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.

A square matrix $A$ is diagonalisable if there is an invertible matrix $P$ such that $P^{-1}AP$ is a diagonal matrix. One can view $P$ as a change of basis matrix so that, if $A$ is viewed as the standard matrix of a linear map $T$ from a vector space to itself in some basis, it is equivalent to say there exists an ordered basis such that the standard matrix of $T$ is diagonal. Diagonal matrices present the eigenvalues of the corresponding linear transformation along its diagonal. A square matrix that is not diagonalizable is called defective.

Not every matrix is diagonalisable over $\mathbb{R}$ (i.e. only allowing real matrices $P$). For example, $$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$$

Diagonalization can be used to compute the powers of a matrix $A$ efficiently, provided the matrix is diagonalizable.

Diagonalization Procedure :

Let $A$ be the $n×n$ matrix that you want to diagonalize (if possible).

  • Find the characteristic polynomial $p(t)$ of $A$.
  • Find eigenvalues $λ$ of the matrix $A$ and their algebraic multiplicities from the characteristic polynomial $p(t)$.
  • For each eigenvalue $λ$ of $A$, find a basis of the eigenspace $E_λ$. If there is an eigenvalue $λ$ such that the geometric multiplicity of $λ$, $dim(E_λ)$, is less than the algebraic multiplicity of $λ$, then the matrix $A$ is not diagonalizable. If not, $A$ is diagonalizable, and proceed to the next step.

  • If we combine all basis vectors for all eigenspaces, we obtained $n$ linearly independent eigenvectors $v_1,v_2,…,v_n$.

  • Define the nonsingular matrix $$P=[v_1\quad v_2\quad …\quad v_n]$$
  • Define the diagonal matrix $D$, whose $(i,i)$-entry is the eigenvalue $λ$ such that the $i^{th}$ column vector $v_i$ is in the eigenspace $E_λ$.
  • Then the matrix A is diagonalized as $$P^{−1}AP=D$$

References:

Diagonal Matrix on Wikipedia

Matrix Diagonalization on Wolfram MathWorld

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Changing base to "diagonalize" a rectangular matrix

I have to say whether exists two bases, respectively, for $\mathbb{R}^5$ and $\mathbb{R}^4$ for which the matrix $$ A = \begin{bmatrix} 1 & 3 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 & -1 \\ -1 & -3 & -1 & -1 & -1 \\ 1 & 5 & -1 & -1 &…
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What does diagonalizing one of the pair of anticommuting matrices do to the other?

If two Hermitian matrices anticommute, $MN=-NM$, then if we diagonalize $M$, does $N$ take a particular form? Block antidiagonal, or strictly antidiagonal? From a few really old answers on the site this is what I have gathered: When…
Cain
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Congruent block diagonalization of an anti-symmetric matrix by indefinite orthogonal matrix

I want to use symmetries of a non-degenerate real symmetric bilinear form, $S$ to find a good basis to represent a non-degenerate real skew-symmetric bilinear form, $A$. Setup Suppose, my symmetric bilinear form can be represented by a $2n\times 2n$…
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Prove matrix with continuous random entries can be diagonalized

Let $X\in\mathbb{R}^{n\times n}$ with $n\in\mathbb{N}^{+}$ be a random matrix with entries $x_{ij}$ continuous random variables for any $i,j\in\{1,...,n\}$ such that $i\neq j$, e.g., $x_{ij}\sim\mathcal{U}(0,1)$ uniformly distributed. The rowsum of…
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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 following situation. Let $\{v_i\}$ be a basis for…
eigenvalue
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Diagonalized matrix not zero on sidelines

I try to diagonalize: $Y = \left(\begin{array}{ccc} X & -X & 0\\ -X & X & 0\\ 0 & 0 & 2\,X \end{array}\right)$ after doing the general diagonalization formula: $Y_{\mathrm{Diagonalized}} = V^{-1}\,\Lambda\,V$ I get: $Y_{\mathrm{Diagonalized}} =…
Leon
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Diagonalize a generic non-symmetric complex matrix via sandwitching with two unitary matrices.

Suppose $M$ is a generic rank-$N$ complex-valued non-symmetric matrix, so its entry $M_{ij} \in \mathbb{C}$. How to prove mathematically rigorously that it is always possible to (1) find unitary matrix $U$ and $W$ (so $U U^\dagger =W W^\dagger…
wonderich
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how to solve this third degree characteristic polynomial?

(this exercise is like the previous question I've written today, but now I have a 3-by-3 matrix (with a real parameter $k$). I need to say where the matrix could be diagonalized, if it's possible. (meaning, find the scalar k, in order to diagonalize…
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I'm having a problem when trying to diagonalize a matrix (with a parameter), how can I solve?

I have this matrix, and I need to say where the matrix could be diagonalized. Alpha is a real parameter. $$ \begin{matrix} 1 & \alpha \\ 3 & 1 \\ \end{matrix} $$ I found characteristic polynomial, $ det(a) = (t*I - A) $ here's the final…
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Unitarity similarity transformation

Prove that $R$ and $R^{\dagger}$ can be diagonalized by a common unitary similarity transformation if $R^{\dagger}$ is commutable with $R$. Let $R = SMS^{-1}$, where $M$ is diagonal and $S$ is unitary. $$R^{\dagger} = (SMS^{-1})^{\dagger} =…
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Does the matrix $U$ that diagonalize another matrix $M$ also diagonalize the derivative of $M$?

I have a $3\times3$ Hermitian matrix $M$. All the elements of $M$ are a function of variable $x$. There is a unitary matrix $U$ that diagonalize $M$, i.e. $$ D=U^\dagger MU $$ I wonder if the same $U$ matrix always diagonalize $\frac{dM}{dx}$?
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Smart way $2\times 2$ JNF

I wanted to find a fast way to construct the JNF (with basis transformation) of a $2\times 2$ Matrix which is not diagonalizable, which means that we need to have one eigenvalue with algebraic multiplicity 2 but geometric multiplicity being only…
user66906
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Error incurred upon truncating a diagonal matrix to only the k largest eigenvalues.

Say I have a diagonal matrix (A) of dim n x n. What is the error incurred if I approximate it with only its k-largest eigenvalues? I am using the 1-norm. I am trying to quantify the error in the following manner: Err=||A||-||A1||, where ||A|| is…
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Set of diagonalizing matrices

Let $M \in \mathbb{R}^{n \times n}$ be a positive definite matrix. I would like to characterize the set $$ \mathcal{A} := \{A \in \mathbb{R}^{n\times n} : AMA' \text{ is diagonal and invertible} \}. $$ Clearly one element of $\mathcal{A}$ is the…
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If $A$ is a real symmetric matrix with full rank and only simple roots, what can be said about $|B\otimes I+I\otimes B|$, for $B = \sum_k a_k A^k$?

I'm considering a situation in which $A, B > 0$ are real $(p\times p)$ symmetric matrices where $A$ has $p$ distinct eigenvalues $\{\lambda_i\}_{i=1}^p$, and $AB=BA$. I'm interested in the level of constraint placed on the matrix $B$ given these…
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