define $T$ as a vector that contains the lowest common denominators shared among the entries of all the $n$ matrices.

I need to find $n$ matrices, picking one from each set, that will minimize the length of $T$, i.e. in this case

$$
A_{1}=begin{pmatrix}
2 & 4 & 18\
5 & 6 & 9\
end{pmatrix}
$$

$$
B_{1}=begin{pmatrix}
14 & 20\
3 & 16\
end{pmatrix}
$$

The resulting $T$ would be

$$
T=begin{bmatrix}
2,3,4,5,9
end{bmatrix}
$$

I know I can bruteforce every possible combination, but is there a more efficient way?

matrices – Reworking matrix to have last row as identity

Given a 3*3 matrix, what is the name and working of the procedure to modify the final row to match an identity matrix?

i.e.
with [ [ a, b, c ], [ d, e, f ], [ h, i, j ] ]

what transform can be performed to get [ [ k, l, m ], [ n, o, p ], [ 0, 0, 1 ] ]

such that when it is applied to a point vector [ x, y, 1 ] I get the same output point.

For context, I have a matrix formed from various operations and wish to use it in a SVG matrix operation which only takes the values of the first two rows, expecting the third to be [ 0, 0, 1 ]

linear algebra – set of invertible diagonal matrices

Let $mathcal{T}$ be the set of invertible diagonal matrices. Show that for any invertible matrix $B$ such that $Bmathcal{T}B^{-1} = mathcal{T}, B=PT$ for some permutation matrix $P$ and invertible diagonal matrix $T.$ Here, $AB := {ab: ain A, bin B}$ when $A$ and $B$ are sets.

I’m not sure how to show this result, though I’m pretty sure I need to consider eigenvalues, diagonalizable matrices, and change of basis matrices. Using the definition alone, I get stuck quite easily; I only know that for any invertible diagonal matrix $T, BTB^{-1}$ is an invertible diagonal matrix and for any invertible diagonal matrix $T’, T’ = BT”B^{-1}$ for some invertible diagonal matrix $T”.$ I know how to show that if an $ntimes n$ matrix has $n$ distinct eigenvalues, then it has $n$ distinct eigenvectors (an eigenvector can correspond to only one eigenvalue) and thus these $n$ eigenvectors are linearly independent, so the matrix is diagonalizable, but I’m not sure if this is useful.

functions – Can we determine higher powers of a matrix in terms of lower powered matrices?

Consider a n-ordered square matrix A. Using Cayley-Hamilton Theorem, I can represent the matrix $A^n$ as a matrix polynomial P(A) of degree n-1.

Further any matrix $A^k$ where $k>n$ can also be represented as follows:

$$A^k= a_{k,n-1} A^{n-1} + a_{k,n-2} A^{n-2}+a_{k,n-3} A^{n-3} … + a_{k,2} A^{2}+a_{k,1} A^{1} + a_{k,0}I$$

What I want to know that is there any way to determine the coefficients as a function of k?

matrices – Matrix derivative w.r.t. a general inverse form: $(A^TA)^{-1/2}D(A^TA)^{-1/2}$

I want to find derivative of matrix $(A^TA)^{-1/2}D(A^TA)^{-1/2}$ w.r.t. $A_{ij}$ where D is a diagonal matrix. Alternatively, it is okay too to have

$$frac{partial}{partial A_{ij}} a^T(A^TA)^{-1/2}D(A^TA)^{-1/2}b$$

Is there any reference for such problem? I have the matrix cookbook which gives results when $D=I$. But how is this general form evaluating to?

To give more information, empirical distribution of diagonal of diagonal matrix D converges to some known distribution.

reference request – rank of a linear combination of matrices

Let $A_1,…, A_s in M_n(mathbb{R})$ be symmetric matrices and suppose they are linearly independent over $mathbb{R}$. This means that
$$
m = min_{(c_1, …, c_s) in mathbb{R}^s backslash {0}} rank( sum_{i=1}^s c_i A_i ) > 0
$$

I am interested in the question how large can $m$ be?
I am not sure where to start looking… if someone could point me to a good reference it’s very appreciated! Also any comments are appreciated!

matrices – Derivative of a Matrix w.r.t. its Matrix Square, $frac{partial text{vec}X}{partialtext{vec}(XX’)}$

Let $X$ be a nonsingular square matrix.

What is
$$
frac{partial text{vec}X}{partialtext{vec}(XX’)},
$$

where the vec operator stacks all columns of a matrix in a single column vector?

It is easy to derive that
$$
frac{partialtext{vec}(XX’)}{partial text{vec}X} = (I + K)(X otimes I),
$$

where $K$ is the commutation matrix that is defined by
$$
text{vec}(X) = Ktext{vec}(X’).
$$

Now $(I + K)(X otimes I)$ is a singular matrix, so that the intuitive solution
$$
frac{partial text{vec}X}{partialtext{vec}(XX’)} = left( frac{partialtext{vec}(XX’)}{partial text{vec}X} right)^{-1}
$$

does not work.

Is the solution simply the Moore-Penrose inverse of $(I + K)(X otimes I)$, or is it more complicated?

matrices – Matrixes whose elements are matrixes

I’ve worked with matrixes whose elements are objects in a field, such that real numbers, complex numbers, inclusive functions in space of functions, but… Today I was talking to a friend and he asked me about something he saw in his PhD in informatic science that was about “matrixes with matrixes in their entries” and I know that we can make an arrange of the blocks of the matrixes in the entries to form a matrix in a nxn space, for some n… but… what use or there is any example of how is useful a matrix with this characteristic?