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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1answer
19 views

property of simultaneous diagonalisable matrices

If A,B and B,C and C,A are simultaneously diagonalisable matrices then is this true that A,B,C are simultaneously diagonalisable? Here $A,B,C\geq{0}$.
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1answer
38 views

How to prove tr($Z^TZ$) is the sum of singular values

Ask a maybe trivious question: We know the product of singular values is |determinant of tha matrix| as in: Singular value proofs Then how to prove: $tr(Z^TZ)^{1/2}$ $=\sigma_1(Z) +...+ ...
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7answers
4k views

Proof: The inverse of the inverse matrix is the matrix.

If $A$ is a square matrix such that it is not singular, then $(A^{-1})^{-1} = A$ How can I prove this property? I would appreciate it if somebody can help me.
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1answer
45 views

Bounding a Symmetric Matrix

Consider the following $n \times n$ matrix $A$, which has 1's on the superdiagonal and subdiagonal and 0's elsewhere, i.e. $$\begin{pmatrix} 0 & 1 & 0 & \cdots & \cdots & \cdots ...
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2answers
95 views

Orthogonal Matrix question

$$A= \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} $$ is an orthogonal matrix. a) Prove that $A^{-1}=A^T$ b) show further that $a^2=d^2$ and that $b^2=c^2$. ...
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2answers
4k views

Proof: The identity matrix is invertible and the inverse of the identity is the identity

How can i show that: $II^{-1} = I = I^{-1}I$ (the identity matrix is invertible) for all cases. And then proof that: $I^{-1} = I$ (The inverse of the identity is the identity). I don't know how start ...
1
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1answer
87 views

Cayley transformation of a skew-symmetric matrix is orthogonal?

If $S$ is skew-symmetric ($S^{T} = -S$), how do I show that $Q$ is orthogonal where $$Q = (I + S)(I - S)^{-1}$$ which is the Cayley transformation of $S$.
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1answer
49 views

Find the Least Integer $k$ such that $B^k=I$

If $A$ and $B$ are two non Singular Matrices such that $B\ne I$, $A^6=I$ and $$AB^2=BA$$ Then what is the Least Integer $k$ such that $B^k=I$ My Try: Given $$AB^2=BA$$ which we can write as ...
3
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2answers
30 views

$f:M(n,\mathbb R) \to \mathbb R$ be , then $\exists ! C \in M(n,\mathbb R)$ such that $f(A)=Trace (AC) , \forall A \in M(n,\mathbb R)$?

Let $f:M(n,\mathbb R) \to \mathbb R$ be a linear function , then does there exist a unique $C \in M(n,\mathbb R)$ such that $f(A)=Trace (AC) , \forall A \in M(n,\mathbb R)$ ?
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0answers
10 views

Estimation of changes in solution x when A change

Suppose I have system $Ax = b$ where A = [${2}$ $-1$ $1$; $-1$ $10^{-10}$ $10^{-10}$; $1$ $10^{-10}$ $10^{-10}$]; b = [$2(1 + 10^{-10})$; $-10^{-10}$; $-10^{-10}$] and x = [$10^{-10}$; $-1$; $1$] ...
0
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2answers
68 views

Unable to calculate pseudo-inverse $A^TA$

I'm trying to calculate the pseudo-inverse of $A^TA$ as described in this paper: The SVD is particularly simple to calculate when the matrix is of the form $A^TA$ because $U=V$ and the rows of ...
0
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1answer
19 views

Storing a real matrix - fl notation

I have two matrix $A$ and $B$ and $fl(A), fl(B)$ denote the stored version them, respectively. Let $fl(AB)$ be the stored version of the product of $A$ and $B$. Is it true that $fl(AB) = fl(A) * ...
37
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6answers
42k views

Is the inverse of a symmetric matrix also symmetric?

Let $A$ be a symmetric invertible matrix, $A^T=A$, $A^{-1}A = A A^{-1} = I$ Can it be shown that $A^{-1}$ is also symmetric? I seem to remember a proof similar to this from my linear algebra class, ...
1
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0answers
34 views

Minimize $f(X)=trace\left(X^{T}\begin{bmatrix} 1&0\\0&16\\ \end{bmatrix}\right)$

Minimize $$f(X)=trace{\left(X^{T}\begin{bmatrix} 1&0\\0&16\\ \end{bmatrix}X\right)}$$ subject to the condition $g(X)=det(X)=1$. Then for taking $X=\begin{bmatrix} ...
4
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1answer
701 views

Transpose a square matrix code

I know it's not programming area , but I think it's more related to math. I have the following function: ...
2
votes
2answers
74 views

Linear Algebra: determine the number of linearly independent columns

$$ A = \left[\begin{array}{cccc}0& 1& 1\\ 1& 2& 3\\ 2& 0& 2\end{array}\right]$$ Clearly first and second columns are linearly independent. The third column is the sum of ...
0
votes
2answers
70 views

Linear Algebra Matrices

Determining the values of a for which the Matrix A has an inverse ! A= \begin{pmatrix} 1 & a & 1 \\ 2 & a+2 & 1 \\ 1 & 2 & a \end{pmatrix} How i solved it: ...
0
votes
1answer
96 views

