Questions tagged [matrices]

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate and adjoint, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), invariant factors, quadratic forms, etc. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

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609
votes
13answers
115k views

What's an intuitive way to think about the determinant?

In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t ...
266
votes
26answers
60k views

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

If $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$. I do not understand anything more than the following. Elementary row operations. Linear ...
246
votes
2answers
13k views

How does one prove the determinant inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

Let $\,A,B,C\in M_{n}(\mathbb C)\,$ be Hermitian and positive-definite matrices such that $A+B+C=I_{n},$ where $I_{n}$ is the identity matrix. Show that $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\...
201
votes
0answers
11k views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ $K_1=\left(\...
196
votes
8answers
339k views

Is a matrix multiplied with its transpose something special?

In my math lectures, we talked about the Gram-Determinant where a matrix times its transpose are multiplied together. Is $A A^\mathrm T$ something special for any matrix $A$?
191
votes
6answers
13k views

Why does this matrix give the derivative of a function?

I happened to stumble upon the following matrix: $$ A = \begin{bmatrix} a & 1 \\ 0 & a \end{bmatrix} $$ And after trying a bunch of different examples, I noticed the ...
175
votes
6answers
9k views

How could we define the factorial of a matrix?

Suppose I have a square matrix $\mathsf{A}$ with $\det \mathsf{A}\neq 0$. How could we define the following operation? $$\mathsf{A}!$$ Maybe we could make some simple example, admitted it makes any ...
168
votes
8answers
90k views

What are the Differences Between a Matrix and a Tensor?

What is the difference between a matrix and a tensor? Or, what makes a tensor, a tensor? I know that a matrix is a table of values, right? But, a tensor?
144
votes
14answers
46k 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 ...
137
votes
1answer
10k views

Is the following matrix invertible?

$$\begin{bmatrix} 1235 &2344 &1234 &1990\\ 2124 & 4123& 1990& 3026 \\ 1230 &1234 &9095 &1230\\ 1262 &2312& 2324 &3907 \end{bmatrix}$$ Clearly, its ...
132
votes
2answers
30k views

What is the geometric interpretation of the transpose?

I can follow the definition of the transpose algebraically, i.e. as a reflection of a matrix across its diagonal, or in terms of dual spaces, but I lack any sort of geometric understanding of the ...
110
votes
10answers
191k 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 $B^{-...
104
votes
20answers
198k views

Calculate Rotation Matrix to align Vector A to Vector B in 3d?

I have one triangle in 3d space that I am tracking in a simulation. Between time steps I have the the previous normal of the triangle and the current normal of the triangle along with both the current ...
96
votes
14answers
7k views

Why, intuitively, is the order reversed when taking the transpose of the product?

It is well known that for invertible matrices $A,B$ of the same size we have $$(AB)^{-1}=B^{-1}A^{-1} $$ and a nice way for me to remember this is the following sentence: The opposite of putting on ...
96
votes
14answers
30k views

Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix

I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I've read the proof at http://en.wikipedia.org/wiki/Rank_of_a_linear_transformation and I understand the ...
90
votes
7answers
108k 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 ...
89
votes
7answers
111k views

Physical meaning of the null space of a matrix

What is an intuitive meaning of the null space of a matrix? Why is it useful? I'm not looking for textbook definitions. My textbook gives me the definition, but I just don't "get" it. E.g.: I think ...
86
votes
6answers
170k 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?
84
votes
3answers
18k 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 ...
79
votes
1answer
50k views

Simultaneously Diagonalizable Proof

Two $n\times n$ matrices $A, B$ are said to be simultaneously diagonalizable if there is a nonsingular matrix $S$ such that both $S^{-1}AS$ and $S^{-1}BS$ are diagonal matrices. a.) Show that ...
78
votes
3answers
17k views

Why did no student correctly find a pair of $2\times 2$ matrices with the same determinant and trace that are not similar?

I gave the following problem to students: Two $n\times n$ matrices $A$ and $B$ are similar if there exists a nonsingular matrix $P$ such that $A=P^{-1}BP$. Prove that if $A$ and $B$ are two ...
78
votes
10answers
5k views

Why does Friedberg say that the role of the determinant is less central than in former times?

I am taking a proof-based introductory course to Linear Algebra as an undergrad student of Mathematics and Computer Science. The author of my textbook (Friedberg's Linear Algebra, 4th Edition) says in ...
78
votes
9answers
121k views

When is matrix multiplication commutative?

I know that matrix multiplication in general is not commutative. So, in general: $A, B \in \mathbb{R}^{n \times n}: A \cdot B \neq B \cdot A$ But for some matrices, this equations holds, e.g. A = ...
76
votes
9answers
112k 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, ...
76
votes
4answers
48k views

Importance of matrix rank

What is the importance of the rank of a matrix? I know that the rank of a matrix is the number of linearly independent rows or columns (whichever is smaller). Why is it a problem if a matrix is rank ...
70
votes
3answers
59k views

Determinant of a non-square matrix

I wrote an answer to this question based on determinants, but subsequently deleted it because the OP is interested in non-square matrices, which effectively blocks the use of determinants and thereby ...
69
votes
7answers
7k views

Why is it important for a matrix to be square?

