Questions tagged [diagonalization]

For questions about matrix diagonalization. Diagonalization is the process of finding a corresponding diagonal matrix for a diagonalizable matrix or linear map. This tag is NOT for diagonalization arguments common to logic and set theory.

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For which values is the matrix diagonalizable?

i have the following 2x2 matrix: $\left( \begin{array}{rrr} 0 & 1 \\ -1 & a \\ \end{array}\right)$, with $a ∈ \mathbb{R}$ and then same matrix but with $a ∈ \mathbb{C}$ For which $a$ is the ...
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Is $((w_i, Hv_j))_{ij}$ diagonalizable?

Let $(V, (\cdot,\cdot))$ be a finite-dimensional inner product space, and let $H$ be a Hermitian operator on $V$. We know that $H$ is diagonalizable, and its eigenvalues are real. We consider the ...
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If diagonalizable matrices are not dense over $\Bbb R$, how common are they?

The link: The diagonalizable matrices are not dense in the square real matrices says that diagonalizable matrices are dense over $\Bbb C$ but not over $\Bbb R$. If that's the case, then the logical ...
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Shouldn't the matrix of eigen vectors of a defective matrix be non-invertible?

Is it true that a defective (non-diagonalizable) square matrix has a set of eigen vectors that don't span the whole space? In that case, if we do its singular value decomposition, shouldn't the matrix ...
120 views

Proving a matrix is diagonalizable given eigenvectors and information about characteristic polynomial ranks

Let $A \in \Bbb R^{5 \times 5}$. Let $$v_1=(1,0,0,1,1), \quad v_2=(1,1,0,0,1), \quad v_3=(-1,0,1,0,0)$$ be eigenvectors of $A$. Also, $$\rho(2I-A) \gt\rho(3I-A)$$ and $$A(1,2,2,1,3)^t=(0,4,6,2,6)^t$$ ...
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Change of basis matrix exactly one row away from being correct

For homework, I'm given a matrix $$A = \begin{bmatrix} 3 & 2i & -2i\\ -2i & 0 & -1\\ 2i & -1 & 0 \end{bmatrix}$$ in an hermitian space. We are trying to find an orthonormal ...
Let $A\in \mathbb R^{n\times n}$. Suppose $B$ is symmetric and positive definite, and \begin{equation}\label{eq:sym} A^TB=BA, \end{equation} then $A$ is diagnolizable by a B-orthonormal basis. ...