Schatten p norm p>1

The Schatten p norm is differentiable away from the origin for p> 1. Does a stronger condition of Lipschitz continuity of the gradient also hold?
0
votes
0answers
38 views

Finding minimal distance between column sums by permuting row values

I'm sure the title is not very clear, that's because I don't know the proper name for this problem - if there is any. Consider a matrix with integers. The task is to find such a permutation of the ...
1
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2answers
85 views

Dynamic update of co-variance matrix upon new sample

I face a problem, where I have a total number of $c$ samples $S^{c\times r}$ of $r$ features. These are split at a position $p\in{1...c}$ into two subsets $S_{left}^{p\times r}$ and ...
2
votes
2answers
246 views

How to put a matrix in Jordan canonical form, when it has a multiple eigenvalue?

I have a question that reads: Put the matrix \begin{bmatrix} 3 & -4\\ 1 & -1 \end{bmatrix} in Jordan Canonical Form. Moreover, in each case, find the appropriate ...
0
votes
1answer
45 views

Why is $ \text{Rank}(A^{215}) = 3 $?

I have a question. Why is $ \text{Rank}(A^{215}) = 3 $, where $$ A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 2 & 0 & 2 \end{bmatrix}? $$ How can I even calculate this? I’m ...
0
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1answer
36 views

Find the value of $λ$ such that $v − u$ is orthogonal to $w$

Let $v$ and $w$ be any two non-zero vectors in $\Bbb{R}^n$. Let $u = λw$ for some real number λ. Find the value of $λ$ such that $v − u$ is orthogonal to $w$. I dont know how to use vector v. There ...
1
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4answers
383 views

How to show that a $4 \times 4$ strictly upper triangular matrix is nilpotent?

If $\displaystyle A$ is a $ \displaystyle 4 \times 4 $-strictly upper triangular matrix, how do you show that $\displaystyle A$ is nilpotent? Would it be sufficient to write out an example of a $ ...
1
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0answers
52 views

A quick question on product of Smith normal form

I am having trouble get a hold on this problem for a couple of days: if I have two $n$ by $n$ matrices $A,B$ with entries in some principle ideal domain $D$, and suppose their Smith normal form is ...
5
votes
2answers
3k views

Prove $(A^T)^{-1}$ = $(A^{-1})^T$

Prove $(A^T)^{-1}$ = $(A^{-1})^T$ for any invertible matrix "A". I actually don't know where to start - I do not think I can just apply index laws. Any help is cool! Thanks.
0
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1answer
95 views

Range and Kernel of a Hermitian Matrix

Is it true that the kernel and range of a complex $n\times n$ Hermitian matrix are always independent? If yes, can we prove it without using inner product? Thanks!
0
votes
1answer
29 views

How are the (syntactic) differences in these two definitions of the characteristic polynomial of a square matrix explained?

I’m used to defining the characteristic polynomial of a square matrix as follows: $$p_A(x) = det(xI_n - A)$$ where $A$ is a square matrix over a field and $I$ is an $n \times n$ identity matrix. The ...
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votes
1answer
71 views

Formula for reflection across a line in $\mathbb{R}^2$? [duplicate]

$\newcommand{\Reals}{\mathbb{R}}$I have an equation of a line: $4x - 3y = 0$. Let $S : \Reals^2 \to \Reals^2$ be reflection through that line, and let $P : \Reals^2 \to \Reals^2$ be projection onto ...
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2answers
35 views

For $X \in M(n,\mathbb R)$ , let $||X||:=\sqrt{Trace(AA^t)}$ , then $||AB|| \le ||A||\space||B|| , \forall A,B \in M(n,\mathbb R)$?

Let $M(n,\mathbb R)$ be the set of all square matrices of size $n$ with real entries . For $A \in M(n,\mathbb R)$ , let $||A||:=\sqrt{Trace(AA^t)}$ , then is it true that $A,B \in M(n.\mathbb R) ...
1
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1answer
150 views

Showing that $\det(AB)=\det A \det B$ with the following identity.

Given the following formulation of the determinant with Levi-Civita permutation symbols, show that $\det(AB)=\det A \det B$. $$\det A = \sum\limits_{ij\cdots l}\epsilon_{ij\cdots l} ...
0
votes
2answers
62 views

Explain rank of matrix

$$ A = \left[\begin{array}{cccc}4& 1& 5& 2\\ 1& 2& 3& 4\\ 2& 0& 2& 0\\ 3& 4& 80& 22\end{array}\right]$$ Can anyone please explain why rank is ...
0
votes
1answer
26 views

Proof of $\operatorname{im}(T) = \operatorname{im}(A)$

There is a theorem that states that if you let $T(x) = Ax$ be a linear transformation from $\mathbb{R}^m$ to $\mathbb{R}^n$, then: $\ker(T) = \ker(A)$ and they are a subspace of $\mathbb{R}^m$. ...
0
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1answer
246 views