I am currently trying to self-study linear algebra. I've noticed that a lot of the definitions for terms (like eigenvectors, characteristic polynomials, determinants, and so on) require a square ...
69
votes
8answers
5k views

Why does this “miracle method” for matrix inversion work?

Recently, I answered this question about matrix invertibility using a solution technique I called a "miracle method." The question and answer are reproduced below: Problem: Let $A$ be a matrix ...
68
votes
8answers
18k views

Why are rotational matrices not commutative?

Is there any intuition why rotational matrices are not commutative? I assume the final rotation is the combination of all rotations. Then how does it matter in which order the rotations are applied?
68
votes
19answers
7k views

What are some applications of elementary linear algebra outside of math?

I'm TAing linear algebra next quarter, and it strikes me that I only know one example of an application I can present to my students. I'm looking for applications of elementary linear algebra outside ...
62
votes
26answers
8k views

What are some mathematically interesting computations involving matrices?

I am helping designing a course module that teaches basic python programming to applied math undergraduates. As a result, I'm looking for examples of mathematically interesting computations involving ...
62
votes
6answers
68k 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!
61
votes
8answers
11k 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 ...
60
votes
11answers
10k views

What is the largest eigenvalue of the following matrix?

Find the largest eigenvalue of the following matrix $$\begin{bmatrix} 1 & 4 & 16\\ 4 & 16 & 1\\ 16 & 1 & 4 \end{bmatrix}$$ This matrix is symmetric and, thus, the ...
57
votes
3answers
81k views

A matrix and its transpose have the same set of eigenvalues

Let $ \sigma(A)$ be the set of all eigenvalues of $A$. Show that $ \sigma(A) = \sigma(A^T)$ where $A^T$ is the transposed matrix of $A$.
57
votes
3answers
3k views

Alice and Bob play the determinant game

Alice and Bob play the following game with an $n \times n$ matrix, where $n$ is odd. Alice fills in one of the entries of the matrix with a real number, then Bob, then Alice and so forth until the ...
54
votes
2answers
16k views

What do eigenvalues have to do with pictures?

I am trying to write a program that will perform OCR on a mobile phone, and I recently encountered this article : Can someone explain this to me ?
53
votes
4answers
6k 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 ...
52
votes
4answers
4k views

Must eigenvalues be numbers?

This is more a conceptual question than any other kind. As far as I know, one can define matrices over arbitrary fields, and so do linear algebra in different settings than in the typical freshman-...
52
votes
6answers
12k views

Intuitive explanation of a positive semidefinite matrix

What is an intuitive explanation of a positive-semidefinite matrix? Or a simple example which gives more intuition for it rather than the bare definition. Say $x$ is some vector in space and $M$ is ...
52
votes
5answers
57k 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\\ ...
51
votes
3answers
31k views

Matrices commute if and only if they share a common basis of eigenvectors?

I've come across a paper that mentions the fact that matrices commute if and only if they share a common basis of eigenvectors. Where can I find a proof of this statement?
51
votes
5answers
51k 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 ...
49
votes
5answers
38k 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 ...
49
votes
9answers
3k views

If the field of a vector space weren't characteristic zero, then what would change in the theory?

In the book of Linear Algebra by Werner Greub, whenever we choose a field for our vector spaces, we always choose an arbitrary field $F$ of characteristic zero, but to understand the importance of the ...
48
votes
7answers
32k views

The relation between trace and determinant of a matrix

Let $M$ be a symmetric $n \times n$ matrix. Is there any equality or inequality that relates the trace and determinant of $M$?
48
votes
10answers
92k views

How to show that $\det(AB) =\det(A) \det(B)$?

Given two square matrices $A$ and $B$, how do you show that $$\det(AB) = \det(A)\det(B)$$ where $\det(\cdot)$ is the determinant of the matrix?
48
votes
1answer
3k views

Why is that if every row of a matrix sums to 1, then the rows of the inverse matrix sums to 1 too?

Why is that if every row of a matrix sums to $1$ then the rows of its inverse matrix sum to $1$ too? For example, consider $$A=\begin{pmatrix} 1/3 & 2/3 \\ 3/4 & 1/4 \end{pmatrix}$$ then ...
47
votes
2answers
2k views

Polynomial equations $p(A, B) = 0$ for matrices that ensure $AB = BA$

Let $k$ be a field with characteristic different from $2$, and $A$ and $B$ be $2 \times 2$ matrices with entries in $k$. Then we can prove, with a bit art, that $A^2 - 2AB + B^2 = O$ implies $AB = BA$,...
46
votes
4answers
7k views

What exactly is a matrix?

I know how basic operations are performed on matrices, I can do transformations, find inverses, etc. But now that I think about it, I actually don't "understand" or know what I've been doing all this ...