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
52 views

What is this (unusual) matrix/vector operation called?

A typographical error let to an unexpected (but, for me, potentially useful) result: $$ \left\{\begin{array} & a & b & c\\ d & e & f \\ g & h & i ...
1
vote
1answer
26 views

Multiple Linear Regression in Matrix Form

I am currently studying for my exams and came across the following question: State the multiple linear regression equation in matrix form. Write down the order of each matrix and explain what the ...
3
votes
2answers
95 views

Find the range for $AX-XA$

I don't know how to prove the following and need help. Let $M_{n\times n}$ be the vector space of matrices over $\mathbb{C}$. Let $A\in M_{n\times n}$ be fixed matrices, and $X\in M_{n\times n}$ ...
-1
votes
0answers
38 views

Finding a basis for $R^2$ with some constraints

Find a Basis $B$ of $R^2$ s.t. (1) $\left(\begin{array}{c}1 \\2\end{array}\right)_B = \left(\begin{array}{c}3 \\5\end{array}\right)$, and (2) $\left(\begin{array}{c}3 \\4\end{array}\right)_B = ...
0
votes
1answer
10 views

A weak form diagonal dominance and positive semidefiniteness

Consider the following nonstandard form of diagonal dominance: Let $A = (a_{ij})$ be symmetric and assume that for all $i$, $$ |a_{ii}| > |a_{ij}|, \quad j \neq i $$ Is there a matrix with such ...
2
votes
0answers
55 views
+50

Hermitian Matrices with Repeated Eigenvalues has Codimension 3?

It is sometimes claimed that the space of $n\times n$ Hermitian matrices with at least one repeated eigenvalue has codimension 3. (See link exercise 10.) The proof of this in dimension two is ...
0
votes
2answers
21 views

Indentifying Linearly Dependent Vectors: Why do constants cancel out?

I am having trouble understanding why the set of vectors (Johnson & Wichern, 2007) $$\mathbf x_1=\left[ \begin{array}{ccc} 1\\2\\1 \end{array} \right] \mathbf x_2=\left[ \begin{array}{ccc} ...
1
vote
1answer
41 views

Determinants and monic polynomials [duplicate]

I wish to show that $$ \det \begin{pmatrix} x & a & a & a\\ a & x & a & a\\ a & a & x & a\\ a & a & a & x \end{pmatrix}=(x-a)^3(x+3a).$$ Obviously, I ...
0
votes
1answer
19 views

Bound of the Minimum Eigenvalue of Product between Matrices

We have two real-valued Positive Semi-definite matrices, $A$ and $B$, both $n$ by $n$. Can the minimum eigenvalue $\lambda_\mathrm{min}(AB)$ be lower-bounded using the following eigenvalue sequences? ...
1
vote
1answer
32 views

What are some Applications of Hermitian Positive Definite matrices?

I am learning about Hermitian positive definite matrices and the way that they can be decomposed with Cholesky decomposition. I have learned that these matrices deserve special attention for how often ...
-3
votes
1answer
23 views

Proof related to $2 \times 2$ matrix [closed]

Let $A$ be a $2 \times 2$ matrix such that $AX = XA$ for all $2 \times 2$ real matrices $X$. Show that $A =kI$ for some $k$ belonging to $\mathbb{R}$
-2
votes
2answers
36 views

Proof related to matrix [duplicate]

Let $A$ and $B$ be $n \times n$ real matrices such that $A^2 = I, B^2 = I$ and $(AB)^2 = I$. Prove that $AB = BA$. Someone help me with this problem
0
votes
1answer
32 views

Proof related to matrix with if and only if condition

Suppose $A$ and $B$ are matrices such that $AB$ and $BA$ are defined. Prove that $$(A+B)^2=A^2+B^2+2AB\quad\text{if and only if}\quad AB=BA.$$ Someone help me with this.
1
vote
1answer
15 views

How to approach specific dimensionality reductions?

I'm having difficulty in rationalizing dimensionality reduction (I've used other sources), and I would appreciate it if someone could help me out with a specific example. Given an $M \times M$ PCA ...
1
vote
1answer
38 views

Making a matrix diagonal with its eigenvectors

I'm trying to make my matrix diagonal. this is my matrix (for matlab and octave) ...
1
vote
2answers
29 views

Prove these matrix-vector products are linearly dependent/independent.

I have two statements that I wanto to prove or disprove. Let $A \in \Bbb R^{n \times n}$ and $b_1,\ldots,b_n \in \Bbb R^n$ be linearly independant. Then, $Ab_1,\ldots,Ab_n$ are also linearly ...
0
votes
0answers
18 views

How to rotate a 2nd derivative gaussian function?

