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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.

A matrix is a rectangular array of elements, usually numbers or variables, arranged in rows and columns. A matrix with $m$ rows and $n$ columns has $m \times n$ elements and is called an $m$ by $n$ matrix. Matrices are a part of .

Matrices can be added and subtracted. Furthermore, if they have compatible shapes, they can be multiplied. More precisely, given two matrices $A$ and $B$, the matrix $AB$ is defined when the number of columns of $A$ is equal to the number of rows of $B$. In particular, given a natural number $n$, any two matrices $A$ and $B$ with $n$ columns and $n$ rows can be multiplied in both ways (that is, both $AB$ and $BA$ exist).

For questions specifically concerning matrix equations, use the tag.

55954 questions
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Right multiplication on $\mathbb{R}^{\infty}$

If $X=(x_1,x_2,\dots)$ is an infinite real row vector and $A=(a_{ij}),0
amir
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Ask a question about the definition of trace norm.

Suppose $X\in \mathbb{R}^{M\times N}$ $\|X\|_*=\mathrm{trace}(\sqrt{X^*X})=\sum_i^{\min{M,N}}\sigma_i$ where $\sigma_i$ is the singular values of $X$. I know that $\mathrm{trace}({X^*X})=\sum_i^{\min{M,N}}\sigma_i^2$. Thus, the following equation…
Vivian
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What are the probability that A is neither symmetric nor skew-symmetric?

A is a 3 × 3 matrix with entries from the set {–1, 0, 1}. Then the probability that A is neither symmetric nor skew-symmetric is: My thoughts: There can be nine members on a $3*3$ matrix and there are three possibilities for each member. Therefore…
Rajath Radhakrishnan
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Unitary equivalence

How to show that: Let $A=QR$ be a $QR$ factorization, then $A$ is normal if and only if $Q$ and $R^*$ can be simultaneously diagonalized by unitary equivalence.
user95640
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Unitarily similarity

I don't know how to show that $A\in M_2$ is unitarily similar to $\begin{bmatrix}\lambda_1&x\\0&\lambda_2\end{bmatrix}$, where $\lambda_1$ and $\lambda_2$ is its eigenvalues, $x^2=\text{tr}AA^*-|\lambda_1|^2-|\lambda_2|^2$.
user95640
  • 405
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Cardinality of lower triangular Matrix contains rational number

Can anybody explain what is the cardinality of lower triangular Matrix contains rational number and size of matrix is lower than $7$? I don't have any idea where to start and what to do?
Mohammad Razeghi
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Rank of matrices that are the products of square and non-square matrices

Can someone give me a proof for the following statement? $B$ is an invertible $n\times n$ matrix, then the rank of $AB$ is the same as the rank of $A$ for every $m\times n$ matrix $A$. Is the converse true? Thank you all
Mark
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invertibility of non square matrix

I was looking at the properties of invertible matrix, and I came across this statement. If $A$ is $m$ by $n$ matrix and the $rank$ of $A$ is equal to $m$, then $A$ has a right inverse. Can any give me a proof for this statement?
Mark
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Find AB where A= matrix and B=matrix

$A=\left[\begin{array}{ccc} 2&1&0\\0&3&-1 \end{array}\right]$ $B=\left[\begin{array}{cc}a&1\\1&b\\b&a\end{array}\right]$ Matrices Find $AB$
tess
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the identity matrix is unique

let A be an $m\times n$ matrix. Prove that there are unique matrices $I_m$ and $I_n$ such that : $$I_mA=AI_n=A$$ Actually I can't prove the uniqueness here,any help is appreciated. Thanks
dimitris
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Given an n x n matrix A, is there an n x n matrix E such that $A\odot E=A$ and $A \odot F=E$?

For two matrices of dimensions $m \times n$ and $n \times k$, define $C=A\odot B$ to be the matrix with entries $$C_{ij}=\max_{k=1}^n A_{ik} + B_{kj}$$. Given an $n \times n$ matrix $A$, is there an $n \times n$ matrix $E$ such that $A\odot E=A$?…
Jess
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A square matrix $A$ such that $AA_{*j}=0$ for every $j$ must be zero

True or False? If the square matrix $A=[A_{*1},A_{*2},...A_{*n}]$ has the property that $AA_{*j}=0$ for every $j$, then $A=0$. Note that $A_{*j}$ denotes the $j$th column in the matrix $A$. Not really sure how to get started.
mike
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Span of a matrix

Let $A$ be an $m \times n$ matrix, and let $A'$ be the result of a sequence of elementary row operations on $A$. Prove that the rows of $A$ span the same space as the rows of $A'$. I know how to prove that A and A' have both the same rank (rank (A)=…
user61913
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Matrix Multiplication Problem

I'm working on the following problem and I can't seem to come up with the right answer. $$ \text{Let}: A^{-1} = \begin{bmatrix} 1 & 2 & 1 \\ 0 & 3 & 1 \\ 4 & 1 & 2 \\ \end{bmatrix} $$ Find a matrix such…
codedude
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Expressing elementary matrices in terms of each another

How can I express an elementary matrix of type 2 in terms of the product of elementary matrices of types 1 and 3? Just for clarity, here are the types: Type 1: \begin{bmatrix}1&a\\0&1\\ \end{bmatrix} or \begin{bmatrix}1&0 \\a&1\end{bmatrix} Type 2:…
user85362
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