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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68
votes
16answers
13k views

If $AB = I$ then $BA = I$

If $A$ and $B$ are square matrices such that $AB = I$ where $I$ is identity matrix. Show that $BA = I$. I do not understand anything more than the following. Elementary row operations. Linear ...
13
votes
8answers
3k views

How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?

Let A and B be two matrices which can be multiplied. Then $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$ I proved $\operatorname{rank}(AB) \leq ...
12
votes
3answers
1k views

Why do the $n \times n$ non-singular matrices form an “open” set?

Why is the set of $n\times n$ real, non-singular matrices an  open subset of the set of all $n\times n$ real matrices? I don't quite understand what "open" means in this context. Thank you.
25
votes
4answers
12k views

How do I tell if matrices are similar?

I have two $2\times 2$ matrices, $A$ and $B$, with the same determinant. I want to know if they are similar or not. I solved this by using a matrix called $S$: $$\left(\begin{array}{cc} a& b\\ ...
4
votes
8answers
911 views

How to calculate the following determinants (all ones, minus $I$)

How do I calculate the determinant of the following $n\times n$ matrices $ \left[ \begin {matrix} 0 & 1 & \ldots & 1 \\ 1 & 0 & \ldots & 1 \\ \vdots & \vdots & ...
14
votes
4answers
564 views

Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$

I was reviewing some matrices and found this interesting if $r = \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}$ then $rr=-I$, also $$\exp{(\theta r)} = \cos\theta I + \sin\theta r$$ No wonder, the ...
7
votes
2answers
468 views

The Center of $\operatorname{GL}(n,k)$

The given question: Let $k$ be a field and $n \in \mathbb{N}$. Show that the centre of $\operatorname{GL}(n, k)$ is $\lbrace\lambda I\mid λ ∈ k^∗\rbrace$. I have spent a while trying to prove this ...
56
votes
9answers
11k views

Intuition behind Matrix Multiplication

If I multiply two numbers, say $3$ and $5$, I know it means add $3$ to itself $5$ times or add $5$ to itself $3$ times. But If I multiply two matrices, what does it mean ? I mean I can't think it ...
6
votes
4answers
201 views

$M,N\in \Bbb R ^{n\times n}$, show that $e^{(M+N)} = e^{M}e^N$ given $MN=NM$

I am working on the following problem. Let $e^{Mt} = \sum\limits_{k=0}^{\infty} \frac{M^k t^k}{k!}$ where $M$ is an $n\times n$ matrix. Now prove that $$e^{(M+N)} = e^{M}e^N$$ given that $MN=NM$, ie ...
19
votes
2answers
6k views

Motivation behind Definition of Matrix Multiplication

I have just watched the first half of the 3rd lecture of Gilbert Strang on the open course ware with link: http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/ It ...
6
votes
2answers
1k views

How to show if $ \lambda$ is an eigenvalue of $AB^{-1}$, then $ \lambda$ is an eigenvalue of $ B^{-1}A$?

Statement: If $ \lambda$ is an eigenvalue of $AB^{-1}$, then $ \lambda$ is an eigenvalue of $ B^{-1}A$ and vice versa. One way of the proof. We have $B(B^{-1}A ) B^{-1} = AB^{-1}. $ Assuming $ ...
11
votes
1answer
1k views

Probability that a random binary matrix is invertible?

What is the probability that a random $\{0,1\}$, $n \times n$ matrix is invertible? Assume the 0 and 1 are each present in an entry with probability $\frac{1}{2}$. Is there an explicit formula as a ...
11
votes
4answers
3k views

Square root of a matrix

Under what conditions does a matrix $A$ have a square root? I saw somewhere that this is true for Hermitian positive definite matrices(whose definition I just looked up). Moreover, is it possible ...
10
votes
4answers
3k views

Prove that $A+I$ is invertible if $A$ is nilpotent [duplicate]

Possible Duplicate: Units and Nilpotents Given $A^{2012}=0$ prove that $A+I$ is invertible and find an expression for $(A+I)^{-1}$ in terms of $A$. ($I$ is the identity matrix).
2
votes
2answers
233 views

Determining eigenvalues, eigenvectors of $A\in \mathbb{R}^{n\times n}(n\geq 2)$.

Let $a$ and $b$ be distinct nonzero real numbers and let $A\in \mathbb{R}^{n\times n}(n\geq 2)$ with each diagonal entry equal to $a$ and each off-diagonal entry equal to $b$. Determine all ...
11
votes
1answer
4k views

Are the eigenvalues of $AB$ equal to the eigenvalues of $BA$? (Citation needed!)

