Anki Deck Changes

Commit: 7e1e9169 - add first chap 8

Author: obrhubr <obrhubr@gmail.com>

Date: 2026-01-16T13:03:33+01:00

Changes: 14 note(s) changed (11 added, 3 modified, 0 deleted)

ℹ️ Cosmetic Changes Hidden: 1 note(s) had formatting-only changes and are not shown below

Note 1: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: C%s}g6HGxU
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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
The algebraic multiplicity of a root is the number of times it appears in the factorisation \(P(z) = a_n (z-\lambda_1)(z - \lambda_2) \dots (z - \lambda_n)\).

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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
The algebraic multiplicity of a root is the number of times it appears in the factorisation \(P(z) = a_n (z-\lambda_1)(z - \lambda_2) \dots (z - \lambda_n)\).
Example: If the algebraic multiplicity of \(\lambda_2\) is \(3\) then \((z - \lambda_2)^3 \ | \ P(z)\).
Field-by-field Comparison
Field Before After
Text The {{c1::algebraic multiplicity of a&nbsp;<i>root</i>}} is {{c2:: the number of times it appears in the factorisation \(P(z) = a_n (z-\lambda_1)(z - \lambda_2) \dots (z - \lambda_n)\)}}.
Extra <div><strong>Example:</strong> If the algebraic multiplicity of \(\lambda_2\) is \(3\) then \((z - \lambda_2)^3 \ | \ P(z)\).</div>
Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers

Note 2: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: DnDRP_uA~N
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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
\(\overline{z_1 + z_2} = {{c1:: \overline{z_1} + \overline{z_2} }}\)

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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
\(\overline{z_1 + z_2} = {{c1:: \overline{z_1} + \overline{z_2} }}\)
Field-by-field Comparison
Field Before After
Text \(\overline{z_1 + z_2} = {{c1:: \overline{z_1} + \overline{z_2} }}\)
Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers

Note 3: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Classic
GUID: DzlMCHq[Im
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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
Prove some \(x \in \mathbb{C}\) is actually in \(\mathbb{R}\)?

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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
Prove some \(x \in \mathbb{C}\) is actually in \(\mathbb{R}\)?
Show that \(x = \overline{x} \implies x \in \mathbb{R}\)
Field-by-field Comparison
Field Before After
Front <div>Prove some \(x \in \mathbb{C}\) is actually in \(\mathbb{R}\)?</div>
Back Show that \(x = \overline{x} \implies x \in \mathbb{R}\)
Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers

Note 4: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: FTH7rOs5Fz
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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
Every polynomial \(P(z) = a_n z^n + a_{n - 1} z^{n - 1} + \dots + a_1 z + a_0\) with \(a_n \neq 0\) has {{c1:: a zero \(\lambda \in \mathbb{C} \)}}

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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
Every polynomial \(P(z) = a_n z^n + a_{n - 1} z^{n - 1} + \dots + a_1 z + a_0\) with \(a_n \neq 0\) has {{c1:: a zero \(\lambda \in \mathbb{C} \)}}
Fundamental theorem of algebra
Field-by-field Comparison
Field Before After
Text Every polynomial&nbsp;\(P(z) = a_n z^n + a_{n - 1} z^{n - 1} + \dots + a_1 z + a_0\) with \(a_n \neq 0\) has {{c1:: a zero&nbsp;\(\lambda \in \mathbb{C} \)}}
Extra Fundamental theorem of algebra
Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers

Note 5: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: F]H_)uK%+@
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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1,9._Closed_Form_Recursion_(Fibonacci)
Express the Fibonacci formula \(f_n = f_{n - 1} + f_{n - 2}\) as a matrix equation?
  1. {{c1::Let  \(\mathbf{g}_{n+1} = M\mathbf{g}_n\), where \(M = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}\)(matrix version of the recursion)}}
  2. The eigenvalues \(\lambda_1 = \frac{1+\sqrt{5{2}\) (golden ratio \(\phi\)) and \(\lambda_2 = \frac{1-\sqrt{5}}{2}\) are found, along with their eigenvectors \(v_1 = \begin{pmatrix} \lambda_1 \\ 1 \end{pmatrix}\) and \(v_2 = \begin{pmatrix} \lambda_2 \\ 1 \end{pmatrix}\) . These eigenvectors are independent since \(\lambda_1 \neq \lambda_2\) and thus they form a basis for \(\mathbb{R}^2\).}}
  3. {{c3::The initial state \(\mathbf{g}_0\) is written as a linear combination of eigenvectors with coefficients \(\pm\frac{1}{\sqrt{5}}\): \(g_0 = \frac{1}{\sqrt{5}}v_1 - \frac{1}{\sqrt{5}}v_2\).}}
  4. {{c4::Since \(M^n\mathbf{v}_i = \lambda_i^n\mathbf{v}_i\), we get the closed form:\[F_n = \frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^n - \left(\frac{1-\sqrt{5}}{2}\right)^n\right]\]}}

