Anki Deck Changes

Note 1: ETH::A&D

Deck: ETH::A&D
Note Type: Horvath Classic
GUID: Qv/bX3RU0v

modified

Before

Front

How does the number of ZHK's change in Boruvka's for each round?

Back

How does the number of ZHK's change in Boruvka's for each round?

The number of component halves in each round, thus \(\log |V|\) iterations worst case.

After

Front

How does the number of ZHK's change in Boruvka's for each round?

Back

How does the number of ZHK's change in Boruvka's for each round?

The number of components halves in each round, thus \(\log |V|\) iterations worst case.

Field-by-field Comparison

Field	Before	After
Back	The number of component halves in each round, thus \(\log \|V\|\) iterations worst case.	The number of components halves in each round, thus \(\log \|V\|\) iterations worst case.

Tags: ETH::1._Semester::A&D::11._Minimum_Spanning_Trees::1._Boruvka's_Algorithm

Note 2: ETH::A&D

Deck: ETH::A&D
Note Type: Algorithms
GUID: hL7UB-)y6N

modified

Before

Front

Runtime of Floyd-Warshall?

Back

Runtime of Floyd-Warshall?

\(O(|V|^3)\)

After

Front

Runtime of Floyd-Warshall?

Back

Runtime of Floyd-Warshall?

\(O(|V|^3)\)

Field-by-field Comparison

Field	Before	After
Requirements	No negative cycles	No negative cycles.
Use Case	All Pairs Shortest Path	All pairs shortest path

Tags: ETH::1._Semester::A&D::12._All-Pair_Shortest_Paths::1._Floyd-Warshall_Algorithm

Note 3: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: Dy6Zv8Tp2B

modified

Before

Front

For formulas \(F\) and \(H\), where \(x\) does not occur free in \(H\), we have:
- \((\forall x \, F) \land H\) \( \equiv\) \( \forall x \, (F \land H)\)
- \((\forall x \, F) \lor H \) \(\equiv\) \(\forall x \, (F \lor H)\)
- \((\exists x \, F) \land H \) \(\equiv\) \(\exists x \, (F \land H)\)
- \((\exists x \, F) \lor H\) \(\equiv\) \(\exists x \, (F \lor H)\)

Back

For formulas \(F\) and \(H\), where \(x\) does not occur free in \(H\), we have:
- \((\forall x \, F) \land H\) \( \equiv\) \( \forall x \, (F \land H)\)
- \((\forall x \, F) \lor H \) \(\equiv\) \(\forall x \, (F \lor H)\)
- \((\exists x \, F) \land H \) \(\equiv\) \(\exists x \, (F \land H)\)
- \((\exists x \, F) \lor H\) \(\equiv\) \(\exists x \, (F \lor H)\)

After

Front

For formulas \(F\) and \(H\), where \(x\) does not occur free in \(H\), we have:

\((\forall x \, F) \land H\) \( \equiv\) \( \forall x \, (F \land H)\)
\((\forall x \, F) \lor H \) \(\equiv\) \(\forall x \, (F \lor H)\)
\((\exists x \, F) \land H \) \(\equiv\) \(\exists x \, (F \land H)\)
\((\exists x \, F) \lor H\) \(\equiv\) \(\exists x \, (F \lor H)\)

Back

For formulas \(F\) and \(H\), where \(x\) does not occur free in \(H\), we have:

\((\forall x \, F) \land H\) \( \equiv\) \( \forall x \, (F \land H)\)
\((\forall x \, F) \lor H \) \(\equiv\) \(\forall x \, (F \lor H)\)
\((\exists x \, F) \land H \) \(\equiv\) \(\exists x \, (F \land H)\)
\((\exists x \, F) \lor H\) \(\equiv\) \(\exists x \, (F \lor H)\)

