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Questions tagged [matrix-decomposition]

Questions about matrix decompositions, such as the LU, Cholesky, SVD (Singular value decomposition) and eigenvalue-eigenvector decomposition.

22
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1answer
13k views

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 &...
3
votes
3answers
1k views

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 $1-r$, ...
8
votes
3answers
5k views

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} ...
2
votes
2answers
305 views

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 ...
9
votes
1answer
3k views

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 ...
9
votes
2answers
5k views

Matrix exponential of a skew symmetric matrix

I have a skew symmetric matrix $$C=\left( \begin{array}{ccc} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0 \\ \end{array} \right).$$. Question 1) How do I compute $e^{...
8
votes
2answers
913 views

Inverse of the sum of a invertible matrix with known Cholesky-decomposion and diagonal matrix

I want to ask a question about invertible matrix. Suppose there is a $n\times n$ symmetric and invertible matrix $M$, and we know its Cholesky decomposion as $M=LL'$. Then do we have an efficient way ...
6
votes
1answer
4k views

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 ...
2
votes
3answers
300 views

diagonalisability of matrix few properties

What are the most important things one should remember to check the diagonalisability of a matrix? Please help, I have exams on next week. Say some best and easy methods,time efficient.
10
votes
1answer
3k views

Every $n\times n$ matrix is the sum of a diagonalizable matrix and a nilpotent matrix.

I would like to prove that every $n\times n$ matrix is the sum of a diagonalizable matrix and a nilpotent matrix. How is this possible? I'm not sure where to begin really- I know that a nilpotent ...
4
votes
2answers
2k views

Comparing LU or QR decompositions for solving least squares

Let $X \in R^{m\times n}$ with $m>n$. We aim to solve $y=X\beta$ where $\hat\beta$ is the least square estimator. The least squares solution for $\hat\beta = (X^TX)^{-1}X^Ty$ can be obtained ...
2
votes
1answer
151 views

How to apply SVD to real data to reduce the number of parameters?

I have a question about applying the Singular Value Decomposition (SVD) to real data. Say I have the equation $$ y= Ax+v$$ where $A \in \mathbb{R}^{m \times n}$, $y \in \mathbb{R}^m$, $x \in \mathbb{...
4
votes
2answers
3k views

How do I find upper triangular form of a given 3 by 3 matrix??

We are asked to find an invertible matrix $P$ and an upper triangular matrix $U$ such that: $P^{-1}\begin{pmatrix} 3 & -1 & 1 \\ 2 & 0 & 0 \\ -1 & 1 & 3 \end{pmatrix}P=U$ I'm ...
2
votes
1answer
130 views

Overlapping positive definite block matrices

Let $A \in \mathbb{R}^{p \times p}$ and $B \in \mathbb{R}^{q \times q}$ be nonsymmetric positive definite matrices in the sense that $v^T A v > 0, \forall v \in \mathbb{R}^p$ and $v^T B v > 0, \...
14
votes
3answers
9k views

null space from SVD

It is said that a matrix's null space can be derived from QR or SVD. I tried an example: $$A= \begin{bmatrix} 1&3\\ 1&2\\ 1&-1\\ 2&1\\ \end{bmatrix} $$ I'm convinced that QR (more ...
4
votes
2answers
2k views

Cholesky factor when adding a row and column to already factorized matrix

I have a positive deifnite, symmetrical, $N\times N$ real matrix $A$ which has 1's on the diagonal and all off-diagonal elements positive and $<1$. Let $A=LL^t$ be the Cholesky decomposition of $A$....
10
votes
1answer
807 views

Singular values of $AB$ and $BA$ where $A$ and $B$ are rectangular matrices

This question is a generalisation of Eigenvalues of $AB$ and $BA$ where $A$ and $B$ are rectangular matrices which itself is a generalisation of Eigenvalues of $AB$ and $BA$ where $A$ and $B$ are ...
6
votes
1answer
1k views

$QR$ decomposition of rectangular block matrix

So I am running an iterative algorithm. I have matrix $W$ of dimensions $n\times p$ which is fixed for every iteration and matrix $\sqrt{3\rho} \boldsymbol{I}$ of dimension $p\times p$ where the $\rho$...
5
votes
1answer
296 views

General Cholesky-like decomposition

For a real positive definite matrix $A$, we can find the Cholesky decomposition $LL^T = A$. If we relax the constraints that L has to be lower triangular, we should be able to find a whole host of ...
6
votes
1answer
463 views

Expressing a quadratic form, $\mathbf{x}^TA\mathbf{x}$ in terms of $\lVert\mathbf{x}\rVert^2$, $A$

EDIT: This question is actually an attempt to solve this. Please take a look. Let $A$ be a symmetric postive-definite $n\times n$ matrix, i.e. $A\in\mathbb{S}_{++}^{n}$ Also, let $\mathbf{x}\in\...
3
votes
2answers
220 views

Why isn't the SVD simplified to USV (no transpose)?

