All Questions
Tagged with matrix-decomposition linear-algebra
1,644
questions
0
votes
0
answers
30
views
How to estimate the inverse of a non-invertible matrix?
So I'm working on a machine learning problem where my solution requires taking the inverse of a matrix at some point. The problem is that this matrix is sometimes non-invertible. In theory the the ...
- 11
0
votes
2
answers
51
views
For any SVD $A = U\Sigma V^T$ of a positive definite, symmetric matrix $A \in \mathbb{R}^{n \times n}$, we have $U = V$.
First of all, I've read through all of the answers here and here, but neither of those threads was able to give completely satisfying answers.
Now, I understand that, if $A$ is symmetric and positive ...
- 23
2
votes
0
answers
34
views
need a Jacobian of an orthonormal basis for the column space of a matrix $\mathbf{X}$
I require to find a linear decomposition of any tall matrix $\mathbf{X}$ $\in \mathbb{R}^{M \times N}$, where $M \geq N$, into some uniquely defined decomposition $\mathbf{X}$ $=$ $\mathbf{A}$ $\...
- 41
0
votes
1
answer
32
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Exponeintal of symmetric triangular matrix
I want to know the exponeintal of given $n \times n$ symmetirc real tridiagonal matrix ${\bf K}_n$, which is defined as
$${\bf K}_n=\begin{bmatrix}
0 & a & 0 & 0 & \dots & 0 & ...
- 530
0
votes
1
answer
52
views
Showing existence of symplectic transformations preserving a quadratic form
Question: I need help to prove the following statement.
Let $W_i:=w_iw_i^T\in\mathbb{R}^{n\times n}$, for $n$ even, be symmetric rank-1 matrices, $J=-J^T$ the canonical symplectic matrix and define ...
- 108
0
votes
0
answers
28
views
Woodbury matrix identity with a minus sign
Is there a form of Woodbury matrix identity
$(A + UCV )^{-1} = A^{-1} - A^{-1}U (C^{-1} + VA^{-1}U )^{-1} VA^{-1}$
But with a minus sign? i.e. $(A - UCV )^{-1}$ It seems like I have to painfully ...
- 1,634
0
votes
0
answers
27
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Absolute value of elements of b=Ax and the minimum singular value of A
For $b=Ax$, is there a way to relate the minimum absolute value of the element of $b$, $\min|b_i|$, and the minimum singular value, $\sigma_{min}$, of $A$?
What I want is something like: $\sigma_{min}$...
- 115
0
votes
3
answers
81
views
Decompose a $3 \times 3$ orthogonal matrix into a product of rotation and reflection matrices
It holds that every orthogonal matrix can be expressed as a product of rotation and reflection matrices.
We can prove that every $2 \times 2$ orthogonal matrix can be represented in the form
$$\begin{...
- 137
20
votes
3
answers
3k
views
Why is there not a test for diagonalizability of a matrix
Let $A$ be square.This question is a bit opinion based, unless there is a technical answer.I think it is helpful tho. Also this question is closely related to this question : quick way to check if a ...
- 477
1
vote
1
answer
30
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Any decomposition of inverse of nonnegative diagonal matrix times a PSD matrix plus lambda times Identity?
I generally have to solve the following system:
$$
(DA + \lambda I)^{-1} v
$$
where $D$ is a diagonal matrix with nonnegative entries, $A$ a symmetric, positive semi-definite (PSD) matrix, $I$ is the ...
- 133
0
votes
1
answer
27
views
Is it true that $D A P D^T = A D P D^T$ if $P$ is symmetrical, positive definite and $D$ is diagonal?
I know in general, matrix multiplication is not commutative, but would it be true in this special case?
$D A P D^T = A D P D^T$ where $A, D, P$ are all $n by n$ matrix. But $P$ is symmetrical and ...
- 149
2
votes
1
answer
44
views
Singular value of a bidiagonal matrix?
Consider a $n\times n$ matrix:
\begin{equation}
X=\begin{bmatrix}
a &1-a & & \\
& a &1-a & \\
& & \ddots &\\
& & & 1-a\\
& & & a
\end{...
- 304
-2
votes
1
answer
81
views
What is the square root of a square matrix squared? [closed]
Admittedly, made the title a little funny, but this is a valid question.
I have come across the following equation
$$
I x^2=AA
$$
where $I$ is a unit matrix, $A$ is a square matrix of the same ...
- 101
0
votes
0
answers
52
views
Suitability of QR factorization for solving a ill-conditioned linear system.
I am trying to solve a linear system $Ax = b$ where
$$
A =\begin{bmatrix}
2& 9& 2& 1& 4& 1& 0& 0& 0 \\
9& 65& 9& 1& 4& 1& 0& 0&...
- 1,261
0
votes
0
answers
25
views
Singular values on streching the vectors
For the following statement: for a vector $x$ and a matrix $A$, if the vector $x$ is not in the null space of $A$, the vector $x$ will at least be stretched by the smallest non-zero singular value, i....
- 115