Field-by-field Comparison

Field	Before	After
Text	For formulas \(F\) and \(H\), where \(x\) <b>does not occur free</b> in \(H\), we have:<br>- {{c1::\((\forall x \, F) \land H\)}} \( \equiv\) {{c2::\( \forall x \, (F \land H)\)}}<~~br>-~~ {{c3::\((\forall x \, F) \lor H \)}} \(\equiv\) {{c4::\(\forall x \, (F \lor H)\)}}<~~br>-~~ {{c5::\((\exists x \, F) \land H \)}} \(\equiv\) {{c6:: \(\exists x \, (F \land H)\)}}<~~br>-~~ {{c7::\((\exists x \, F) \lor H\)}} \(\equiv\) {{c8:: \(\exists x \, (F \lor H)\)}}	For formulas \(F\) and \(H\), where \(x\) <b>does not occur free</b> in \(H\), we have:<br><ol><li>{{c1::\((\forall x \, F) \land H\)}} \( \equiv\) {{c2::\( \forall x \, (F \land H)\)}}</li><li>{{c3::\((\forall x \, F) \lor H \)}} \(\equiv\) {{c4::\(\forall x \, (F \lor H)\)}}</li><li>{{c5::\((\exists x \, F) \land H \)}} \(\equiv\) {{c6:: \(\exists x \, (F \land H)\)}}</li><li>{{c7::\((\exists x \, F) \lor H\)}} \(\equiv\) {{c8:: \(\exists x \, (F \lor H)\)}}</li></ol>

Tags: ETH::1._Semester::DiskMat::6._Logic::6._Predicate_Logic_(First-order_Logic)::5._Some_Basic_Equivalences_Involving_Quantifiers

Note 4: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: fd?4%T(3|z

modified

Before

Front

A function is injective (or one-to-one) if for \(a \ne b\) we have \(f(a) \ne f(b)\), i.e. no "collisions"

Back

A function is injective (or one-to-one) if for \(a \ne b\) we have \(f(a) \ne f(b)\), i.e. no "collisions"

Example: \(f(x) = x\), counterexample: \(f(x) = x^2, x \in \mathbb{R}\)

After

Front

A function is injective (or one-to-one) if for \(a \ne b\) we have \(f(a) \ne f(b)\), i.e. no "collisions".

Back

A function is injective (or one-to-one) if for \(a \ne b\) we have \(f(a) \ne f(b)\), i.e. no "collisions".

Example: \(f(x) = x\), counterexample: \(f(x) = x^2, x \in \mathbb{R}\)

Field-by-field Comparison

Field	Before	After
Text	A function is {{c1::injective (or one-to-one)}} if {{c2::for \(a \ne b\) we have \(f(a) \ne f(b)\), i.e. no "collisions"}}	A function is {{c1::injective (or one-to-one)}} if {{c2::for \(a \ne b\) we have \(f(a) \ne f(b)\), i.e. no "collisions"}}.

Tags: ETH::1._Semester::DiskMat::3._Sets,_Relations,_and_Functions::6._Functions

Note 5: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: ujCuoEmotl

modified

Before

Front

If \(p\) is a prime which divides the product \(x_1 x_2 \dots x_n\) of some integers, then \(p\) divides at least one of them: \[p | (x_1 x_2 \dots x_n) \Rightarrow \exists i \ p | x_i\]

Back

If \(p\) is a prime which divides the product \(x_1 x_2 \dots x_n\) of some integers, then \(p\) divides at least one of them: \[p | (x_1 x_2 \dots x_n) \Rightarrow \exists i \ p | x_i\]

After

Front

If \(p\) is a prime which divides the product \(x_1 x_2 \dots x_n\) of some integers, then \(p\) divides at least one of them: \[p | (x_1 x_2 \dots x_n) \Rightarrow \exists i \ p | x_i\]

Back

If \(p\) is a prime which divides the product \(x_1 x_2 \dots x_n\) of some integers, then \(p\) divides at least one of them: \[p | (x_1 x_2 \dots x_n) \Rightarrow \exists i \ p | x_i\]

Field-by-field Comparison

Field	Before	After
Text	If \(p\) is a prime which divides the product \(x_1 x_2 \dots x_n\) of some integers, then \(p\) {{c1:: divides at least one of them: \[p \| (x_1 x_2 \dots x_n) \Rightarrow \exists i \ p \| x_i\]}}<br>	If \(p\) is a prime which divides the product \(x_1 x_2 \dots x_n\) of some integers, then \(p\) {{c1::divides at least one of them: \[p \| (x_1 x_2 \dots x_n) \Rightarrow \exists i \ p \| x_i\]}}<br>

Tags: ETH::1._Semester::DiskMat::4._Number_Theory::3._Factorization_into_Primes::2._Proof_of_the_Fundamental_Theorem_of_Arithmetic_*

Note 6: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: CPAR6ayFL2

modified

Before

Front

\(A \in \mathbb{R}^{n \times n}\) needs to {{c1:: have \(n\) eigenvectors that form a basis of \(\mathbb{R}^n\)}} to be diagonalisable.

