There’s a Matrix X wholes several columns were chosen as basis to construct new columns by linear combination with positive coefficients (weighted mean). These new columns had been joined to X to form Y.

How would you compute X using Y?

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# Tag: matrix

## Identification of mix signatures/columns in a matrix

## pr.probability – Expectation of the determinant of the inverse of non-central Wishart matrix

## Of the numbers 2,3 and 5 , which are eigenvalues of the matrix (353,173,128)

## performance tuning – Mean values of skew diagonals of a $(n+1,n)$ matrix

## linear algebra – A matrix Riccati differential equation with constant coefficients? Is there a solution for this in closed form?

## linear algebra – Problem with a rank of symmetric matrix

## matrix – Logarithm of singular matrices

## patterns and practices – How does an Application Security Risk Matrix look like?

## What I found

## Minimizing the Frobenius norm of a quadratic matrix expression

## matrices – Matrix rank only adding single row

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There’s a Matrix X wholes several columns were chosen as basis to construct new columns by linear combination with positive coefficients (weighted mean). These new columns had been joined to X to form Y.

How would you compute X using Y?

The joint distribution of the eigenvalues $lambda_i$, $i=1,2,ldots n$ of $A$ is known,

$$P(lambda_1,lambda_2,ldotslambda_n)=c_{k,n}prod_{i<j}|lambda_i-lambda_j|prod_m e^{-lambda_m/2}lambda_m^{(k-n-1)/2},$$

with $c_{k,n}$ a normalization constant.

The desired expectation value is given by

$$U_{n,k}=mathbb E(det(I+(I+A)^{-1}))=int_0^infty dlambda_1int_0^infty dlambda_2cdots int_0^infty dlambda_n,prod_{i<j}|lambda_i-lambda_j|prod_m e^{-lambda_m/2}lambda_m^{(k-n-1)/2}left(1+(1+lambda_m)^{-1}right).$$

For small $n$ the integrals can be done by quadrature, but the integrals quickly become unwieldy.

For example, for $n=1$, $k>0$ I find

$$U_{1,k}=2^{-frac{k}{2}} left(2^{k/2}+sqrt{e} ,Gamma left(1-tfrac{1}{2}k,tfrac{1}{2}right)right),$$

with $Gamma$ the incomplete Gamma function.

If you are satisfied with the expectation value of the *logarithm* of the determinant, then you can use the Marchenko-Pastur distribution to obtain an accurate result for large $n$.

Of the numbers 2,3 and 5, which are eigenvalues of the matrix

( 3 5 3 )

( 1 7 3 )

( 1 2 8 )

How mean values of skew diagonals of a $(n+1,n)$ matrix can be computed efficiently?

Here is my naive implementation:

```
ClearAll(build) ;
build(matrix_) := Block(
{col,row,signal},
{col,row} = Dimensions(matrix) ;
signal = ConstantArray(0,2*row) ;
Do(
signal((i)) = Table(If(q+p==i+1,matrix((q,p)),Nothing),{q,1,col},{p,1,row}) ;
signal((i)) = Mean(Flatten(signal((i)))) ;
,{i,1,2*row,1}
) ;
signal
)
```

Looks like it’s time complexity is $O(n^3)$, can it be reduced?

Example:

```
n = 4 ;
ncols = n + 1 ;
nrows = n ;
matrix = Array(m,{ncols,nrows}) ;
matrix
build(matrix)
(* {{m(1,1),m(1,2),m(1,3),m(1,4)},{m(2,1),m(2,2),m(2,3),m(2,4)},{m(3,1),m(3,2),m(3,3),m(3,4)},{m(4,1),m(4,2),m(4,3),m(4,4)},{m(5,1),m(5,2),m(5,3),m(5,4)}} *)
(* {m(1,1),1/2 (m(1,2)+m(2,1)),1/3 (m(1,3)+m(2,2)+m(3,1)),1/4 (m(1,4)+m(2,3)+m(3,2)+m(4,1)),1/4 (m(2,4)+m(3,3)+m(4,2)+m(5,1)),1/3 (m(3,4)+m(4,3)+m(5,2)),1/2 (m(4,4)+m(5,3)),m(5,4)} *)
n = 4 ;
ncols = n + 1 ;
nrows = n ;
data = Range(1,2*n) ;
data = Partition(data,n,1) ;
data
build(data)
(* {{1,2,3,4},{2,3,4,5},{3,4,5,6},{4,5,6,7},{5,6,7,8}} *)
(* {1,2,3,4,5,6,7,8} *)
```

The following is a matrix Riccati differential equation with constant coefficient matrices.

$$Dfrac{partial{C(t)}}{partial{t}}S + frac{1}{n}C(t)QDC(t)S – EC(t)Q = 0$$ or

$$Ddot{C}(t)S + frac{1}{n}C(t)QDC(t)S – EC(t)Q = 0$$

given initial condition $C(0) = C_0$.

