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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2
votes
1answer
73 views

Criteria for positive semi-definiteness - zero diagonal

I am currently doing a bit of background reading on some fundamental topics in preparation for a talk, and came across a question relating to positive definiteness. It is taken from Horn and Johnson's ...
1
vote
0answers
69 views

Properties of matrix functions

Can I say that a certain matrix function is absolutely continuous or monotonically increasing in $\lambda$(assuming that the matrix is a function of $\lambda$)? In other words, are these ...
1
vote
0answers
41 views

Finding matrix with given special eigenvalues

Given an arbitrary matrix $A=\left[a_{i,j}\right]\in\mathbb{C}^{2\times2} $ Now define the set $$ \mathcal K = \left\lbrace z\in\mathbb C : ...
2
votes
1answer
105 views

bound on trace-norm of product of matrices

Is it true that $$ \|ABA^\dagger\|_1\leq \|A\|^2\|B\|_1, $$ where $\|A\|_1$ is the trace norm, $\|A\|$ is the spectral norm, and $A$ and $B$ are square matrices?
0
votes
1answer
162 views

eigenvalues of the sum of a diagonal and diagonalizable matrix

Let $A$ be a matrix with a nonzero entry $t_0$ in the (1,1) entry and padded with zeros everywhere else; and let $B$ be a diagonalizable matrix, such that $VBV^T$ is diagonal. Is there anything we can ...
2
votes
0answers
329 views

How Would Arnold Explain the Jordan Normal Form to a 6 Year Old?

How would Vladimir Arnold explain the Jordan normal form, to a six year old, in full detail starting from nothing in a way that somehow explains everything in a deeper way, probably including topology ...
7
votes
4answers
355 views

Largest singular value of non square matrix

Let $B$ be an $m\times n$ matrix with complex number as its element. Let $\sigma$ denotes the largest singular value of $B$ Prove that \begin{equation} \sigma = \max\limits_{\|u\|_2=1,\|v\|_2=1} ...
1
vote
0answers
44 views

Solving a linear matrix equation with respect to the maximum of the euclidian distances between rows.

With $n>m$, real number matrices $A$, $B$, $C$ are shaped like: $$A=\left( \begin{array}{ccc} a_{1,1} & \cdots & a_{1,m} \\ \vdots & \ddots & \vdots \\ a_{n,m} & \cdots ...
0
votes
1answer
94 views

number of ways to fill a 2D grid

We have a 2D grid with n rows and m columns, we can fill it with numbers between 1 and k (both inclusive). Only condition is that for each r such that 1<=r<=k ,no two rows must have exactly the ...
1
vote
1answer
55 views

Inequality of two symmetric matrices

Given a full rank $n \times m$ matrix $K$ with $m<n$ and an invertible symmetric matrix $J$. Let $A$ be a symmetric positive semi-definite $n \times n$ matrix such that \begin{equation} (K^T ...
1
vote
2answers
157 views

Can someone explain span and basis in matrices?

I did not really understand my textbook. So would someone mind simply explaining what exactly is span and basis and how do you find it? Thanks guys :)
1
vote
2answers
33 views

Hermitian - Using the Spectral Theorem

Ok so I am trying to show that if A is normal and has real eigenvalues, then A is hermitian. It was suggested that I try using the spectral theorem. So if we assume A is normal and has real ...
3
votes
1answer
91 views

Prove this matrix is invertible for $n < m-1$

Prove this $(n+1)\times (n+1)$ matrix $\bf{A}$ is invertible for $n < m-1$ and the $x_k$ distinct, \begin{bmatrix} m &\sum_{k=1}^mx_k &\sum_{k=1}^mx_k^2 &\cdots ...
1
vote
0answers
104 views

Diagonalization of matrices and linear transformations

I'm reading Lawrence Perko's Differential Equations and Dynamical Systems, and he writes the following : Theorem. If the eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_n$ of an $n \times n$ ...
1
vote
0answers
180 views

Hermitian positive semi-definite matrix is a Gram matrix

I showed that every Gram matrix, i.e. a $n \times n$ matrix $A$ with $A_{ij} = <x_i,x_j>$ where $x_1,...,x_n$ are vectors in an inner product vector space $V$, is Hermitian and positive ...
1
vote
1answer
60 views

Positive definitness of infinite dimensional matrices

Assume $M=(b_{i,j})_{ i,j=1}^{\infty}$ is an infinite dimensional matrix such that $b_{i,i}>0$ and $b_{i,j}=b_{j,i}$ for all $i,j\in\mathbb{N}$ (i.e., $M$ is symmetric with positive diagonal ...
6
votes
5answers
290 views

If $A^T=-A$, then A is not invertible

Let $n \in \mathbb{N}$ be odd and $A \in$Mat$(n,\mathbb{R})$ with $A^T=-A$. Show that $A$ is not invertible. I have no idea how to start this...
2
votes
1answer
72 views

$A+A^T=I$, $\lambda$ is an eigenvalue of $A$, show that $\lambda=\frac{1}{2}+\alpha i$

I tried to solve it but I got $\lambda =\frac{1}{2}$ without the complex part, I'd like to know where my logic is flawed. Assume $v$ is the eigenvector associated with lambda, then: $(A+A^T)v=Iv$ ...
4
votes
1answer
868 views

Inverse of a symmetric tridiagonal matrix.