Back

\(A \in \mathbb{R}^{n \times n}\) 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 \(i \neq j\)).

After

Front

\(A \in \mathbb{R}^{n \times n}\) needs to {{c1:: have \(n\) eigenvectors that form a basis of \(\mathbb{R}^n\)}} to be diagonalisable.

Back

\(A \in \mathbb{R}^{n \times n}\) 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 \(i \neq j\)).

Field-by-field Comparison

Field	Before	After
Extra	\[ A = V \Lambda V^{-1} \]where \(\Lambda\) is a diagonal matrix with \(\Lambda_{ii} = \lambda_i\) (and \(\Lambda_{ij} = 0\) for all \(i \neq j\)).	\[ A = V \Lambda V^{-1} \]where \(\Lambda\) is a <b>diagonal</b> matrix with \(\Lambda_{ii} = \lambda_i\) (and \(\Lambda_{ij} = 0\) for all \(i \neq j\)).

Tags: blank::1._Diagonalisation ETH::1._Semester::LinAlg::9._Diagonalisable_Matrices_and_the_SVD::1._Diagonalisation

Note 7: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: EVK,*aoX3m

modified

Before

Front

The eigenvectors of an eigenvalue are those and exactly those vectors \(v \neq 0\) in \(v \in N(A - \lambda I)\) .

Back

The eigenvectors of an eigenvalue are those and exactly those vectors \(v \neq 0\) in \(v \in N(A - \lambda I)\) .

After

Front

The eigenvectors of an eigenvalue are those and exactly those vectors \(v \neq 0\) in \(v \in N(A - \lambda I)\).

Back

The eigenvectors of an eigenvalue are those and exactly those vectors \(v \neq 0\) in \(v \in N(A - \lambda I)\).

Field-by-field Comparison

Field	Before	After
Text	The <b>eigenvectors</b> of an eigenvalue are <b>those and exactly those</b> vectors {{c1::\(v \neq 0\)}} in {{c1::\(v \in N(A - \lambda I)\)~~ ~~:: subspace}}.	The <b>eigenvectors</b> of an eigenvalue are <b>those and exactly those</b> vectors {{c1::\(v \neq 0\)}} in {{c1::\(v \in N(A - \lambda I)\)::subspace}}.

Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::5._Extra

Note 8: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: K-XRXZqOV,

modified

Before

Front

If \(AB = BA\) then \(A,B\) share an EV .

Back

If \(AB = BA\) then \(A,B\) share an EV .

Assume \(AB = BA\).

If \(\lambda, v\) an EW-EV pair of \(A\) then \(A(Bv) = (AB)v = B(Av) = \lambda Bv\) thus \(Bv\) is an eigenvector of \(A\).
Then \(Bv\) is a multiple of some \(v\) of that EW \(\lambda\) (easiest to see for \(A\) complete set of real EVs) \(\implies\) \(Bv = \lambda'v\) thus that \(v\) is also an EV of \(B\)

After

Front

If \(AB = BA\), then \(A,B\) share an EV.

Back

If \(AB = BA\), then \(A,B\) share an EV.

Assume \(AB = BA\).

If \(\lambda, v\) an EW-EV pair of \(A\) then \(A(Bv) = (AB)v = B(Av) = \lambda Bv\) thus \(Bv\) is an eigenvector of \(A\).

Then \(Bv\) is a multiple of some \(v\) of that EW \(\lambda\) (easiest to see for \(A\) complete set of real EVs) \(\implies\) \(Bv = \lambda'v\) thus that \(v\) is also an EV of \(B\).

