For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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1
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1answer
623 views

Using permutation matrix to get LU-Factorization with $A=UL$

Let $Q$ be the $n$x$n$ permutation matrix $$Q= \begin{bmatrix} 0&0&...&0&1\\ 0&0&...&1&0\\ .& \\ .&\\ .&\\ 0&0&...&0&0\\ ...
1
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2answers
107 views

Calculate the rank of matrix $B-C$ while $AB=AC$ and $\operatorname{rank}(A) = r$?

$A$ is an $n\times n$ matrix and rank$(A)=r$.,$B,C$ are both $n \times n$ matrices and $AB=AC.$ Calculate the maximun possible rank of the matrix $(B-C)$. This question is a part of my homework in ...
3
votes
1answer
292 views

Checking if one “special” kind of block matrix is Hurwitz

I have the next block matrix $$ J = \begin{bmatrix}A & B \\ K &0\end{bmatrix} $$ all matrices are square, where $A < 0$ (definite negative), $B$ has all its eigenvalues with positive real ...
4
votes
0answers
192 views

estimation of transition probabilities from aggregate data

Please, O mathematicians, help me understand the approach to the problem of estimating transition probabilities given only aggregate data in Kalbfleisch & Lawless' 1984 paper "Least-Squares ...
5
votes
1answer
75 views

Is there any connection between this matrices

Matrices I discuss are all $N\times N$ hermitian matrices. Define two positive (semi)definite matrices $H_1$ and $H_2$. Define the following matrices \begin{align} P_1&=H_1+(I+H_2)^{-1} \\ ...
1
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1answer
48 views

Does spectral normalization preserve eigenvalue ratios?

Let $A$ be a non-negative square matrix. Normalize $A$ by its spectral radius $\sigma(A)$, and call it $A_2 = A/\sigma(A)$. Does this normalization preserve the ratio between the two largest ...
4
votes
1answer
55 views

Finding an idempotent that satisfies certain conditions in a matrix ring.

I've been stuck on a problem, and I was wondering if anyone could help me out. The problem is: Let $R$ be the $2 \times 2$ matrix ring over the reals $\mathbb{R}$ of the form $$ \begin{bmatrix}a ...
0
votes
1answer
146 views

question about schur canonical form

If $A$ is an $n\times n$-matrix and it has $n$ orthonormal eigenvectors, is it true that $U^*AU$ is diagonal? $U$ is an unitary matrix and $U^*$ is the conjugate transpose of $U$. If it is true, ...
3
votes
0answers
377 views

What is a good metric to compare matrices?

I have a matrix that I obtained from theoretical computation and I have another matrix which I obtained by actual data collection. How do I compare the two matrices? How do I state that one matrix is ...
2
votes
1answer
59 views

Rules on replacing a solution to one system by another

Suppose a linear system $$Ax=b \tag 1$$ is given, $A\in\mathbb{R}^{n\times n}$, $b\in\mathbb{R}^n$, and a solution exists. Now, suppose the system is multiplied from the left by some ...
0
votes
1answer
55 views

matrix completion by rank minimization

In matrix completion, the starting point is often stated as: the optimization problem for matrix completion: min(X): (1/2) ||X-M||^2 s.t. rank(X)<= r Where X is the reconstructed matrix and M ...
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1answer
115 views

An algorithm for creating a matrix with certain properties

I'm searching for an algorithm that efficiently does the following Input: $n\in \mathbb{N}$ $j_1,j_2,\ldots ,j_n$ with the property $j_1<j_2<\cdots<j_n$ $b_1,\ldots ,b_n \in \mathbb{N}$ ...
7
votes
1answer
162 views

Why would $(A^{\text T}A+\lambda I)^{-1}A^{\text T}$ be close to $A^{\dagger}$ when $A$ is with rank deficiency?

In many applications that is not with high requirements, it is common to use $(A^{\text T}A+\lambda I)^{-1}A^{\text T}$ or $A^{\text T}(AA^{\text T}+\lambda I)^{-1}$ ($\lambda$ is small) to ...
0
votes
1answer
90 views

Finding a basis

Finding the basis for the kernel of: \begin{pmatrix} a & b \\c & d\end{pmatrix} $which$ $maps$ $to:$ \begin{pmatrix} a \\a\\3a + b \end{pmatrix} It's all complex, but I'm not sure if ...
0
votes
1answer
62 views

Counting commuting Pauli Strings of a certain weight

Let a $n-$length pauli string represent any tensor product of finitely many pauli matrices, Ex: $X\otimes Z\otimes \mathbb{I}\otimes Y\otimes \mathbb{I}\otimes X\otimes\cdots\otimes Z$ where the ...
0
votes
1answer
450 views

Solving equation of motion differential equation by using matlab

I have an equation of motion differential equation: $$M\;x''(t) + C\;x'(t) + K\;x(t) = 0$$ I know $M,C,$ and $K$ (constant $4\times 4$ matrices) and also the eigenvalue-eigenvector pairs. What ...
3
votes
0answers
92 views

QR decomposition help

What do Q and R stand for? Why must the diagonal entries of R be positive instead of just nonzero?
0
votes
1answer
70 views

Simple matrix problem basis vectors

Why does the standard basis vector $e_2$ lie in the kernal of this matrix? Doesn't $e_1$ also lie in it too? $$\pmatrix{0&0&1\\0&0&0\\0&0&0}$$
2
votes
1answer
2k views

how to prove a symmetric matrix is positive semidefinite?

