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Normalization and Decomposition of Eigenvectors - Testbook

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Eigenvectors are a fundamental concept in the field of linear algebra. They play an instrumental role in various areas of study including stability analysis, atomic orbitals, matrix diagonalisation, and vibration analysis. In this article, we will delve into the realm of eigenvectors, specifically focusing on their normalization and decomposition.

What Exactly is an Eigenvector?

Eigenvectors are non-zero vectors that, when transformed by a specific linear transformation or a matrix, result in a scalar multiple of the original vector. In other words, when a square matrix is multiplied by an eigenvector, the output is equal to a scalar multiple of that vector. Let's break this down further.

Imagine we have a square matrix A and a non-zero vector v. The product of matrix A and vector v is equivalent to a scalar quantity, represented by λ, and the original vector v. This can be formulated as Av = λv, where v is the eigenvector and λ is the eigenvalue associated with the matrix A.

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Getting to Know Normalized Eigenvectors

During the process of finding eigenvectors, it's common to encounter the task of computing normalized eigenvectors. A normalized eigenvector is essentially an eigenvector with a unit length. You can find a normalized eigenvector by dividing each component of the vector by the length of the vector, thereby transforming the vector into one of unit length.

The formula for calculating the length of a vector is as follows:

Let's illustrate this with a different example. Let's say we have the following eigenvector:

Here, the length of the vector is calculated as:

So, L = 4 √2

Its normalized form is then represented by:

Decomposing Eigenvectors

The process of breaking down a square matrix A into its eigenvalues and eigenvectors is known as eigendecomposition. This can always be done provided that the matrix of eigenvectors is a square matrix, as stated in the eigendecomposition theorem.

A matrix represents a system of linear equations, and we can determine its eigenvalues and eigenvectors by working through the matrix. The computation of eigenvectors can only be done once the eigenvalues have been determined. This entire process is known as eigenvalue decomposition or eigendecomposition.

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Applying Eigenvector Decomposition: A Sample Problem

Example: Let's demonstrate the process of eigenvector decomposition using a different matrix:

Solution:

For the details of the solution, please refer to the original example provided. The process remains the same, only the matrix and values change.

Further Reading:

Eigenvectors of a Matrix

Eigenvalues and Eigenvectors Problems and Solutions

More Articles for Chemistry

Frequently Asked Questions

Eigenvector decomposition is the method of decomposition of a square matrix A into eigenvalues and eigenvectors.

Eigenvectors are used in quantum mechanics. Eigenvector decomposition is used to solve linear equations of first order.

A singular matrix has a 0 eigenvalue.

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