Field-by-field Comparison

Field	Before	After
Text	If \(AB = BA\) then {{c1::~~ ~~\(A,B\) share an EV :: EVs of A, B}}.	If \(AB = BA\), then {{c1::\(A,B\) share an EV::EVs of A, B}}.
Extra	Assume \(AB = BA\).<br><br>If \(\lambda, v\) an EW-EV pair of \(A\) then \(A(Bv) = (AB)v = B(Av) = \lambda Bv\) thus \(Bv\) is an eigenvector of \(A\).<br>Then \(Bv\) is a multiple of some \(v\) of that EW \(\lambda\) (easiest to see for \(A\) complete set of real EVs) \(\implies\) \(Bv = \lambda'v\) thus that \(v\) is also an EV of \(B\)	Assume \(AB = BA\).<br><br>If \(\lambda, v\) an EW-EV pair of \(A\) then \(A(Bv) = (AB)v = B(Av) = \lambda Bv\) thus \(Bv\) is an eigenvector of \(A\).<br><br>Then \(Bv\) is a multiple of some \(v\) of that EW \(\lambda\) (easiest to see for \(A\) complete set of real EVs) \(\implies\) \(Bv = \lambda'v\) thus that \(v\) is also an EV of \(B\).

Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::5._Extra

Note 9: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: KAa7x3sA#0

modified

Before

Front

The eigenvalues of \(A + B\) are not the eigenvalues of \(A\) plus those of \(B\).

Back

The eigenvalues of \(A + B\) are not the eigenvalues of \(A\) plus those of \(B\).

They aren't correlated.

After

Front

The eigenvalues of \(A + B\) are not correlated.

Back

The eigenvalues of \(A + B\) are not correlated.

Field-by-field Comparison

Field	Before	After
Text	The eigenvalues of \(A + B\) are {{c1::<b>not</b> ~~the eigenvalues of \(A\) plus those of \(B\)~~}}.	The eigenvalues of \(A + B\) are {{c1::<b>not</b> correlated}}.
Extra	~~They aren't correlated.~~

Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::4._Warnings

Note 10: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Classic
GUID: Lr(&c[;1SI

modified

Before

Front

An linear combination of \(\lambda_1\textbf{v}_1 + \lambda_2\textbf{v}_2 + \dots + \lambda_n\textbf{v}_n\) is conic if

Back

An linear combination of \(\lambda_1\textbf{v}_1 + \lambda_2\textbf{v}_2 + \dots + \lambda_n\textbf{v}_n\) is conic if

\(\lambda_j \geq 0\) for \(j = 1, 2, \dots, n\)

After

Front

A linear combination of \(\lambda_1\textbf{v}_1 + \lambda_2\textbf{v}_2 + \dots + \lambda_n\textbf{v}_n\) is conic if

Back

A linear combination of \(\lambda_1\textbf{v}_1 + \lambda_2\textbf{v}_2 + \dots + \lambda_n\textbf{v}_n\) is conic if

\(\lambda_j \geq 0\) for \(j = 1, 2, \dots, n\)

Field-by-field Comparison

Field	Before	After
Front	An linear combination of  \(\lambda_1\textbf{v}_1 + \lambda_2\textbf{v}_2 + \dots + \lambda_n\textbf{v}_n\) is <b>conic</b> if	A linear combination of  \(\lambda_1\textbf{v}_1 + \lambda_2\textbf{v}_2 + \dots + \lambda_n\textbf{v}_n\) is <b>conic</b> if

Tags: ETH::1._Semester::LinAlg::1._Vectors::1._Vectors_and_linear_combinations::4._Affine,_conic,_and_convex_combinations PlsFix::ClozeThatBish

Note 11: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: doaX+*9B4:

modified

Before

Front

For an eigenvalue \(\lambda\) of \(M\), \(\lambda + c\) is a real eigenvalue of the matrix \(M + cI\).

Back

For an eigenvalue \(\lambda\) of \(M\), \(\lambda + c\) is a real eigenvalue of the matrix \(M + cI\).

Intuitively this makes sense as by adding \(cI\) were increasing the values on the diagonal, meaning we have to increase the value of \(\lambda\) by the same amount so it makes \(0\) again.

After

Front

For an eigenvalue \(\lambda\) of \(M\), \(\lambda + c\) is a real eigenvalue of the matrix \(M + cI\).

Back

For an eigenvalue \(\lambda\) of \(M\), \(\lambda + c\) is a real eigenvalue of the matrix \(M + cI\).

Intuitively this makes sense as by adding \(cI\) we're increasing the values on the diagonal, meaning we have to increase the value of \(\lambda\) by the same amount so that we get \(0\) again.