I have a symmetric matrix where the diagonals are all positive. I need to prove the matrix is positive semidefinite. The matrix is made up of a bunch of constants and I tried getting the eigenvalues ...
1
vote
1answer
924 views

How to prove exponential of every square matrix is invertible?

For a square matrix $A$. Define $exp(A)=I+\sum_{n}A^{n}/(n!)$ . I need to prove two things exp(A) converges and is invertible. Its inverse is given by exp(-A). Second part is straightforward. Can ...
3
votes
2answers
2k views

Minimum and Maximum eigenvalue inequality from a positive definite matrix.

I got a positive definite matrix $B,$ that is, $V(x) = x^TBx > 0$ for any vector $ x \neq 0.$ I want to show that $ \lambda_\min \|x\|_2^2 \leq V(x) \leq \lambda_\max \|x\|_2^2$ for any $x \neq ...
1
vote
3answers
142 views

How to find an example of matrix $A$ that satisfies $A^{-1} = \frac{1}{n} A$, where $A = [a_{ij}]_{n \times n}$?

How to find an example of matrix $A$ that satisfies $A^{-1} = \frac{1}{n} A$, where $A \in n \times n$? For example if $A= \left( \begin{array}{ccc} 1 & 1 & 1\\ 1 & i & i^2\\ ...
3
votes
2answers
176 views

Conditions for Schur decomposition and its generalization

Let $M$ be a $n$ by $n$ matrix over a field $F$. When $F$ is $\mathbb{C}$, $M$ always has a Schur decomposition, i.e. it is always unitarily similar to a triangular matrix, i.e. $M = U T U^H$ where ...
1
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1answer
58 views

what can a matrix be similar to if and only if there exists a generalized eigenvector?

Let $M$ be a $n$ by $n$ matrix over a field $F$. $M$ is diagonizable, i.e. $M=P DP^{-1}$ for some invertible matrix $P$ and some diagonal matrix $D$, if and only if there exists an eigenbasis. I ...
2
votes
0answers
113 views

Spectrum of Transition Matrix for Random Walk

Consider the symmetric random walk on $\{0, 1, \dots, n\}$ with transition probabilities $P(j \to j \pm 1) = 1/2$ for $1 \le j \le n-1$ and $P(0 \to 0) = P(0 \to 1) = P(n \to n) = P(n \to n-1) = 1/2$. ...
0
votes
1answer
101 views

Eigen of $AA^H$ and of $A^HA$

If I am correct, for a complex square matrix $A$, $AA^H$ and $A^HA$ are positive semidefinite, and therefore their eigenvalues are all nonnegative. They share the same set of nonzero eigenvalues, and ...
2
votes
1answer
230 views

Geometric multiplicities of the same eigenvalue of $A$ and of $A^T$

For a square complex/real matrix $A$, $A$ and $A^T$ have the same set of eigenvalues, each with same algebraic multiplicities, since their characteristic polynomials are the same. I wonder for each ...
1
vote
3answers
110 views

Eigenvalue theorem

I was reading my differential equation book and there is a theorem I am having trouble understanding. What do they mean by this? An $n\times n$ matrix $A$ has at least one and at most $n$ ...
2
votes
1answer
304 views

Need help with relative and absolute errors?

Lets assume I have $Ax=b$ equation, where $A$ is $2$x$2$ matrix. 1) I want to find an A, x, and b such that relative error in x is small but absolute error in x is large 2) Also want to find A, x, ...
1
vote
3answers
575 views

Positive-definiteness of block diagonal matrix

Given a block diagonal matrix $A$ like so: $$A= \begin{bmatrix} B & & \\ & C & \\ & & D \\ \end{bmatrix} $$ Given $B$ is ...
3
votes
2answers
299 views

Require brilliant resources to self teach.

I'm far from the level of mathematical knowledge every user on this website posseses, however I am very much determined to get there as my love for mathematics increases. These are the topics: ...
2
votes
1answer
98 views

Positive definite semi-ordering

I wanted to ask why is positive definite semi-ordering is well defined only for Hermitian matrices (or symmetric matrices if restricted to the reals)? I saw an extension to the definition of positive ...
1
vote
1answer
158 views

Equation involving Moore-Penrose inverse

Given a Laplacian matrix $A\in\mathbb{R}^{n\times n}$ (symmetric, positive semi-definite matrix with positive diagonal elements and non-positive off-diagonal), and its Moore-Penrose pseudoinverse ...
3
votes
2answers
112 views

Characterize triangular matrices by its eigenvalues?