I've looked at a lot of explanations of the SVD, and they all phrase it as $A = USV^T$ (or $V^*$ for complex matrices), and where V is unitary. But the conjugate of a unitary matrix is always unitary, ...
2
votes
3answers
2k views

How to prove whether a matrix has rank $1$

If $u$ $∈ \mathbb R^{m \times 1}$ and $v ∈ \mathbb R^{n \times 1}$ how do you show that the $(m \times n)$ matrix $uv^T$ has rank $1$? Would providing an example be sufficient to prove it?
2
votes
4answers
203 views

Finding the $LU$ factorization of the matrix

Find the $LU$ factorization of the matrix: $$\begin{bmatrix} 1 & 1 & 1 \\ 3 & 5 & 6 \\ -2 & 2 & 7 \end{bmatrix}$$ I am aware that I need to find $L=\begin{bmatrix} 1 & 0 ...
2
votes
3answers
4k views

New proof about normal matrix is diagonalizable.

I try to prove normal matrix is diagonalizable. I found that $A^*A$ is hermitian matrix. I know that hermitian matrix is diagonalizable. I can not go more. I want to prove statement use only this ...
8
votes
1answer
156 views

Solving the matrix equation $X^tA+A^tX=0$ for $X$ in terms of $A$

Suppose that I know $A$. And all matrices in the equation are square matrices. I want to solve for $X$ given that $$X^tA + A^tX = 0$$ I'm not really good at matrix calculus. Is it possible to solve ...
4
votes
1answer
809 views

If $Tr(A)=0$ then $T=R^{-1}AR$ has all entries on its main diagonal equal to $0$

Prove that if $A$ is a square matrix and $Tr(A)=0$, then there exixts an invertible matrix $R$ such that the matrix $T=R^{-1}AR$ has all entries on its main diagonal equal to $0$. It seems like the ...
2
votes
1answer
112 views

Iwasawa Matrix Decomposition Proof

Iwasawa Decomposition (special case): Let $G=SL_n(\Bbb{R})$, $K=$ real unitary matrices, $U=$ upper triangular matrices with $1$'s on the diagonal (called unipotent), and $A=$ diagonal matrices with ...
1
vote
1answer
100 views

Order of $LU$ factorisation

Can someone tell me how to calculate the order of a) $LU$ decomposition as well as b)the gaussian elimination of a square matrix $A$? I am at a loss ... Given:: $A$ is a $n\times n$ matrix and ...
0
votes
1answer
652 views

If two matrices have the same characteristic polynomials, determinant and trace, are they similar?

If two $n \times n$ matrices have the same characteristic polynomials, determinant and trace, are they similar, EVEN if ($ \lnot \#Spec= 0$)?
0
votes
2answers
68 views

Diagonalisable or not? [closed]

Let $$A = \begin{pmatrix}a & b\\c & d\end{pmatrix}.$$ Show that 1) $A$ is diagonalisable if $(a - d)^2 + 4 bc > 0$ 2) $A$ is not diagonalisable if $(a - d)^2 + 4 bc < 0$
11
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5answers
776 views

Prove $\rm AB = BA = 0$ if the set of nonzero eigenvalues of $\rm A + B$ is union of set of nonzero eigenvalues of $\rm A$ and $\rm B$.

$A$ and $B$ are $n×n$ symmetric matrices, their non-zero eigenvalues are $(\lambda_{1},\ldots,\lambda_{r} )$,$( \mu_{1},\ldots,\mu_{s} )$. If the nonzero eigenvalues of $A + B$ are $(\lambda_{1},\...
9
votes
1answer
3k views

Decompose a 2D arbitrary transform into only scaling and rotation

Related to: Newly Developed With Details - Describing orthographic projection using simple 2D transformations Given an arbitrary 2D linear transform (which may include shear, i.e. the vectors or the ...
8
votes
2answers
2k views

Inverse of the sum of a symmetric and diagonal matrices

I have two matrices $A$ and $B$ with quite a few notable properties. They are both square. They are both symmetric. They are the same size. $A$ has $1$'s along the diagonal and real numbers in $(0 ...
7
votes
1answer
3k views

Why is the largest element of symmetric, positive semidefinite matrix on the diagonal?