Field-by-field Comparison

Field	Before	After
Extra	Intuitively this makes sense as by adding \(cI\) were increasing the values on the diagonal, meaning we have to increase the value of \(\lambda\) by the same amount so ~~it makes~~ \(0\) again.	Intuitively this makes sense as by adding \(cI\) we're increasing the values on the diagonal, meaning we have to increase the value of \(\lambda\) by the same amount so that we get \(0\) again.

Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::5._Extra

Note 12: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: krdY1rh*f]

modified

Before

Front

If \(AB = BA\) then they share an EV and thus \(A + B\) also has that EV .

Back

If \(AB = BA\) then they share an EV and thus \(A + B\) also has that EV .

If both \(A\) and \(B\) share an EV:
\((A + B)v = Av + Bv = \lambda v + \lambda' v = (\lambda + \lambda')v\) then \(A + B\) also has that EV.

After

Front

If \(AB = BA\) then they share an EV and thus \(A + B\) also has that EV.

Back

If \(AB = BA\) then they share an EV and thus \(A + B\) also has that EV.

If both \(A\) and \(B\) share an EV:
\((A + B)v = Av + Bv = \lambda v + \lambda' v = (\lambda + \lambda')v\) then \(A + B\) also has that EV.

Field-by-field Comparison

Field	Before	After
Text	If \(AB = BA\) {{c1::then they share an EV and thus \(A + B\) also has that EV :: sum}}.	If \(AB = BA\) {{c1::then they share an EV and thus \(A + B\) also has that EV::sum}}.

Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::5._Extra

Note 13: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Classic
GUID: lh.Ty3ypWO

modified

Before

Front

How do we find a basis for the row space \(R(A) = C(A^\top)\)?

Back

How do we find a basis for the row space \(R(A) = C(A^\top)\)?

The first \(r\) columns of \(R^\top\) where \(R\) is the RREF of \(A\) form a basis of the row space (the non-zero rows). In particular \(\dim(\textbf{R}(A)) = r\)

This works because as noted before, multiplying by and invertible matrix \(M\) does not change the row-space of \(MA\) on the left.

After

Front

How do we find a basis for the row space \(R(A) = C(A^\top)\)?

Back

How do we find a basis for the row space \(R(A) = C(A^\top)\)?

The first \(r\) columns of \(R^\top\) where \(R\) is the RREF of \(A\) form a basis of the row space (the non-zero rows). In particular \(\dim(\textbf{R}(A)) = r\)

This works because as noted before, multiplying by an invertible matrix \(M\) does not change the row-space of \(MA\) on the left.

Field-by-field Comparison

Field	Before	After
Back	The first \(r\) columns of \(R^\top\) where \(R\) is the RREF of \(A\) form a basis of the row space (the non-zero rows). In particular \(\dim(\textbf{R}(A)) = r\)<br><br><div>This works because as noted before, multiplying by and invertible matrix \(M\) does not change the row-space of \(MA\) on the left.</div>	The first \(r\) columns of \(R^\top\) where \(R\) is the RREF of \(A\) form a basis of the row space (the non-zero rows). In particular \(\dim(\textbf{R}(A)) = r\)<br><br><div>This works because as noted before, multiplying by an invertible matrix \(M\) does not change the row-space of \(MA\) on the left.</div>

Tags: ETH::1._Semester::LinAlg::4._The_Three_Fundamental_Subspaces::3._Computing_the_three_fundamental_subspaces::2._Row_Space

Note 14: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: pk}0Q0,75,

modified

Before

Front

The trace is commutative

Back

The trace is commutative

This makes sense as addition is element-wise.

After

Front

The trace is commutative.

Back

The trace is commutative.

This makes sense as addition is element-wise.

Field-by-field Comparison

Field	Before	After
Text	The trace is {{c1::commutative}}	The trace is {{c1::commutative}}.

Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::3._Properties

Note 15: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: rvqzRY-*GX

modified

Before

Front

A diagonal matrix \(D\) has eigenvalues which are the diagonals and a full set of eigenvectors \(e_1, \dots, e_n\).

Back

A diagonal matrix \(D\) has eigenvalues which are the diagonals and a full set of eigenvectors \(e_1, \dots, e_n\).

After

Front

A diagonal matrix \(D\) has eigenvalues which are the diagonals and a full set of eigenvectors \(e_1, \dots, e_n\).

Back

A diagonal matrix \(D\) has eigenvalues which are the diagonals and a full set of eigenvectors \(e_1, \dots, e_n\).

