Eigenvector - Revision history

Rice: /* Eigenspace */

2024-06-10T17:21:36Z

Eigenspace

← Older revision		Revision as of 17:21, 10 June 2024
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	This is the ''characteristic polynomial'' of the A, showing that for any A, there are at most n eigenvalues.		This is the ''characteristic polynomial'' of the A, showing that for any A, there are at most n eigenvalues.

			= Symmetric matrices =

			For any symmetric matrix, one can make an orthonormal basis out of their eigenvectors.

2024-06-10T06:46:13Z

← Older revision		Revision as of 06:46, 10 June 2024
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	(A-\lambda I)\vec{x}=0		(A-\lambda I)\vec{x}=0
	</math>		</math>

			Eigenvectors are the foundation of the [[diagonalization]] technique.

	= Intuition =		= Intuition =

2024-06-10T06:43:40Z

Eigenspace

← Older revision		Revision as of 06:43, 10 June 2024
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	= Eigenspace =		= Eigenspace =
	Given A and <math>\lambda</math>, the '''eigenspace''' is all the eigenvectors ~~with that particular eigenvalue~~.		Given A and <math>\lambda</math>, the '''eigenspace''' is all the eigenvectors.

	Because any linear combination of eigenvectors will yield another eigenvector of the same eigenvalue, the eigenspace is a vector space.		Because any linear combination of eigenvectors will yield another eigenvector of the same eigenvalue, the eigenspace is a vector space.

2024-06-10T06:43:12Z

← Older revision		Revision as of 06:43, 10 June 2024
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	= Intuition =		= Intuition =
	If we think of a matrix as a linear transformation, eigenvectors do not change direction. Instead, they simply scale by an eigenvalue.		If we think of a matrix as a linear transformation, eigenvectors do not change direction. Instead, they simply scale by an eigenvalue.

			= Eigenspace =
			Given A and <math>\lambda</math>, the '''eigenspace''' is all the eigenvectors with that particular eigenvalue.

			Because any linear combination of eigenvectors will yield another eigenvector of the same eigenvalue, the eigenspace is a vector space.

			The following definition of eigenvectors help explore the eigenspace:

			<math>
			(A-\lambda I)\vec{x}=0
			</math>

			We can find the determinant of the preceding matrix:

			<math>
			det(A-\lambda I)=(-1)^n[\lambda^n+c_{n-1}\lambda^{n-1}...+c_0]
			</math>

			This is the ''characteristic polynomial'' of the A, showing that for any A, there are at most n eigenvalues.

2024-06-10T06:35:16Z

← Older revision		Revision as of 06:35, 10 June 2024
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	[[Category:Linear Algebra]]		[[Category:Linear Algebra]]

			Given a [[matrix]], its '''eigenvectors''' are special vectors that satisfy the following property:

			<math>
			A\vec{x}=\lambda\vec{x}
			</math>

			where <math>\lambda</math> is the '''eigenvalue''' associated with the eigenvector <math>\vec{x}</math>

			The definition of eigenvectors are also frequently written in this form:

			<math>
			(A-\lambda I)\vec{x}=0
			</math>

			= Intuition =
			If we think of a matrix as a linear transformation, eigenvectors do not change direction. Instead, they simply scale by an eigenvalue.

2024-06-10T06:31:36Z

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[[Category:Linear Algebra]]