Questions tagged [matrix-decomposition]
Questions about matrix decompositions, such as the LU, Cholesky, SVD (Singular value decomposition) and eigenvalue-eigenvector decomposition.
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Calculating SVD by hand: resolving sign ambiguities in the range vectors.
When calculating the SVD of the matrix
$$A = \begin{bmatrix}3&1&1\\-1&3&1\end{bmatrix}$$
I followed these steps
$$A A^{T} = \begin{bmatrix}3&1&1\\-1&3&1\end{bmatrix} ...
- 1,101
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4
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How unique are $U$ and $V$ in the singular value decomposition $A=UDV^\dagger$?
According to Wikipedia:
A common convention is to list the singular values in descending order. In this case, the diagonal matrix $\Sigma$ is uniquely determined by $M$ (though the matrices $U$ and ...
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Computing the Smith Normal Form
Let $A_R$ be the finitely generated abelian group, determined by the relation-matrix
$$R := \begin{bmatrix}
-6 & 111 & -36 & 6\\
5 & -672 & 210 & 74\\
0 & -255 &...
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If $\mathrm{Tr}(A)=0$ then $T=R^{-1}AR$ has all entries on its main diagonal equal to $0$
Prove that if $A$ is a square matrix and $\mathrm{Tr}(A)=0$, then there exists an invertible matrix $R$ such that the matrix $T=R^{-1}AR$ has all entries on its main diagonal equal to $0$.
It seems ...
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Polar decomposition of real matrices
Can we decompose real matrices in a way that is analogous to polar decomposition in the complex case?
What I mean is: given an invertible real matrix $M$, can we always write:
$$
M = OP,
$$
maybe ...
- 8,127
12
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A normal matrix with real eigenvalues is Hermitian
$A$ is a normal matrix (i.e. $AA^*=A^*A$, where * denotes the hermitian conjugate). If all its eigenvalues are real, prove that it is Hermitian (i.e. $A^*=A$).
I have tried many things but could not ...
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Problem with Singular Value Decomposition
I have a very trivial SVD Example, but I'm not sure what's going wrong.
The typical way to get an SVD for a matrix $A = UDV^T$ is to compute the eigenvectors of $A^TA$ and $AA^T$. The eigenvectors of ...
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5
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Find the inverse of a submatrix of a given matrix
I have a $A$ matrix $4 \times 4$. By delete the first row and first column of $A$, we have a matrix $B$ with sizes $3 \times 3$. Assume that I have the result of invertible A that denote $A^{-1}$ ...
- 802
14
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Why is the non-negative matrix factorization problem non-convex?
Supposing $\mathbf{X}\in\mathbb{R}_+^{m\times n}$, $\mathbf{Y}\in\mathbb{R}_+^{m\times r}$, $\mathbf{W}\in\mathbb{R}_+^{r\times n}$, the non-negative matrix factorization problem is defined as:
$$\...
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Find the eigenvalues of a matrix with ones in the diagonal, and all the other elements equal [duplicate]
Let $A$ be a real $n\times n$ matrix, with ones in the diagonal, and all of the other elements equal to $r$ with $0<r<1$.
How can I prove that the eigenvalues of $A$ are $1+(n-1)r$ and $1-r$, ...
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Intuitively, what is the difference between Eigendecomposition and Singular Value Decomposition?
I'm trying to intuitively understand the difference between SVD and eigendecomposition.
From my understanding, eigendecomposition seeks to describe a linear transformation as a sequence of three basic ...
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Prove that the polar decomposition of normal matrices, $A=SU$, is such that $SU=US$
Assume $A$ is a normal matrix. Suppose $A=SU$ is a polar decomposition of $A$. Prove that $SU=US$.
I have no idea to prove this.
$A$ is normal then $AA^*=A^*A$. And then we have
$$
SS^*=U^*S^*SU.
$$
...
- 1,589
8
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3
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How to calculate the cost of Cholesky decomposition?
The cost of Cholesky decomposition is $n^3/3$ flops (A is a $n \times n$ matrix). Could anyone show me some steps to get this number? Thank you very much.
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How can you explain the Singular Value Decomposition to non-specialists?
In two days, I am giving a presentation about a search engine I have been making the past summer. My research involved the use of singular value decompositions, i.e., $A = U \Sigma V^T$. I took a high ...
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When does a Square Matrix have an LU Decomposition?
When can we split a square matrix (rows = columns) into it’s LU decomposition? The LUP (LU Decomposition with pivoting) always exists; however, a true LU decomposition does not always exist. How do ...
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