I have a 2nd gaussian derivative in y and a normal gaussian in x, which results in the function: $$ f\left ( x,y \right ) =\frac{- \exp ^{-\left (\frac{x^{2} +y^{2}}{2\cdot \sigma^{2} } \right ...
0
votes
0answers
13 views

Expansion formula for the permanent of a sum of two matrices

Given two squared matrices $A$ and $B$, what is the relationship between the permanent of the sum $A+B$ and the permanents of $A$ and $B$? I found in a reference that this expansion is reported in ...
0
votes
1answer
17 views

Inconsistency of column representation with orthogonality of vectors

Let's say I have two vectors $v_{1}$ and $v_{2}$ which form a basis for $\mathbb{R}^2$. Any vector $v$ in $\mathbb{R}^2$ can be represented as $$v = av_{1} + bv_{2}$$ for some $a,b \in \mathbb{R}^2$. ...
1
vote
0answers
39 views

Branching $U(2)$ with respect to $SU(2)$

By construction $SU(2)$ is contained in $U(2)$, the special unitary and unitary groups respectively. Thus, any representation of $U(2)$ will induce a representation of $SU(2)$. The irreducible irreps ...
0
votes
1answer
21 views

How to prove Cholesky decomposition for positive-semidefinite matrices?

According to Cholesky decomposition $A$ is a Hermitian positive-definite matrix if and only if $A=T^*T$ for some upper triangular matrix $T$. When $A$ is positive-semidefinite we have such ...
2
votes
2answers
71 views

matrix inequality proof [completion of squares]

Can someone help me to prove this? $\begin{bmatrix} 0 & B^\top W^\top \\ WB & 0 \end{bmatrix} \leq \begin{bmatrix} B^\top Q B & 0 \\0 & W^\top Q^{-1}W \end{bmatrix}$ with $Q$ ...
1
vote
2answers
17 views

Proving generalized form of Laplace expansion along a row - determinant

Definition: Let $A$ be an ($n \times n$)-matrix. Let $M_{ij}$ denote the matrix acquired from $A$ by deleting row $i$ and column $j$. For $n \geq 2$ we define the determinant of $A$ inductively as ...
4
votes
2answers
103 views

Prove that $\dim(V)$ is even

Let $V$ be a finite dimensional vector space. Let $A_1,A_2: V\rightarrow V$ be commuting linear operators such that $A_1+A_2=-I$ where $I$ is the identity operator. Also $A_1,A_2$ have no negative ...
0
votes
0answers
28 views

Matrix Calculation .

$$P\pi=\pi$$ $$ \begin{bmatrix} 0&0&0&0&1 \\ 1&0&0&0&0 \\ \frac{1}{2}&\frac{1}{2}&0&0&0 \\ ...
0
votes
0answers
16 views

How much similar two large matrices are-a practical approach

I'm wondering is there any method to check how much similar two matrices are? For example the following three matrices $A = \left[ {\begin{array}{*{20}{c}} 1&1\\ 1&1 \end{array}} \right],B = ...
0
votes
1answer
14 views

Such a matrix: Except the main diagonal, the rest are Hermitian

In a matrix, all the entries are complex numbers. If we set the main diagonal entries to be zero, then the matrix will be Hermitian. Does this matrix has a name or some nice properties, especially ...
0
votes
2answers
74 views

Explain how the proof is done

A solution of matrix problem appears to be as follows some one explain the following in the solution why is A cube is eliminated and fourth power of A is obtained? In the seventh line In the ...
1
vote
1answer
16 views

Understanding representation of permutation matrix as vector

I hope this question is relevant here: I'm using some external software that does an LU decomposition of a square $(n\times n)$ matrix; the result is returned as three matrices L, U and P where P is ...
4
votes
1answer
64 views
+50

Determinant of a Certain Block Structured Positive Definite Matrix

Is there a lower bound for the determinant or minimum eigenvalue of the following $d$ by $d$ matrix in terms of $d$? $$\Gamma=\left( {\begin{array}{cc} I & B \\ B^{*} & I \\ \end{array} ...
1
vote
1answer
19 views

Tightness of inequalities for various matrix norms

For a general inequality involving matrix norms, does the choice of the norm influence the tightness of the inequality? Eg. In $\|AB\| \leq \|A\| \|B\|$, Does the choice of the norm affect the ...
1
vote
0answers
42 views

How do we find the eigenvalues of the matrix?

We have the $m \times m$ matrix $$A=\begin{bmatrix} -2 & 1 & & & 0 \\ 1 & -2 & 1 & & \\ & \dots & \dots & \dots & \\ & & 1 & -2 ...
0
votes
3answers
32 views

What is $[M_1,M_2]$ equal to? ($M_1$ and $M_2$ are matrices)

This is an old exercise that I had a year ago: $$M_1 = \dfrac{1}{\sqrt{2}} \begin{bmatrix}0 & 1 &0\\1 & 0 & 1\\0 & 1 & 0\end{bmatrix}$$ $$M_2 = \dfrac{1}{\sqrt{2}} ...
1
vote
0answers
17 views

Product expression maximization given 4 matrices as function of 2 column vectors

Hypothesis: - we are given 4 complex matrices denoted with $H_1, H_2, G_1$ and $G_2$. - the 4 matrices are not necessarily square so their size is $N$ by $M$. - we denote with $w_1$ and $w_2$ two ...
0
votes
0answers
30 views

Help understanding a theorem about diagonalizable matrices

So while studying for my Linear Algebra test, I'm required to study some theorems and their proofs, and I have trouble understanding a particular part of the proof for the following (I'm translating ...
0
votes
0answers
11 views

The image of a a vector in the edge space when multiplied by it's incidence matrix.