First of all, am I being crazy in thinking that if $\lambda$ is an eigenvalue of $AB$, where $A$ and $B$ are both $N \times N$ matrices (not necessarily invertible), then $\lambda$ is also an ...
18
votes
2answers
2k views

necessary and sufficient condition for trivial kernel of a matrix over a commutative ring

In answering Do these matrix rings have non-zero elements that are neither units nor zero divisors? I was surprised how hard it was to find anything on the Web about the generalization of the ...
7
votes
2answers
889 views

Proof: $\mathrm{adj}(\mathrm{adj}(A)) = (\mathrm{det}(A))^{n-2} \cdot A$ for $A \in \mathbb{R}^{n\times n}$

I had my exam of linear algebra today and one of the questions was this one. Given $ A \in \mathbb{R}^{n \times n}$, prove that: $$\mathrm{adj}(\mathrm{adj}(A)) = (\mathrm{det}(A))^{n-2} \cdot A.$$ ...
2
votes
2answers
213 views

Computing determinant of a specific matrix.

How to calculate the determinant of $$ A=(a_{i,j})_{n \times n}=\left( \begin{array}{ccccc} a&b&b& \cdots & b\\ b& a& b& \cdots& b\\ \vdots& \vdots& \vdots& ...
2
votes
3answers
1k views

I don't understand why the inverse is this?

my question is related to matrix inverting and Hill cipher(you don't have to know what it is to help me) My teacher gave me an example. First we have a matrix (the key matrix) that multiplied by a ...
31
votes
6answers
2k views

How many connected components does $\mathrm{GL}_n(\mathbb R)$ have?

I've noticed that $\mathrm{GL}_n(\mathbb R)$ is not a connected space, because if it were $\det(\mathrm{GL}_n(\mathbb R))$ (where $\det$ is the function ascribing to each $n\times n$ matrix its ...
7
votes
5answers
808 views

Fast(est) and intuitive ways to look at matrix multiplication?

Most of the time I see matrix multiplication presented and defined, as a seemingly arbitrary sequence of operations. For example, the textbook I'm currently reading for a linear algebra course defines ...
4
votes
3answers
839 views

Why does a diagonalization of a matrix B with the basis of a commuting matrix A give a block diagonal matrix?

I am trying to understand a proof concerning commuting matrices and simultaneous diagonalization of these. It seems to be a well known result that when you take the eigenvectors of A as a basis and ...
22
votes
5answers
33k views

Inverse of the sum of matrices

I have two square matrices - $A$ and $B$. $A^{-1}$ is known and I want to calculate $(A+B)^{-1}$. Are there theorems that help with calculating the inverse of the sum of matrices? In general case ...
17
votes
6answers
10k views

if eigenvalues are positive, is the matrix positive definite?

If the matrix is positive definite, then all its eigenvalues are strictly positive. Is the converse also true? That is, if the eigenvalues are strictly positive, then matrix is positive definite? Can ...
21
votes
5answers
8k views

Is the rank of a matrix the same of its transpose? If yes, how can I prove it?

I am auditing a Linear Algebra class, and today we were taught about the rank of a matrix. The definition was given from the row point of view: "The rank of a matrix A is the number of non-zero ...
4
votes
2answers
782 views

Jordan Canonical Form determination

Why and how is the Jordan Canonical form of a matrix in $M_3(\mathbb C)$ fully determined by its characteristic and minimal polynomials? And why does it fail for $n >3$?  Thanks.
15
votes
5answers
679 views

Every integer vector in $\mathbb R^n$ with integer length is part of an orthogonal basis of $\mathbb R^n$

Suppose $\vec x$ is a (non-zero) vector with integer coordinates in $\mathbb R^n$ such that $\|\vec x\| \in \mathbb Z$. Is it true that there is an orthogonal basis of $\mathbb R^n$ containing $\vec ...
8
votes
3answers
647 views

Trace of powers of a nilpotent matrix

Let $A$ be an $n\times n$ complex nilpotent matrix. Then we know that because all eigenvalues of $A$ must be 0, it follows that $\text{tr}(A^n)=0$ for all positive integers $n$. What I would like to ...
14
votes
5answers
2k views

$\sin(A)$, where $A$ is a matrix

If $A$ is an $n\times n$ matrix with elements $a_{ij}$ $i=$i'th row, $j=$j'th column. Then $e^A$ is also a matrix as can be seen by expanding it in a power series.Is $e^A$ always convergent and ...
3
votes
4answers
456 views

A be a $3\times 3$ matrix over $\mathbb {R}$ such that $AB =BA$ for all matrices $B$. what can we say about such matrix $A$

Let $A$ be a $3\times 3$ matrix over $\mathbb {R}$ such that $AB =BA$ for all matrices $B$ over $\mathbb {R}$ then what can we say about such matrix $A$. or such matrix $A$ must be orthogonal matrix? ...
14
votes
1answer
510 views

Why does the inverse of the Hilbert matrix have integer entries?