Hello, everyone! I am trying to find the inverse of an $N\times N$ matrix with ones on the diagonal and $-\frac{1}{2}$ in all entries of the subdiagonal and superdiagonal. For example, with $N=3$, ...
0
votes
1answer
454 views

Calculating eigenvectors and eigenvalues of a 2x2 complex matrix

I've previously asked elsewhere, http://stackoverflow.com/questions/21118820/non-trivial-eigenvectors-of-a-22-matrix-in-code, how to calculate the eigenvectors and eigenvalues of a 2x2 matrix in a ...
2
votes
1answer
55 views

power series for matrix with elements smaller than 1

If I have a square matrix A such that all elements $|a_{ij}| < 1$ does this guarantee that all my eigenvalues will also be less than 1 and that the power series $S = I - A + A^2 - A^3...$ will ...
0
votes
1answer
94 views

Matrix identities (for invertible matrices)

I've got a little question concerning matrix operations. I am supposed to prove the following equation: $(I + CBC^T)^{-1} = I - C(C^TC + B ^{-1})^{-1}C^T$ B and C are assumed to be invertible real ...
1
vote
0answers
34 views

Understanding of a formula with matrix summation

We are a quite a few students at in the class struggling to compute this would anyone be able to help? Also note --> ' means transpose. Sorry for the misunderstanding, when I said that it is not ...
1
vote
2answers
64 views

is (I+P) invertible when row sum in P = 0

I have a $n$x$n$ matrix P where the sum of each row = 0 (the individual entries are real but can be negative). Clearly P is not invertible. Can we show that I+P is invertible? thanks
1
vote
0answers
29 views

Is there a method to calculate Mahalanobis distance incrementally?

By incrementally (or recursively) I mean to update the pool/group of values/vectors as soon as more vectors are available without having to recompute the entire covariance and inverse matrices. I ...
6
votes
1answer
217 views

Is the product of three positive semidefinite matrices positive semidefinite

Is the product of three positive semidefinite matrices positive semidefinite if the product is symmetry? If so, any proof or reference? Thanks Paper - on weakly positive matrices, from Wigner 1963, ...
0
votes
1answer
108 views

Completeness of eigenvectors of Hermitian Matrix.

How do you show that eigenvectors of a Hermitian matrix form a complete set of basis?
17
votes
8answers
6k views

Do all square matrices have eigenvectors?

I came across a video lecture in which the professor stated that there may or may not be any eigenvectors for a given linear transformation. And, so far I thought every matrix has eigenvectors. Please ...
2
votes
1answer
128 views

for any ring $A$ the matrix ring $M_n(A)$ is simple if and only if $A$ is simple

Let integer $n\geq 1$. I have obtained that for any field $k$, the matrix ring $M_n(k)$ is simple, i.e., $M_n(k)$ contains no nonzero proper two sided ideals. Now I want to prove that: for any ring ...
1
vote
1answer
110 views

condition number after scaling matrix

Maybe a well-known question. Let $\Sigma$ represent a real symmetric positive definite matrix, i.e. a covariance matrix. Which diagonal matrix $D$ with positive diagonal minimizes the condition ...
2
votes
1answer
126 views

The dimension of the subvariety of matrices of rank 3 in M(n, m)

Consider the space $M(m, n)$ of matrices of size $m \times n$ over field $K$. Let $X \subset M(m, n)$ be the subset of matrices of rank $3$. Show that $X$ is an algebraic subvariety of $M(m, n)$. ...
2
votes
2answers
92 views

Linear Combinations

I am currently trying to learn linear combinations and am stuck on two different problems. The first states: ...
0
votes
1answer
59 views

Compare row eigenvectors and column eigenvectors of a square matrix

I want to know if there is any difference between row eigenvectors and column eigenvectors? By mentioning "difference", I don't mean the exact value of the vectors. I mean the maximal number of ...
3
votes
2answers
62 views

Proof that diagonally dominant matrices are regular - Reference request

I know that it is easy to proof that diagonally dominant matrices are regular (non-singular) by the gershgorin circle theorem. But the theorem that diagonally dominant matrices are regular was ...
1
vote
2answers
61 views

What is this form of 'notation' called?