I stumbled upon this from some other problem and I don’t have any background in matrix differential equations and I’d like to know if there is any way to solve this equation. I read it can be reduced to an algebraic Riccati equation. Is there any closed form expression for solution of this equation? Or anything that is closest to solving this equation?

**Matrix dimensions**

$C(t)$———-> $(m+1)times n$

$S$————–>$ntimes 1$

$Q$————–>$ntimes(m+1)$

$D$————–>$(m+1)times(m+1)$ diagonal matrix. (it is also singular, as there is a diagonal entry that is 0).

$E$————–>$1times (m+1)$

If its useful to know, $n>>m$ and $mge 3$

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I wanted to calculate following quantity:

$X=Tr(rho_1 log(rho_2))$,

as in the relative entropy. Here, $rho_1, rho_2$ are positive semidefinite matrices with non-orthogonal support (so that the thing does not diverge – terms like $0 log(0)$ should be taken to zero, which is standard assumption) and $Tr$ is trace.

There is a similar SE question, the given function

```
MatrixLogSafe(x_) := MatrixFunction(Piecewise({{Log(#1), #1 > 0}}) &, x)
```

which should deal with the matrix logarithm, it behaves, however, strange.

For e.x. I assume that $rho_1 =rho_2 ={{0.33,0,0},{0,0,0},{0,0,0.66}}$. The quantity $X$ should then be

$X=0.33 log(0.33) + 0.66 log(0.66)= -0.640099$.

However, using MatriLogSafe in the definition gives different output:

```
In(402):= Tr({{0.33, 0, 0}, {0, 0, 0}, {0, 0, 0.66}}.MatrixLogSafe({{0.33, 0, 0}, {0, 0, 0}, {0, 0, 0.66}}))
Out(402)= -0.731717
```

The problem is, that MatrixLogSafe sometimes “switch the eigenvectors”,

```
In(403):= MatrixLogSafe({{0.33, 0, 0}, {0, 0, 0}, {0, 0, 0.66}})
Out(403)= {{0., 0., 0.}, {0., -0.415515, 0.}, {0., 0., -1.10866}}
```

(so $log(0.33)= -1.10866$ and $log(0.66)=-0.415515$, but the output should be { { -1.10866, 0, 0.}, {0., 0., 0.}, {0., 0.,-0.415515}}).

(Somehow I think the problem is that the I use numerical values, but I want that the function works for both numerical and “exact” (?) numbers)

How one can fix it?

I am looking into “Secure Development Lifecycle” (SDLC) and I found a resource that say one should create a “Application Security Risk Matrix”. I guess that matrix has one row for each thread (e.g. SQL Injection), and the columns might be something like severity / likelihood / mitigation / comments. But I don’t really know. And I also would appreciate more examples of threads.

Is there a template / a reputable example for an Application Security Risk Matrix? Is it the same as a “Thread Matrix” (example)?

INFORMATION SECURITY RISK ANALYSIS – A MATRIX-BASED APPROACH: A lot of different matrices which seem to weight already found issues.

ICT risk matrix / Managing Cybersecurity Risks Using a Risk Matrix: More like a chart. I found a lot of tose via image search, but most of them were in private blogs.

$$min_{X in mathbb R^{p times m}} | X R R^T X^T -Y Y^T |_F$$

where matrix $R$ is full rank. I am guessing the solution could be $X=YR^+$, but I cannot prove it.

I have a dense matrix and a set of rows. I would like to check if adding any single row from the set to the original matrix would make the new matrix rank deficient. Right now I am doing a full LU decomposition each time. This feels wasteful, and I have a hunch that I should be able to keep some information between iterations. Does anyone know of a way to speed this up?

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