Let $A$ be the $n\times n$ matrix given by $ A_{ij}=\frac{1}{i + j - 1}$. I need to show that $A$ is invertible and the inverse has integer entries. I was able to show that $A$ is invertible. How do I ...
5
votes
2answers
8k views

extracting rotation, scale values from 2d transformation matrix

How can I extract rotation and scale values from a 2D transformation matrix? ...
7
votes
4answers
11k views

Similar matrices have the same eigenvalues with the same geometric multiplicity

Suppose $A$ and $B$ are similar matrices. Show that $A$ and $B$ have the same eigenvalues with the same geometric multiplicities. Similar matrices: Suppose $A$ and $B$ are $n\times n$ matrices ...
6
votes
3answers
907 views

Cayley-Hamilton theorem on square matrices

Can anyone help me by giving the proof of the Cayley-Hamilton theorem? It states that every square matrix $A$ satisfies its own characteristic equation: $p_{A}(A)=0$. I could prove it when $A$ has ...
0
votes
1answer
266 views

Constructing a graph from a degree sequence

Let's say I'm given several degree sequences like {4,3,3,2,2} {3,3,3,3} {5,3,3,2,2,1} I can find the number of edges using the handshaking lemma But how do I construct a graph just given these ...
24
votes
5answers
10k views

Importance of rank of a matrix

What is the importance of rank of a matrix ? I know that rank of a matrix is the number of linearly independent rows/columns (whichever is smaller). Why is it a problem if a matrix is rank ...
30
votes
4answers
2k views

Solutions to the matrix equation $\mathbf{AB-BA=I}$ over general fields

Some days ago, I was thinking on a problem, which states that $AB-BA=I$ does not have a solution in $M_{n\times n}(\mathbb R)$ and $M_{n\times n}(\mathbb C)$. (Here $M_{n\times n}(\mathbb F)$ denotes ...
30
votes
3answers
1k views

Why, historically, do we multiply matrices as we do?

Multiplication of matrices — taking the dot product of the $i$th row of the first matrix and the $j$th column of the second to yield the $ij$th entry of the product — is not a very ...
17
votes
4answers
18k views

Inverse of an invertible triangular matrix (either upper or lower ) is triangular of the same kind

How can we prove that the inverse of an upper (lower) triangular matrix is upper (lower) triangular?
21
votes
4answers
766 views

Prove that the set of $n$-by-$n$ real matrices with positive determinant is connected

Math people: In the fourth edition of Strang's "Linear Algebra and its Applications", page 230, he poses the following problem (I have changed his wording): show that if $A \in \mathbf{R}^{n \times ...
14
votes
3answers
979 views

Why is the determinant of a symplectic matrix 1?

suppose $A \in M_{2n}(\mathbb{R})$. and$$J=\begin{pmatrix} 0 & E_n\\ -E_n&0 \end{pmatrix}$$ where $E_n$ represents identity matrix. if $A$ satisfies $$AJA^T=J$$ How to figure out ...
4
votes
3answers
691 views

What is step by step logic of pinv (pseudoinverse)?

So we have a matrix $A$ size of $M \times N$ with elements $a_{i,j}$. What is a step by step algorithm that returns the Moore-Penrose inverse $A^+$ for a given $A$ (on level of ...
18
votes
3answers
720 views

Determinant of transpose?

$$\det(A^T) = \det(A)$$ Using the geometric definition of the determinant as the area spanned by the columns could someone give a geometric interpretation of the property? Thanks!
13
votes
4answers
4k views

Show that the determinant of $A$ is equal to the product of its eigenvalues.

Show that the determinant of a matrix $A$ is equal to the product of its eigenvalues $\lambda_i$. So I'm having a tough time figuring this one out. I know that I have to work with the characteristic ...
6
votes
4answers
244 views

how many unique patterns exist for a NxN grid

I'm trying to figure out if there is a way to determine how many unique patterns exist for a given NxN grid if you choose N points on the grid. For example, for a 2x2 grid we can get two unique ...
6
votes
1answer
283 views

Inequality concerning inverses of positive definite matrices

I don't find a way to prove this: given $A$, $B$, symmetric and positive definite: $$A>B \Rightarrow A^{-1} < B^{-1},$$ where $A>B$ means that $A-B$ is positive definite.
9
votes
2answers
16k views

Transpose of inverse vs inverse of transpose

I can't seem to find the answer to this using Google. Is the transpose of the inverse of a square matrix the same as the inverse of the transpose of that same matrix? Thanks!
7
votes
1answer
8k views

Elegant proofs that similar matrices have the same characteristic polynomial?

It's a simple exercise to show that two similar matrices has the same eigenvalues and eigenvectors (my favorite way is noting that they represent the same linear transformation in different bases). ...
6
votes
3answers
4k views

Is there a 3-dimensional “matrix” by “matrix” product?

Is it possible to multiply A[m,n,k] by B[p,q,r]? Does the regular matrix product have generalized form? I would appreciate it if you could help me to find out some tutorials online or mathematical ...