I was reading some of Max Tegmark's lecture materials and I found this little thing. Is there a name for it? Specifically, I am talking about $S_1$ R $S_2$ & $S_1$ R $S_2$ and the matrix. Is ...
0
votes
1answer
94 views

How do I answer this question during a test

Let v1 =\begin{bmatrix} 1\\ 0 \\ 0 \end{bmatrix} , v2 = \begin{bmatrix} 0 &\\ 1\\ 0 \end{bmatrix} , H ={\begin{bmatrix} s &\\ s\\ ...
1
vote
1answer
54 views

Deriving with matrices, matrice equation

Can someone do this stepwise so that I am able to see what is going on. I am sure there is not many steps to take, but I am struggling to see the logic that goes from the left hand side to the right.! ...
1
vote
1answer
133 views

Matrix Algebra, Proof of some Trigonometric Identities

Please Refer to the image to see the problem. This was the easiest way to input the question as it has some difficult symbols to input from a keyboard. [Edit: Image with task replaced by $\LaTeX$:] ...
0
votes
2answers
65 views

If A is symmetric positive definite, is -A symmetric negative definite and why?

I ask this because I'm programming a function that does only take a symmetric positive definite matrix as input. But now I'm told give to the function the negation of such a matrix. That makes no ...
1
vote
1answer
95 views

What's wrong with this argument? (Diagonalizable matrices and spectral theorem)

Please consider all matrices to be in $M_n(\mathbb{R})$. Let $A$ be a positive definite symmetric matrix and $B$ a symmetric matrix. Then $A$ represent a positive definite scalar product $\Phi$ in an ...
1
vote
0answers
62 views

For $2 \times 2$ matrix, if $AB = -BA$, what properties would $A$ and $B$ have with regards to each other?

As title says, assume that $A$ and $B$ are two-by-two matrices. I want them to satisfy $AB = -BA$. What properties do these two matrices have with regards to each other? Sufficient properties would be ...
1
vote
4answers
669 views

Find the values of $x$ which makes $\det (A)=0$ without expending determinant

Find the values of $x$ which makes $\det(A)=0$ without expending determinant: Let $A$ : $$\begin{bmatrix}1 & -1 & x \\2 & 1 & x^2\\ 4 & -1 & x^3 \end{bmatrix} $$ How can I ...
13
votes
3answers
231 views

If $A$ is positive definite, then $\int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}\text{d}x=\left|\det\left({\pi}^{-1}A\right)\right|^{-1/2}$

Let $A$ be a positive definite real $n\times n$ matrix. How can I prove that $$ \int_{\mathbb{R}^n}\mathrm{e}^{-\langle ...
2
votes
7answers
452 views

Does $\{u_1, u_2, u_3, u_4\}$ spanning $\mathbb R^3$ mean that $\{u_1,u_2,u_3\}$ also does? Since it's a subset?

Does $\{u_1, u_2, u_3, u_4\}$ spanning $\mathbb R^3$ mean that $\{u_1,u_2,u_3\}$ also does? Since it's a subset? A little unclear about this...
0
votes
1answer
116 views

Prove that an upper triangular matrix $A$, such that $A^*A = AA^*$, must be diagonal.

Let $A \in \mathbb{C}^{n \times n}$ be an upper triangular matrix that satisfies $A^{*}A=AA^{*}$. Prove that $A$ must be diagonal. My attempt is to partition $A$ as follows: $$ A = ...
1
vote
0answers
40 views

Similarity of a Companion Matrix to a Diagonal Matrix

Let $A$ be the companion matrix of a monic polynomial $f \in K[x]$ with $deg f=n$. Show that $A-xI_{n}$ is similar to a diagonal matrix with main diagonal $1,1,1, \dots , 1, f$. I tried my proof in ...
0
votes
1answer
28 views

Proving the equality: $D^{-1}+D^{-1}CPBD^{-1}=(D-CA^{-1}B)^{-1}$ where $P=(A-BD^{-1}C)^{-1}$

I am trying to prove the following equality that I need to use as an intermediate step to solve one of my problems. The equality is the following: $D^{-1}+D^{-1}CPBD^{-1}=(D-CA^{-1}B)^{-1}$, where ...
0
votes
2answers
62 views

If $\operatorname{rank}(AB)=n$, what are the $\operatorname{rank}(A)$ and $\operatorname{rank(B)}$?

$A$ and $B$ are $n\times n$ matrices. Any hints on how to solve this or where to find the answer are welcome
2
votes
3answers
155 views

Find $\exp(D)$ where $D = \begin{bmatrix}2& -1 \\ 1 & 2\end{bmatrix}. $

$$C = \begin{bmatrix}2& -1 \\ 0 & 2\end{bmatrix}\quad $$ I break it down into two matrices $$A = \begin{bmatrix}2& 0 \\ 0 & 2\end{bmatrix}\quad \text{and}\quad B =\begin{bmatrix}0 ...
1
vote
4answers
78 views

How to solve for matrix $A$ in $AB = I$

Given $B$ = $\begin{bmatrix} 1 & 0 & 0\\ 1 & 1 & 0\\ 1 & 1 & 1 \end{bmatrix}$ I know that $B$ is equal to inverse of $A$, how can I go backwards to solve for $A$ in $AB = ...