Proving subspace conditions from subsets of vector spaces

Let n>=2. Which of the conditions defining a subspace are satisfied for the following subsets of the vector space Mnxn(R) of real (nxn)-matrices? (Proofs of counterexamples needed). U={A is an ...
0
votes
1answer
29 views

Matrix factorization and its rank

Ask a fundamentla problem: Suppose a matrix $A \in R^{n \times m}$ can be factored into $A = UV'$, with $U \in R^{n \times k}$ and $V \in R^{m \times k}$ If $m,n \geq k$, what the rank of matrix $A$ ...
0
votes
2answers
92 views

Coefficient Matrix of $T:\mathbb{R}^{3}\rightarrow\mathbb{R}$

Consider a vector $\vec{u}=\begin{pmatrix}u_1\\u_2\\u_3\end{pmatrix}$, such that $\left|\vec{u}\right| =1$. We define a linear transformation $T:\mathbb{R}^{3}\rightarrow\mathbb{R}$ given by ...
1
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1answer
55 views

Inverse of Matrix of Standard Vectors

Having a bit of trouble with the reasoning behind this question: "Consider a $n\times n$ matrix $A$. Assume that for each standard vector $\vec{e}_i$, there exists another vector $\vec{v_i}$ such ...
0
votes
1answer
40 views

Understanding matrix with greater than or equal

I'm reading a book and it states the following: $$ \begin{bmatrix} v\\ u\\ \end{bmatrix} \ge k $$ Does this mean that BOTH v and u are greater than or equal to k? If ...
1
vote
2answers
60 views

If $A^n$ is normal, is $A$ normal?

My question is : Given an invertible matrix $A$ ( with complex entries ) , if $A^n$ is normal,is $A$ normal? This is related to the question : If $A$ is an invertible $n\times n$ complex matrix and ...
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0answers
35 views

Solutions to linear equations

Am I right in thinking that the following augmented matrix equation only has one solution: $\begin{bmatrix} 0 & 1 & 0 & 4\\ 0 & 0 & 1 & 10 \end{bmatrix} $ i.e., if the ...
1
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0answers
148 views

Linear Algebra Proof for matrices

Could someone possibly help me in proving this: Let $A$ be the augmented $m \times (n + 1)$ matrix of a system of m linear equations with $n$ unknowns. Let $B$ be the $m \times n$ matrix obtained ...
0
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0answers
58 views

Column spaces in matrix equations

I am having trouble with the second part of this problem from an exam paper: I was easily able to obtain the rank, a basis for the column space and a basis for the null space by reducing the matrix ...
7
votes
2answers
5k views

Span of an empty set is the zero vector

I am reading Nering's book on Linear Algebra and in the section on vector spaces he makes the comment, "We also agree that the empty set spans the set consisting of the zero vector alone". Is Nering ...
2
votes
2answers
83 views

Minimal polynomial of an $n\times n$ matrix $A$ is $x^3+2x+2$; then $3$ divides $n$

Let $A$ be an $n × n$ matrix with rational entries such that the minimal polynomial of $A$ is $x^3 + 2x+2$. Prove that $3$ divides $n$. I think there is no rational root of this polynomial but ...
0
votes
0answers
41 views

Comparing Bases in $\mathbb{R}^{n}$

A bit of trouble with the following question: Let $\mathcal{B}$ be the basis of $\mathbb{R}^{n}$ consisting of the vectors $\vec{v_1},\vec{v_2},\cdots,\vec{v_n}$, and let $\mathcal{E}$ be some other ...
6
votes
3answers
268 views

Square root-related calculations with matrices

If $\mathbf A$ is an $n \times n$ matrix such that $\mathbf A^6 = \mathbf I_n$ (the identity matrix), is it true that either $\mathbf A^3 = \mathbf I_n$ or $\mathbf A^3 = \mathbf -I_n$? I'm ...
0
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0answers
74 views

Equivalence proofs concerning matrices and systems of linear equations

Let $\mathbf A$ be the augmented $m × (n + 1)$ matrix of a system of $m$ linear equations with $n$ unknowns. Let $\mathbf B$ be the $m × n$ matrix obtained from $\mathbf A$ by removing the last ...
0
votes
1answer
32 views

Is the observed widest-width of an oblate sphere constant under all rotations?

This is something which I feel intuitively is true but I'm having trouble finding a way of proving it mathematically. Given an oblate sphere, or ellipsoid, with equation $$x^2+y^2+(z^2 / c^2)=1, ...
0
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0answers
22 views

Geometric Interpretation of B-Matrix

I'm stuck a bit at explaining the "geometry" behind the B-matrix that results from the following transformation: Let $\mathcal{B} =(\vec{v_1}, \vec{v_2}, \vec{v_3})$ be any basis of $\mathbb{R}^{3}$ ...
0
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1answer
242 views

Constructing the Matrix B “column by column”

I'm going through the various ways to construct a B-matrix of a linear transformation and I'm hitting a snag with one of the methods. We have $A = \begin{pmatrix} -3 & 4 \\ 4 & 3 \\ ...