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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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Product of matrices; MAPLE giving a strange answer

Either my brain is seriously fried up right now or the computer is wrong. If I have a matrix $\begin{bmatrix} 4 & -2\\ 2 & -1 \\ 0 & 0 \end{bmatrix}$ multiply by its transpose $\begin{bmatrix} 4 &2 &0 \\ -2&-1 &0 \end{bmatrix}$, I…
Hed
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Show orthogonal upper triangular matrix is diagonal

How do I go about showing that an upper triangular matrix with orthogonal columns has to be a diagonal matrix? I know that the property $M^\text{T}M = I$ should be used but I'm not sure how.
quantum
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Two basic examples of trace diagrams?

In the wikipedia entry on Trace Diagrams (see http://en.wikipedia.org/wiki/Trace_diagram), the statement is made that "The simplest trace diagrams represent the trace and determinant of a matrix". Could anyone provide me with the graphic…
Javier Arias
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Is $\left(\sum_{n=0}^\infty\frac{1}{n!}A^n\right)v=\sum_{n=0}^\infty\frac{1}{n!}(A^nv)$?

Suppose we have a convergent power series of matrices $$A=\sum_{n=0}^\infty a_nX^n,$$ for $X\in M_n(\mathbb{C})$. Is it true that if $v\in\mathbb{R}^n$ then $$Av=\sum_{n=0}^\infty a_n(X^nv)?$$ If no, under what conditions is it true? I am…
Grain
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Are there real solutions to $\exp(X)=-I$?

As we know, the equation $$e^x=-1,\quad x\in\mathbb{C}$$ has no real solution (in fact $x=i\pi+2ki\pi$, $k\in\mathbb{Z}$). I am now considering the generalization of this question to $2\times 2$ matrices: Question: Is there a real matrix $X\in…
Scott
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What is the matrix used to find the reflected (x, y) coordinate in the line y=mx?

I hope this makes sense, I'm essentially looking for a matrix in which you can just substitute in the gradient m from y=mx and find the reflected coordinates? If this doesn't make any sense please say why? Regards Tom
Thomas Winkworth
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Matrix norm question, normal matrices

Let $A^*$ denote the complex conjugate transpose of a matrix $A$. In the Euclidean norm (operator norm), if $$\|A^*A+AA^*\|=2\,\|A^*A\|$$ prove/disprove that $A$ is normal.
PeterA
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Commutativity of the square root of matrices

Let $A, B \in \mathbb{R}^{n \times n}$ two positive definite matrices such that $AB = BA$, that is $A$ commutes with $B$. It is easy to prove that $A^{1/2}$ commutes with $A$, indeed $AA^{1/2} = A^{3/2}=A^{1/2}A$, but I am wondering whether it is…
gosbi
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A question about matrices such that the elements in each row add up to $1$.

Let $A$ be an invertible $10\times 10$ matrix with real entries such that the sum of each row is $1$. Then is the sum of the entries of each row of the inverse of $A$ also $1$? I created some examples, and found the proposition to be true. I also…
algebraically_speaking
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What is the expected root mean square determinant of an $n\times n$ matrix?

The expected mean determinant of random $n\times n$ matrices of $0$'s and $1$'s is $0$. What is the expected root mean square determinant? e.g. $\frac{\sqrt{3}}{2\sqrt{2}}$ for a $2\times 2$
Angela Pretorius
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When do two $n \times n$ matrices $A, B$ have the property that $AB = BA$?

As we all know that two $n \times n$ matrices $A, B$ need not have the relation $AB = BA.$ But when do two $n \times n$ matrices have such a property?
Yes
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Oscillation matrix whose $n-1$ power is totally positive

Let $A$ be a $n\times n$ real matrix. If all the sub-determinant of $A$ is $\geq0$, then $A$ is called totally non-negative. If all the sub-determinant of $A$ is $>0$, then $A$ is called totally positive. If for a totally non-negative matrix $A$,…
xldd
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Is there a 3D equivalent of a 2D matrix?

Just thinking, is there a 3D 'equivalent' of a matrix. I know it's possible to get matrices that only have one row or column (i.e. vectors) thus making there a sort of 1D equivalent, but is there a 3D equivalent of a matrix?
Pharap
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How to simplify a determinant

If we have a matrix like: $A_{12\times 12}=\left[ \begin{matrix} {{\Gamma }_{1}} & \mathbf{0_{5\times 6}} \\ {{\Gamma }_{2}} & {{\mathbf{G}}_{2}} \\ \mathbf{0_{5\times6}} & {{\mathbf{G}}_{1}} \\ \end{matrix} \right]$ where the matrices…
Saj_Eda
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derivative of projector onto positive semidefinite cone is independent of the choice of orthogonal matrices

We know that when projector onto positive semidefinite cone $K$ is differentiable at $X$ then its derivative is $D \Pi_K(X)[H]= Q f^{(1)}(\lambda) \circ (Q^T H Q) Q^T$, where $X=Q \lambda Q^T$ is eigenvalue decomposition of $X$, $f(x)=[x]_+$, $…
Daisy
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