Questions tagged [matrix-decomposition]
Questions about matrix decompositions, such as the LU, Cholesky, SVD (Singular value decomposition) and eigenvalue-eigenvector decomposition.
2,713
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Deduce $\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ \end{bmatrix}$ or the simple parity-check matrix just from the final matrix-vector product equations
A linear combination like $A_{1}+ A_{2}$ serves as a backup of $A_{1}$ when $A_{2}$ is known, and serves as a backup of $A_{2}$ when $A_{1}$ is known. As a result of linearity, any two out of $A_{1}x$,...
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How to compute $RA$ in $O(n^2)$ operations instead of $O(n^3)$ using Householder reflection
I am currently writing a program that performs QR decomposition on a matrix $A$.
The guidelines to my assignment tell me that once I calculate $R$ using Householder reflections, there is a way to ...
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Projection of vector
The projection of a vector $x$
onto a vector $u$ is
$proj_u(x) =\frac{\langle x, u \rangle}{\langle u, u \rangle}u.$
Projection onto $u$
is given by matrix multiplication
$proj_u(x)=Px$ where $P=\frac{...
user1290851
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Symplectic approximation to a given matrix
I am interested in understanding methods to rigorously compute the closest symplectic matrix approximation to an arbitrary matrix $A$. A symplectic matrix $S$ satisfies the condition $ S^T J S = J $, ...
- 61
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Interpretation of QR "values" (a la singular values)?
Let $A\in\mathbb{R}^{m\times n}$ with $m>n$ and $\text{rank}A=n$.
There exists $\hat{Q}\in\mathbb{R}^{m\times n}$ with orthonormal columns and $\hat{R}\in\mathbb{R}^{n\times n}$ upper triangular ...
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1
answer
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Equation for Counting Unique RREF "Cases" for any (m x n) Matrix
Given a real matrix $A$ of size $(m \times n)$, I am seeking to determine the number of unique reduced row echelon form (RREF) "Cases" that exist for a given matrix size. Two RREF matrices ...
- 23
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How to derive the relation about Jordan decomposition of a matrix?
Assume that $v$ is an eigenvector with an eigenvalue of $0$ in matrix $H$, and its Jordan decomposition $H^J=SHS^{-1}$ satisfies
$$
H^J=\left( \begin{matrix}
0& 1& \cdots& 0\\
0& ...
- 194
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Distance between subspaces with spectral norm
I was trying to prove this following theorem ,
Let
$$ W=\begin{bmatrix}
\underset{\scriptscriptstyle n\times k} {W_1} && \underset{\scriptscriptstyle n\times (n-k)} {W_2}
\end{bmatrix} $$
$...
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Norm inequality for eigendecomposition and oblique projectors
Consider a stochastic matrix R which permits an eigendecomposition into oblique projectors,
$$R = \sum_{\lambda} \lambda C_{\lambda}$$
I've observed the following 2-norm inequality in a number of $R$ ...
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Inverting a specific symmetric matrix preserves its zero entries
Suppose $S$ is an invertible symmetric matrix with the following property:
For the entry in the $i$th row and $j$th column, if $|i-j|$ is an odd number then $S_{ij} = 0$; if $|i-j|$ is an even number ...
- 225
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Is polar decomposition commutative for diagonal matrices?
I did a error while understanding about the polar decompositon.
I thought polar decomposition is PU, but it is UP. While trying ...
- 111
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Finding the eigenvalues of a tridiagonal block matrix of special form
Consider the following symmetric tridiagonal block matrix: $$\begin{bmatrix}
2I_{N \times N} & -I_{N \times N} & O & \dots & O &O \\
-I_{N \times N} & 2I_{N \times N} &...
- 292
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In what sense are similar matrices "the same," and how can this be generalized?
I sort of intuitively see why we care about similar matrices, i.e., when $A=S^{-1}BS$ for some invertible matrix $S$. But I want to make this intuition more precise and abstract.
Matrices: First of ...
- 6,672
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Inverse $T$ matrix of a 3*3 matrix. [closed]
I have the matrix
$$ A=
\begin{bmatrix}
1 & 4 & 1\\
0 & 2 & 5\\
0 & 0 & 5
\end{bmatrix}.
$$
I have found the
$$ T=
\begin{bmatrix}
1 & 4 & 1\\
0 & 0 & 1\\
0 &...
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Orthogonal procrustes with kernel constraint
Given a matrix $M \in \mathbb{R}^{n \times m}$ with $n > m$ and an arbitrary vector $v \in \mathbb{R}^n$, I am looking for an analytical solution for the orthogonal matrix $R \in \mathbb{R}^{n \...
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