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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1,9._Closed_Form_Recursion_(Fibonacci)
Express the Fibonacci formula \(f_n = f_{n - 1} + f_{n - 2}\) as a matrix equation?
  1. {{c1::Let  \(\mathbf{g}_{n+1} = M\mathbf{g}_n\), where \(M = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}\)(matrix version of the recursion)}}
  2. The eigenvalues \(\lambda_1 = \frac{1+\sqrt{5{2}\) (golden ratio \(\phi\)) and \(\lambda_2 = \frac{1-\sqrt{5}}{2}\) are found, along with their eigenvectors \(v_1 = \begin{pmatrix} \lambda_1 \\ 1 \end{pmatrix}\) and \(v_2 = \begin{pmatrix} \lambda_2 \\ 1 \end{pmatrix}\) . These eigenvectors are independent since \(\lambda_1 \neq \lambda_2\) and thus they form a basis for \(\mathbb{R}^2\).}}
  3. {{c3::The initial state \(\mathbf{g}_0\) is written as a linear combination of eigenvectors with coefficients \(\pm\frac{1}{\sqrt{5}}\): \(g_0 = \frac{1}{\sqrt{5}}v_1 - \frac{1}{\sqrt{5}}v_2\).}}
  4. {{c4::Since \(M^n\mathbf{v}_i = \lambda_i^n\mathbf{v}_i\), we get the closed form:\[F_n = \frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^n - \left(\frac{1-\sqrt{5}}{2}\right)^n\right]\]}}
Field-by-field Comparison
Field Before After
Text Express the Fibonacci formula \(f_n = f_{n - 1} + f_{n - 2}\) as a matrix equation?<br><ol><li>{{c1::Let&nbsp;&nbsp;\(\mathbf{g}_{n+1} = M\mathbf{g}_n\), where&nbsp;\(M = \begin{bmatrix} 1 &amp; 1 \\ 1 &amp; 0 \end{bmatrix}\)(matrix version of the recursion)}}</li><li>{{c2::The eigenvalues&nbsp;\(\lambda_1 = \frac{1+\sqrt{5}}{2}\)&nbsp;(golden ratio&nbsp;\(\phi\)) and&nbsp;\(\lambda_2 = \frac{1-\sqrt{5}}{2}\)&nbsp;are found, along with their eigenvectors&nbsp;\(v_1 = \begin{pmatrix} \lambda_1 \\ 1 \end{pmatrix}\)&nbsp;and&nbsp;\(v_2 = \begin{pmatrix} \lambda_2 \\ 1 \end{pmatrix}\)&nbsp;. These eigenvectors are independent since&nbsp;\(\lambda_1 \neq \lambda_2\)&nbsp;and thus they form a basis for&nbsp;\(\mathbb{R}^2\).}}</li><li>{{c3::The initial state&nbsp;\(\mathbf{g}_0\)&nbsp;is written as a linear combination of eigenvectors with coefficients&nbsp;\(\pm\frac{1}{\sqrt{5}}\):&nbsp;\(g_0 = \frac{1}{\sqrt{5}}v_1 - \frac{1}{\sqrt{5}}v_2\).}}</li><li>{{c4::Since&nbsp;\(M^n\mathbf{v}_i = \lambda_i^n\mathbf{v}_i\), we get the closed form:\[F_n = \frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^n - \left(\frac{1-\sqrt{5}}{2}\right)^n\right]\]}}</li></ol>
Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1,9._Closed_Form_Recursion_(Fibonacci)

Note 6: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Classic
GUID: Gp{ARt+OX,
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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
If you have some expression \(x_1 + i x_2 = y_1 + i y_2\) we can separate this into?

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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
If you have some expression \(x_1 + i x_2 = y_1 + i y_2\) we can separate this into?
  1. \(x_1 = y_1\)
  2. \(x_2 = y_2\)
because of the \(i\)
Field-by-field Comparison
Field Before After
Front If you have some expression \(x_1 + i x_2 = y_1 + i y_2\) we can separate this into?
Back <ol><li>\(x_1 = y_1\)</li><li>\(x_2 = y_2\)</li></ol>because of the \(i\)<br>
Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers

Note 7: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Classic
GUID: d[`]Ch#J!c
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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
Fundamental Theorem of Algebra?

