All Questions
Tagged with matrix-decomposition determinant
47
questions
0
votes
0
answers
41
views
Inverse and Determinant of Matrix $Axx^TA+cA$
Fix $c \in \mathbb{R}$, a symmetric (if needed, positive definite) $n \times n$ real matrix $A$, and $x \in \mathbb{R}^{n \times 1}$. I need help computing the determinant and inverse of the $n \times ...
- 2,497
7
votes
1
answer
240
views
Prove $\det((AB)^{n}-(BA)^{n})$ is a perfect cube.
We have $A,B$ two $3×3$ matrices with integer numbers. We know that $(AB)^{2}+BA=(BA)^2+AB$.
a) Show that $\det((AB)^{n}-(BA)^{n})$ is divisible by $det(AB-BA)$.
b) Show that if $\det(AB-BA)=1$, then $...
- 1,311
7
votes
3
answers
386
views
Show that $\det(AB-BA) × \det(AC-CA) \geq 0$ if $A^2 = -BC$
We have $A,B,C$ three $n×n$ matrices with real entries. We know that
$$
A^2 = -BC
$$
and we want to show that
$$
\det(AB-BA) × \det(AC-CA) \geq 0 \,.
$$
We can easily show that for $n=2k$ we have $...
- 1,311
1
vote
1
answer
50
views
Determinant of product of matrix and nullspace
Assume I have a symmetric, positive-definite matrix $S \in \mathbb{R}^{p \times p}$. Assume that there is some matrix $L \in \mathbb{R}^{n \times p}$ that has full row-rank, i.e., has rank $n$ and ...
- 27
0
votes
2
answers
73
views
Determinant of basis of nullspace of matrix
Assume we have a matrix $A \in \mathbb{R}^{n \times p}$ that has rank $n$ (or perhaps $k$ more generally). We can define a matrix $N \in \mathbb{R}^{p \times (p-n)}$ that is a basis for the nullspace ...
- 27
0
votes
1
answer
86
views
LDU and principal minors of symmetric positive definite matrix
Suppose $A$ is an $n\times n$ symmetric positive definite matrix. We know that $A$'s leading principal minors $m_1,m_2,\dots,m_n$ are positive.
Now suppose that $A$ has LDU decomposition $A=LDU$, and ...
- 421
2
votes
1
answer
113
views
Decompose matrix in blocks with det = 0
I have a linear system which I know $\left[A\right]$ and $\left[B\right]$ and want to find $\left[X\right]$
$$
\left[A\right]_{(n+m) \times(n+m) } \cdot \left[X\right]_{n+m} =
\left[B\right]_{n+m }
$...
- 1,300
0
votes
1
answer
776
views
Determine if LU decomposition is possible on a matrix?
I am trying to understand how you determine if LU decomposition is possible on a given matrix. I believe the way to calculate this is to check if the leading-matrices have non-zero determinants. I ...
- 147
0
votes
0
answers
60
views
Determinant of a matrix with 0 elements above the second diagonal
Good day to all!
I am calculating the determinant of the following $(K + 1) \times (K + 1)$ matrix?
$$
{\bf A} = \left(\begin{matrix}
a_{11} & a_{12} & 0 & \dots & 0 & 0 \\
a_{21} ...
1
vote
0
answers
232
views
Integral of Exponent of L1 Norm of Linear Function
Let $A \in \mathbb{R}^{m \times n}$ be a matrix with $m \geq n$ of full rank. I'm trying to solve for the integral
\begin{equation}
f(\vec{x})=\int_{\mathbb{R}^n} \exp(- \lVert A \vec{x} \rVert_1) \, ...
- 301
34
votes
6
answers
2k
views
Proof that $\text{det}(AB) = \text{det}(A)\text{det}(B)$ without explicit expression for $\text{det}$
Overview
I am seeking an approach to linear algebra along the lines of Down with the determinant! by Sheldon Axler. I am following his textbook Linear Algebra Done Right. In these references the ...
- 1,461
1
vote
0
answers
77
views
A question to matrices over real quaternion division ring
We denote by $\mathbb{H}$ the real quaternion division ring, that is, $\mathbb{H}=\{a+bi+cj+dk\mid a,b,c,d\in\mathbb{R}\}$ where the following multiplication conditions are imposed: $i^2=j^2=k^2=-1,ij=...
- 727
1
vote
0
answers
146
views
While solving a determinant, can't we use an affected row for further operations?
Book's statement: "If more than one operation like $R_{i} \rightarrow R_{i}+kR_{j}$
is done in one step, care should be taken to see that a row
that is affected in one operation should not be ...
- 674
1
vote
1
answer
47
views
Determinant of sandwiched square matrix
Suppose that $A$ is an $m\times m$ matrix, and $S$ a $m\times n$ matrix. Is it possible for the relation
$$ \det S^{\dagger}A S= \det A \det S^{\dagger}S$$ to hold also for $n\neq m$?
For instace, if $...
- 173
1
vote
2
answers
166
views
$\det(I - A) \ge \prod_{i} [1 - \sigma_i(A)]$ if $\|A\|_2\le 1$
I would like to prove
$$
\det(I - A) \ge \prod_{i=1}^n[1-\sigma_i(A)]
$$
for any $A\in\mathbb{R}^{n\times n}$ such that $\|A\|_2 \le 1$. Here, as usual, $\sigma_i(\cdot)$ denotes the $i$-th sigular ...
- 668