For a triangular matrix, its diagonal entries are eigenvalues repeated with algebraic multiplicities. I wonder if the reverse is true. In other words, a matrix whose diagonal entries are ...
2
votes
2answers
274 views

Characterize unitary matrices by their eigenvalues and/or eigenvectors?

Every eigenvalue of a unitary matrix has absolute value 1. I was wondering whether a matrix whose eigenvalues all have absolute value 1 must be unitary? Thanks!
2
votes
2answers
82 views

A Quadratic Problem (which looks very simple)

This arises as a part of my work. \begin{align} \min_{x^{H}x=1}~&x^{H}A_1x \\ subject~to~&x^{H}A_2x=0 \end{align} $A_1$ and $A_2$ are $N\times N$ hermitian matrices and $x$ is a unit norm ...
2
votes
1answer
249 views

unknown matrices multiplication

I am having a algebric problem in my thesis work. It is some how like this ... I have to find $X$, $Y$, $X'$ and $Y'$, where these are unknown $2\times 2$-matrices and $A$, $B$, $C$, $I$, $J$, $K$ ...
0
votes
1answer
55 views

what matrix operation has notation $\oplus$?

From Wikipedia, a Jordan canonical form can be written in terms of its Jordan blocks as : $$J=J_{a_1}(\lambda_1)\oplus J_{a_2}(\lambda_2)\oplus\cdots\oplus J_{a_n}(\lambda_n), $$ I was wondering ...
6
votes
4answers
2k views

Are the matrix products $AB$ and $BA$ similar?

Given two matrices $A,B.$ On what conditions does $AB \sim BA$ hold?
0
votes
1answer
151 views

Uniqueness of a (generalized) (orthonormal) eigenbasis when it exists

In Jordan decomposition of a complex square matrix $M = P J P^{-1}$, the Jordan canonical form $J$ is unique up to permutation of the diagonal Jordan blocks $J_i$'s along the ...
2
votes
2answers
254 views

Entries of a positive definite symmetric matrix as inner products

For any $nxn$ positive definite symmetric matrix $A$ is it possible to write it's entries $a_{ij}$ as inner products of vectors $v_1,v_2,....,v_n$, that is $a_{ij}=\langle v_i,v_j\rangle$? Is there a ...
0
votes
2answers
124 views

How to express this using matrix operations?

I have a matrix $A$: $$A=\begin{pmatrix} 1 &3 &1\\ 7 &5 &2\\ 4& 3& 7\\ 8& 2& 1\\ 3& 9& 6\\ 4 &5 &2 \end{pmatrix}$$ and a matrix $B$: ...
5
votes
5answers
611 views

Is a square matrix whose diagonal and antidiagonal elements are all zero always singular?

Consider an $n\times n$ matrix whose primary and secondary diagonal elements are all zero. Does it necessarily follow that the determinant vanishes for these matrices? When $n=1,2,3,4$, the matrix is ...
2
votes
1answer
1k views

Hat Matrix Identities in Regression

I need to show that $\bar h= \sum{h_{ii}/n} = \operatorname{Tr}[H]/n = (p+1)/n$ Using the fact that $\operatorname{Tr}[AB]=\operatorname{Tr}[BA]$ and $H=X(X^TX)^{-1}X^T$. But I have no idea how to ...
0
votes
1answer
220 views

What is a Jordan Cell?

Google has been surprisingly unhelpful for me. A homework problem from my algebra class asks me to Calculate p(A) where A is a Jordan cell and p is a polynomial. ...
0
votes
2answers
456 views

Finding a row-reduced echelon matrix.

$A =\begin{pmatrix} 1& 2& 1& 0\\ -1 &0 &3 &5\\ 1& -2& 1& 1\end{pmatrix}$. I should find a row-reduced echelon matrix $R$ which is row equivalent to $A$ and an ...
1
vote
3answers
3k views

Proving that the matrix is not invertible.

A is a 2x3 matrix and B is a 3x2. How can i prove that the matrix D = AB is not invertible. I could not go further in this problem. The only thing that i have found is the multiply of these two matrix ...
0
votes
1answer
47 views

Decompose matrix

The following $m$ x $n$ matrix, decompose the first standard basis vector $e_1 = w + z \in \mathbb{R^n}$, where $w \in$ rowspace(A) and $z\in kerA$. Verify your answer by expressing $w$ as a ...
1
vote
1answer
46 views

How can I simplify this equation?

How can I simplify this equation : $$P=s^{T}_1A^{-1}y-\frac{1}{2}s^{T}_1A^{-1}s_1-s^{T}_0A^{-1}y+\frac{1}{2}s^{T}_0A^{-1}s_0 \tag1$$ as below equation: ...
2
votes
0answers
69 views

An identity involves Hilbert matrix

While thinking this problem,I found an identity: $b_n^TH_n^{-1}b_n+\frac{1}{n^2}=2$,where $b_n=(\frac{1}{1^2} \frac{1}{2^2} ... \frac{1}{n^2})^T$,and $H_n$ is the n-th Hilbert matrix.However,I cannot ...