\(A \in \mathbb{R}^{n \times n}\) needs to {{c1:: have \(n\) eigenvectors that form a basis of \(\mathbb{R}^n\)}} to be diagonalisable.
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\(A \in \mathbb{R}^{n \times n}\) needs to {{c1:: have \(n\) eigenvectors that form a basis of \(\mathbb{R}^n\)}} to be diagonalisable.
\[ A = V \Lambda V^{-1} \]where \(\Lambda\) is a *diagonal* matrix with \(\Lambda_{ii} = \lambda_i\) (and \(\Lambda_{ij} = 0\) for all \(i \neq j\)).
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| Text | \(A \in \mathbb{R}^{n \times n}\) needs to {{c1:: have \(n\) eigenvectors that form a basis of \(\mathbb{R}^n\)}} to be diagonalisable. | |
| Extra | \[ A = V \Lambda V^{-1} \]where \(\Lambda\) is a *diagonal* matrix with \(\Lambda_{ii} = \lambda_i\) (and \(\Lambda_{ij} = 0\) for all \(i \neq j\)). |
Note 2: ETH::LinAlg
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\(A \in \mathbb{R}^{n \times n}\) and \(B \in \mathbb{R}^{n \times n}\) are similar matrices. The matrix \(A\) has a complete set of real eigenvectors if and only if \(B\) does .
Back
\(A \in \mathbb{R}^{n \times n}\) and \(B \in \mathbb{R}^{n \times n}\) are similar matrices. The matrix \(A\) has a complete set of real eigenvectors if and only if \(B\) does .
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| Text | \(A \in \mathbb{R}^{n \times n}\) and \(B \in \mathbb{R}^{n \times n}\) are similar matrices. The matrix \(A\) has {{c1::a complete set of real eigenvectors if and only if \(B\) does :: EVs}}. |
Note 3: ETH::LinAlg
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We say that \(A \in \mathbb{R}^{n \times n}\) and \(B \in \mathbb{R}^{n \times n}\) are similar matrices if {{c2::there exists an invertible matrix \(S\) such that \(B = S^{-1}AS\)}}.
Back
We say that \(A \in \mathbb{R}^{n \times n}\) and \(B \in \mathbb{R}^{n \times n}\) are similar matrices if {{c2::there exists an invertible matrix \(S\) such that \(B = S^{-1}AS\)}}.
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| Text | We say that \(A \in \mathbb{R}^{n \times n}\) and \(B \in \mathbb{R}^{n \times n}\) are {{c1::similar matrices}} if {{c2::there exists an invertible matrix \(S\) such that \(B = S^{-1}AS\)}}. |
Note 4: ETH::LinAlg
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\(A\) has a complete set of real eigenvectors if {{c2::we can build a basis of \(\mathbb{R}^n\) with the eigenvectors}}.
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\(A\) has a complete set of real eigenvectors if {{c2::we can build a basis of \(\mathbb{R}^n\) with the eigenvectors}}.
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| Text | \(A\) has {{c1::a complete set of real eigenvectors}} if {{c2::we can build a basis of \(\mathbb{R}^n\) with the eigenvectors}}. |
Note 5: ETH::LinAlg
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Similar matrices \(A\) and \(B = S^{-1}AS\) have the same eigenvalues .
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Similar matrices \(A\) and \(B = S^{-1}AS\) have the same eigenvalues .
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| Text | Similar matrices \(A\) and \(B = S^{-1}AS\) have {{c1::the same eigenvalues :: shared property}}. |
Note 6: ETH::LinAlg
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A diagonal matrix \(D\) has eigenvalues which are the diagonals and a full set of eigenvectors \(e_1, \dots, e_n\).
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A diagonal matrix \(D\) has eigenvalues which are the diagonals and a full set of eigenvectors \(e_1, \dots, e_n\).
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| Text | A diagonal matrix \(D\) has eigenvalues {{c1::which are the diagonals :: where are they?}} and {{c1::a full set of eigenvectors \(e_1, \dots, e_n\)::EVs?}}. |
Note 7: ETH::LinAlg
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Let \(P\) be the projection matrix onto the subspace \(U \subset \mathbb{R}^n\). Then \(P\) has two eigenvalues, \(0\) and \(1\).
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Let \(P\) be the projection matrix onto the subspace \(U \subset \mathbb{R}^n\). Then \(P\) has two eigenvalues, \(0\) and \(1\).
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| Text | Let \(P\) be the <i>projection matrix</i> onto the subspace \(U \subset \mathbb{R}^n\). Then \(P\) has {{c1::two eigenvalues, \(0\) and \(1\):: EW, EVs, and a complete set of real eigenvectors}}. |
Note 8: ETH::LinAlg
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Give an example of a non-diagonalisable matrix that does not have a full set of eigenvectors.
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Give an example of a non-diagonalisable matrix that does not have a full set of eigenvectors.
\[ A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \]
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| Front | Give an example of a non-diagonalisable matrix that does <b>not</b> have a full set of eigenvectors. | |
| Back | \[ A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \] |