Field-by-field Comparison

Field	Before	After
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?}}.	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?}}.

Tags: blank::1._Diagonalisation ETH::1._Semester::LinAlg::9._Diagonalisable_Matrices_and_the_SVD ETH::1._Semester::LinAlg::9._Diagonalisable_Matrices_and_the_SVD::1._Diagonalisation

Note 16: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: s},fJ+;VIc

modified

Before

Front

\(A\) has an EW \(0\) \(\Longleftrightarrow\)\(A\) is not invertible Proof Included

Back

\(A\) has an EW \(0\) \(\Longleftrightarrow\)\(A\) is not invertible Proof Included

We know the EVs are in the \(N(A - \lambda I)\) because they solve the formula \(Av = \lambda v \implies Av - \lambda v \)\(= 0 \implies (A - \lambda I)v = 0\). Thus \(v\) is in the nullspace of \((A - \lambda I)\).

If \(A\) is not invertible, then it will have an EV of \(0\), which means that there is an EV solving \((A - 0 \cdot I)v = 0 \implies Av = 0\). Thus \(v \neq 0\) is in the nullspace of \(A\), i.e. the nullspace is not empty.

After

Front

\(A\) has an EW \(0\) \(\Longleftrightarrow\)\(A\) is not invertible Proof Included

Back

\(A\) has an EW \(0\) \(\Longleftrightarrow\)\(A\) is not invertible Proof Included

We know the EVs are in the \(N(A - \lambda I)\) because they solve the formula \(Av = \lambda v \implies Av - \lambda v \)\(= 0 \implies (A - \lambda I)v = 0\). Thus \(v\) is in the nullspace of \((A - \lambda I)\).

If \(A\) is not invertible, then it will have an EV of \(0\), which means that there is an EV solving \((A - 0 \cdot I)v = 0 \implies Av = 0\). Thus \(v \neq 0\) is in the nullspace of \(A\), i.e. the nullspace is not empty.

Field-by-field Comparison

Field	Before	After
Text	{{c1::\(A\) has an EW \(0\)}} \(\Longleftrightarrow\){{c2::\(A\) is not invertible}}<i> Proof Included</i>	{{c1::\(A\) has an EW \(0\)::EW}} \(\Longleftrightarrow\){{c2::\(A\) is not invertible}}<i> Proof Included</i>

Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::5._Extra

Note 17: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: ww{@M/WDmP

modified

Before

Front

\(A\) and \(A^\top\) share eigenvalues not eigenvectors .

Back

\(A\) and \(A^\top\) share eigenvalues not eigenvectors .

After

Front

\(A\) and \(A^\top\) share eigenvalues not eigenvectors.

Back

\(A\) and \(A^\top\) share eigenvalues not eigenvectors.

Field-by-field Comparison

Field	Before	After
Text	<div>\(A\) and \(A^\top\) share {{c1::eigenvalues <b>not eigenvectors</b>~~ ~~:: EWs, EVs}}.</div>	<div>\(A\) and \(A^\top\) share {{c1::eigenvalues <b>not eigenvectors</b>::EWs, EVs}}.</div>

Tags: ETH::1._Semester::LinAlg::8._Eigenvalues_and_Eigenvectors::4._Warnings

Note 18: ETH::LinAlg

Deck: ETH::LinAlg
Note Type: Horvath Cloze
GUID: y;U[Cn>&)o

modified

Before

Front

Let \(P\) be the projection matrix onto the subspace \(U \subset \mathbb{R}^n\). Then \(P\) has two eigenvalues, \(0\) and \(1\).

Back

Let \(P\) be the projection matrix onto the subspace \(U \subset \mathbb{R}^n\). Then \(P\) has two eigenvalues, \(0\) and \(1\).

After

Front

Let \(P\) be the projection matrix onto the subspace \(U \subset \mathbb{R}^n\). Then \(P\) has two eigenvalues, \(0\) and \(1\).

Back

Let \(P\) be the projection matrix onto the subspace \(U \subset \mathbb{R}^n\). Then \(P\) has two eigenvalues, \(0\) and \(1\).

Field-by-field Comparison

Field	Before	After
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}}.	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}}.

Tags: ETH::1._Semester::LinAlg::9._Diagonalisable_Matrices_and_the_SVD::1._Diagonalisation