Consider a graph $G=(V,E)$ and it's incidence matrix $M$. Let $\textbf{x}$ be the characteristic vector for a standard basis vector in $\mathcal{E}$ (a vector corresponding to the one element edges ...
0
votes
1answer
19 views

Write the equation in cartesian form

I can't remember what method to use, help would be appreciated. The question: A line is given by $r = 2i +3J -k + \lambda$. Write the equation in cartesian form. Thanks
0
votes
1answer
31 views

Problem on singular value and trace of matrix

Let $A,B\in \mathbb{R}^{n\times n}$ show that there exists (1) Orthogonal matrix U,V satisfying $|trace(AB)|\leq\sum_{i=1}^{n}\sum_{j=1}^{n}(\sigma_{i}(A)\sigma_{j}(B)|u_{ij}v_{ij}|)$ (2) Double ...
0
votes
1answer
35 views

Help me proving a property of the determinant

I'm trying to prove the following property using cofactor expansion along the first row. Not sure if my proof is correct (don't really know what to do with the induction hypothesis), and I got trouble ...
1
vote
0answers
28 views

Is my proof true or not ? $rk(A) + rk(B) \ge rk(A+B)$

I know that my question has already an answer here, but I have proved it another way and I want to see whether my proof is true or not? If we assume $A$ , $B$ and $A+B$ are respectively the matrices ...
0
votes
1answer
14 views

Matrix entropy measure

I have a matrix (its dimension is $n$ x $m$) where each cell can be $0$ or $1$. I would like to calculate an "entropy" measure on it that tells me how close are the ones together or how spread they ...
1
vote
1answer
23 views

Finding minimal polynomial of big blocks diagonal matrix [duplicate]

Consider the following matrix: \begin{bmatrix} -3 & 1 & -1 & & & & & \\ -7 & 5 & -1 & & 0 & & & 0\\ -6 & 6 & -2 & & ...
1
vote
1answer
21 views

Question about the signature of a matrix

Let $A$ and $B$ be real symmetric $n \times n$ matrices with the same rank such that $B$ differs from $A$ only by two sufficiently small nondiagonal entries. Can we say that $B$ has the same signature ...
0
votes
1answer
19 views

Strict inequality of vector norms

Given a non orthogonal projection $p$ and non zero vector $x$. I am going to prove that $$\|Px\|<c\|x\|$$ for some $c<1$, where $\|\cdot\|$ is the usual Euclidean norm. I can only have the ...
0
votes
1answer
23 views

How to solve for matrix $X$ in $Y=X(X^TDX)^{-1/2}$

Let $Y \in \mathbb{R}^{n \times n}$ be any matrix such that $Y^T D Y = I$ for some positive diagonal matrix $D$ and $I$ the identity matrix. Further it is known that $Y=X(X^TDX)^{-1/2}$ for some ...
1
vote
0answers
23 views

Ideas for expressing the inverse of matrix quadratic form $CAC^T$

I want to find an expression for the inverse of the matrix system $Z=CAC^T$, where $A \in \mathbb{C}^{n \times n}$ is block diagonal with dense blocks, and $C \in \{-1,0,1\}$ with dimension $m \times ...
1
vote
1answer
42 views

Jacobian of matrix product

I understand from Wikipedia (https://en.wikipedia.org/wiki/Matrix_calculus#Matrix-by-scalar) that if two Matrices $M \in \mathbb{R}^{m+k}$ and ${N \in \mathbb{R}^{k+t}}$ whose elements are real ...
1
vote
1answer
72 views

If $A$ is a $12 \times 12$ real matrix such that $A^{17}=I$ , is $A$ diagonalizable ? Are all eigenvalues of $A$ real ?

If $A$ is a $12 \times 12$ real matrix such that $A^{17}=I$ , is $A$ diagonalizable ? Are all eigenvalues of $A$ real ?
1
vote
3answers
46 views

Rows of an orthogonal matrix form an orthonormal basis

Q: A matrix $A \in \text{Mat}(n \times n, R)$ is said to be orthogonal if its columns are orthonormal relative to the dot product on $\Bbb R^n$. By considering $A^TA$, show that $A$ is ...
0
votes
1answer
31 views

Algorithm to find positive definite matrix given conditions.

I want an algorithm that always find one solution for the given problem below: Given a positive n length vector, b, a n vector of 1 values, u: I want to determine B, matrix n * n, positive definite ...