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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
Fundamental Theorem of Algebra?
Any degree \(n\) non-constant \(n \geq 1\) polynomial \(P(z) = a_n z^n + a_{n - 1} z^{n - 1} + \dots + a_1 z + a_0\) with \(a_n \neq 0\) has a (existance) zero: \(\lambda \in \mathbb{C}\) such that \(P(\lambda) = 0\).
Field-by-field Comparison
Field Before After
Front Fundamental Theorem of Algebra?
Back <div>Any degree \(n\) non-constant \(n \geq 1\) polynomial \(P(z) = a_n z^n + a_{n - 1} z^{n - 1} + \dots + a_1 z + a_0\) with \(a_n \neq 0\) has a (existance) zero:&nbsp;\(\lambda \in \mathbb{C}\) such that \(P(\lambda) = 0\).</div>
Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers

Note 8: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: g/|!95
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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
\(\frac{1}{z} = {{c1:: \frac{\overline{z} }{|z|^2} :: \text{in terms of z} }}\)

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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
\(\frac{1}{z} = {{c1:: \frac{\overline{z} }{|z|^2} :: \text{in terms of z} }}\)
Field-by-field Comparison
Field Before After
Text \(\frac{1}{z} = {{c1:: \frac{\overline{z} }{|z|^2} :: \text{in terms of z} }}\)
Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers

Note 9: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Classic
GUID: gC,90MH_~p
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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
Was ist eine konjugiert-transponierte (auch: Hermitesch-transponierte) Matrix?

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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
Was ist eine konjugiert-transponierte (auch: Hermitesch-transponierte) Matrix?
\( \mathbf{A}^H = (\overline{\mathbf{A}})^\top = \overline{\mathbf{A}^\top}\)

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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
Was ist eine konjugiert-transponierte (auch: Hermitesch-transponierte) Matrix?

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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
Was ist eine konjugiert-transponierte (auch: Hermitesch-transponierte) Matrix?
\( \mathbf{A}^* = (\overline{\mathbf{A}})^\top = \overline{\mathbf{A}^\top}\)
Field-by-field Comparison
Field Before After
Back \( \mathbf{A}^H = (\overline{\mathbf{A}})^\top = \overline{\mathbf{A}^\top}\)<br> \( \mathbf{A}^* = (\overline{\mathbf{A}})^\top = \overline{\mathbf{A}^\top}\)<br>
Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers

Note 10: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Classic
GUID: g[av;3n%%l
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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
Wann ist eine Matrix hermitesch?

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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
Wann ist eine Matrix hermitesch?
Falls \( \mathbf{A}^H = A\)

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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
Wann ist eine Matrix hermitesch?

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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
Wann ist eine Matrix hermitesch?
Falls \( \mathbf{A}^* = A\)
Field-by-field Comparison
Field Before After
Back Falls&nbsp;\( \mathbf{A}^H = A\) Falls&nbsp;\( \mathbf{A}^* = A\)
Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers

Note 11: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: s!I_p%w(=W
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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
Any degree \(n\) non constant (\(n \geq 1\)) polynomial \(P(z)\) has {{c1:: \(n\) zeros: \(\lambda_1, \dots, \lambda_n \in \mathbb{C}\)}}, perhaps with repetitions such that \[ P(z) = a_n (z-\lambda_1)(z - \lambda_2) \dots (z - \lambda_n) \]

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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
Any degree \(n\) non constant (\(n \geq 1\)) polynomial \(P(z)\) has {{c1:: \(n\) zeros: \(\lambda_1, \dots, \lambda_n \in \mathbb{C}\)}}, perhaps with repetitions such that \[ P(z) = a_n (z-\lambda_1)(z - \lambda_2) \dots (z - \lambda_n) \]
Field-by-field Comparison
Field Before After
Text Any degree \(n\) non constant (\(n \geq 1\)) polynomial \(P(z)\) has {{c1::&nbsp;\(n\) zeros: \(\lambda_1, \dots, \lambda_n \in \mathbb{C}\)}}, perhaps {{c1:: with repetitions}} such that \[ P(z) = a_n (z-\lambda_1)(z - \lambda_2) \dots (z - \lambda_n) \]<br>
Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers

Note 12: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: w9]Tx{V20J
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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
For a complex vector \(v\) we have \(||v|| =\) {{c1:: \(v^*v = \overline{v}^\top v = \sum_{i = 1}^n \overline{v_i}v_i = \sum_{i = 1}^n |v_i|^2\)}}.

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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
For a complex vector \(v\) we have \(||v|| =\) {{c1:: \(v^*v = \overline{v}^\top v = \sum_{i = 1}^n \overline{v_i}v_i = \sum_{i = 1}^n |v_i|^2\)}}.
Field-by-field Comparison
Field Before After
Text For a complex vector&nbsp;\(v\)&nbsp;we have&nbsp;\(||v|| =\)&nbsp;{{c1::&nbsp;\(v^*v = \overline{v}^\top v = \sum_{i = 1}^n \overline{v_i}v_i = \sum_{i = 1}^n |v_i|^2\)}}.
Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers

Note 13: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: zC(mt0fx<^
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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
\(z\overline{z} = |z|^2 \)

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ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
\(z\overline{z} = |z|^2 \)
Field-by-field Comparison
Field Before After
Text \(z\overline{z} = {{c1:: |z|^2 }} \)
Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::1._Complex_Numbers
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