Questions tagged [symmetric-matrices]

A symmetric matrix is a square matrix that is equal to its transpose.

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix $A$ is symmetric if $A^T=A$.

The sum and difference of two symmetric matrices is again symmetric, but this is not always true for the product: given symmetric matrices $A$ and $B$, then $AB$ is symmetric if and only if $A$ and $B$ commute, i.e., if $AB = BA$. So for integer $n$, $A^n$ is symmetric if $A$ is symmetric. If $A^{−1}$ exists, it is symmetric if and only if $A$ is symmetric.

The complex generalization is a hermitian matrix, a square matrix equal to its conjugate transpose. This is often denoted $A=A^{H}$ or $A=\overline{A^T}$; see for more information.

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Linear Algebra - Symmetric matrices and bilinear forms

Let $A \in K^{n,n}$ be symmetric. Define $<\cdot,\cdot> : K^n \times K^n \rightarrow K$ by $$ = x^TAy.$$ Prove that $<\cdot,\cdot>$ is a symmetric bilinear form. Please can anyone help me out here?
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Example of 4x4 symmetric matrix that is not diagonalizable

Is the mentioned matrix possible? If so, what is an example of one?
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Why eigenvector of Symmetric Matrix to be orhogonal for repeated eigenvalues.

I know that for a symmetric matrix, its eigenvector are orthogonal. But for a repeated eigenvalue for a symmetric matrix, why still its eigenvalue must be still orthogonal. I read somewhere that symmetric matrix can not have defective eigenvalue, so…
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A simple $n\times n$ system of linear equations

Can anyone provide the analytical solution of the following $n\times n$ system of linear equations $\underbrace{\begin{pmatrix} a & -b & -b & ... & -b\\ -b & a & -b & ... & -b\\ -b & -b & a & ... & -b\\ : & : & : & ... & : \\ : & : & : & ...…
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Let $A$ be an $n \times n$ matrix. If $A$ is invertible, then $|A| \neq 0$?

Let $A$ be an $n \times n$ matrix. If $A$ is invertible, then $|A| \neq 0$. How I can justifying the answer?
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Are the corner hypercubera polytopes self-dual?

Motivation: The polyhedron whose vertices are seven of the vertices of a cube (four on the bottom and three on top) - called a cubera - is self-dual. Does an analogous construction produce a self-dual polytope in higher dimensions? V-definition of…
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How to calculate the generator matrix,parity check matrix and the maximum likelihood decoding

An unknown encoding device for a binary linear block code has 4 bits of input-data pins and 8 bits of output data pins. If we send the messages $$u_1=(1 1 0 0),\, u_2=(1010),\, u_3=(1001),\, u_4=(0001).$$ to the input, we can observe the following…
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prove that the symmetric projection is always orthogonal.

prove that the symmetric projection is always orthogonal. answer : we have: null(A)=ran(At) then A=At so null(A)=ran(A)
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Product of two symmetric matrices is similar to a symmetric matrix

Let $A,B$ be symmetric real matrices. Is $AB$ similar to a symmetric matrix? This is a problem in my exam. Not a conjecture :v
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How do i prove whether a vector is a symmetric matrix?

How do I prove $ \langle Ax, x \rangle \geq 0 $ if $ A $ is a symmetric matrix? Here $ \langle \cdot, \cdot \rangle $ denotes the dot product.
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Prove that there is a positive real number $\lambda$, so that $A =\lambda B$, for two positive definite square matrices

A and B are positive definite square $n$ $\times$ $n$ matrices. The thereby defined dot products define the same orthogonality relation. Which means: $∀v, w ∈ R^n : v · Aw = 0 ⇔ v · Bw = 0.$ Show that there is a positive real number $\lambda$, so…
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Is $P+(I-P)C$ positive definite?

$P=\frac{1}{n}\mathbf{1}\mathbf{1}^T$($\mathbf{1}=[1,1,\dots,1]^T$), $I\in R^n$ is a indentity matrix, $C$ is a positive definite symmetric matrix. Is $P+(I-P)C$ positive definite? How to prove that? With a random positive definite symmetric matrix…
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How can I prove that $Tr(A^TA) = \sum_{i=1}^n\sum_{j=1}^m a_{ij}^2$ using sigma sum notation?

See above^. Original question is to show that the Trace is equal to the sum of the squares of all the elements I particularly do not understand, in the following proof, the the steps 2 and 3 and how we know that $\sum_{j=1}^{n}(A^TA)_{jj} =…
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If $x$ and $y\in\Bbb{R}^{n}$ are eigenvectors for $\lambda\neq\mu$, respectively, show $x^{T}\cdotp y = 0$

For $x^{T}\cdotp y = 0$, I understand that I can either look at it through matrix multiplication $x^{T}y^{T} = 0$ as you can't do that multiplication. I'm very sure this isn't the right way of looking at it but am unsure how else to think about…
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How to prove idempotent and find the rank

How to solve these three parts Let X be a n×p matrix such that X'X has an inverse. Let A = X(X'X)^-1X'. a) Show that A is idempotent. b) Show that P = In - A is also idempotent. c) Find the rank of P. Thanks :)
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