I know the very well know equivalence of the properties of a positive, semidefinite matrix: $A$ is positive semidefinite, $A = U^T U$ for some matrix $U$, $\mathbf{x}^T A \mathbf{x}\geq 0$ for every $...
6
votes
3answers
104 views

Matrices that Differ only in Diagonal of Decomposition

Suppose that $\mathbf A_1$ and $\mathbf A_2$ are $n \times n$ matrices. Are there necessary and sufficient conditions such that there exists $n \times n$ matrices $\mathbf U$ and $\mathbf V$ and $n \...
3
votes
2answers
276 views

Prove that matrix is symmetric and positive definite given the fact that $A+iB$ is.

I have some questions regarding the following problem Let $ A + iB $ - hermitian and positive definite, where $A, B \in \mathbb R^{n\ \times\ n} $ show that the real matrix $$C =\begin{pmatrix} A &...
6
votes
1answer
5k views

Decomposition of a positive semidefinite matrix

Let $Y \in \mathbb{R}^{n \times n}$ be a symmetric, positive semidefinite matrix such that $Y_{kk} = 1$ for all $k$. This matrix is supposed to be factorized as $Y = V^T V$, where $V \in \mathbb{R}^{...
13
votes
1answer
2k views

Cholesky of Matrix plus Identity

I have a positive definite matrix $A$ ($n \times n$ dimension) for which I have the Cholesky decomposition $A=LL^{'}$. I want to use this to compute a) The cholesky decomposition of $A+c^2\times I $ ...
4
votes
2answers
475 views

Decompose a matrix into diagonal term and low-rank approximation

For a matrix $A$ the Singular Values Decomposition allows getting the closest low-rank approximation $$A_K=\sum_i^K\sigma_i \vec{v}_i \vec{u}_i^T$$ so that $\|A-A_k\|_F$ is minimal. I'd like to do ...
4
votes
2answers
36k views

How to use LU decomposition to solve Ax = b

Using LU Decomposition how can I solve for vector $x$ in the system $Ax = b$, given $A$ and $b$. For simplicities sake where $A$ is a 3x3 matrix and $b$ is a vector of size 3. For example how to find ...
3
votes
1answer
1k views

Cholesky decomposition when deleting one row and one and column.

I've thought about this problem for days but could not find a good answer. Given Cholesky decomposition of a symmetric positive semidefinite matrix $A = LL^T$. Now, suppose that we delete the $i$-th ...
2
votes
1answer
373 views

Counter example or proof that $\kappa(AB) \leq \kappa(A)\kappa(B) $

I am stuggling to find a counter example or either proof (for general matrices $A, B \in \mathbb{C}^{m \times n}$) that $$\kappa(AB) \leq \kappa(A)\kappa(B)\,,$$ where the condition number $\kappa(\...
1
vote
0answers
66 views

Given generalized eigenvalues/eigenvectors obtain standard eigenvalues/eigenvectors

Suppose I have the following decomposition of a symmetric matrix $S$, $S = U \Lambda U^T$, where we have that $\Lambda$ is a diagonal matrix and $U U^T = D$ where $D$ is another diagonal matrix. In ...
7
votes
4answers
4k views

Find diagonal of inverse matrix

I have computed the Cholesky of a positive semidifinite matrix $\Theta$. However I wish to know the diagonal elements of the inverse of $\Theta^{-1}_{ii}$. Is it possible to do this using the cholesky ...
6
votes
3answers
4k views

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 $A^{-1}$ ...
5
votes
1answer
387 views

QR factorization of a special structured matrix

A friend asked me the following interesting question: Let $$A = \begin{bmatrix} R \\ \xi{\rm I} \end{bmatrix},$$ where $R \in \mathbb{R}^{n \times n}$ is an upper triangular and ${\rm I}$ ...
3
votes
1answer
483 views

rank of block triangular matrix and its relation to the rank of its diagonal blocks

Prove that $$rank\begin{pmatrix}A & X \\ 0 & B \\ \end{pmatrix}\ge rank(A) + rank(B)$$ where $$A,B\in \mathbb C^{m \times m}$$. I know the intuition behind it (i.e. maximal independent rows,...
3
votes
2answers
3k views

Block-diagonalizing an antisymmetric matrix

I was wondering how to block-diagonalize a $10 \times 10$ antisymmetric matrix into block matrices along the diagonal. Can I just diagonalize each non-diagonal block? Thanks!
3
votes
2answers
5k views

Is the Square Root of an Inverse Matrix Equal to the Inverse of the Square Root Matrix?

I know in general that if a matrix $A$ is positive definite, then there exists a (unique?) square root matrix $B$, which is also positive definite, such that $BB=A$. Therefore, suppose $A$ is ...
3
votes
2answers
471 views

Matrix decomposition definition

Wikipedia says "In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different ...