Are the following equations concerning matrices and their eigenvectors equivalent?
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Let $A$ be a matrix, $a,b$ its eigenvectors and $lambda_a, lambda_b$ their corresponding eigenvalues. Define $a^prime = Aa$ and $b^prime = Ab$. Is it true that
beginequation
lambda_a = fracVert a^primeVertVert aVert iff Vert aVert = frac1sqrtlambda_a
endequation
and
beginequation
lambda_b = fracVert b^primeVertVert bVert iff Vert bVert = frac1sqrtlambda_b ?
endequation
linear-algebra eigenvalues-eigenvectors linear-transformations
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Let $A$ be a matrix, $a,b$ its eigenvectors and $lambda_a, lambda_b$ their corresponding eigenvalues. Define $a^prime = Aa$ and $b^prime = Ab$. Is it true that
beginequation
lambda_a = fracVert a^primeVertVert aVert iff Vert aVert = frac1sqrtlambda_a
endequation
and
beginequation
lambda_b = fracVert b^primeVertVert bVert iff Vert bVert = frac1sqrtlambda_b ?
endequation
linear-algebra eigenvalues-eigenvectors linear-transformations
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up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let $A$ be a matrix, $a,b$ its eigenvectors and $lambda_a, lambda_b$ their corresponding eigenvalues. Define $a^prime = Aa$ and $b^prime = Ab$. Is it true that
beginequation
lambda_a = fracVert a^primeVertVert aVert iff Vert aVert = frac1sqrtlambda_a
endequation
and
beginequation
lambda_b = fracVert b^primeVertVert bVert iff Vert bVert = frac1sqrtlambda_b ?
endequation
linear-algebra eigenvalues-eigenvectors linear-transformations
Let $A$ be a matrix, $a,b$ its eigenvectors and $lambda_a, lambda_b$ their corresponding eigenvalues. Define $a^prime = Aa$ and $b^prime = Ab$. Is it true that
beginequation
lambda_a = fracVert a^primeVertVert aVert iff Vert aVert = frac1sqrtlambda_a
endequation
and
beginequation
lambda_b = fracVert b^primeVertVert bVert iff Vert bVert = frac1sqrtlambda_b ?
endequation
linear-algebra eigenvalues-eigenvectors linear-transformations
linear-algebra eigenvalues-eigenvectors linear-transformations
asked Sep 7 at 11:10
TheSodesa
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968
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3 Answers
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No. First of all, note that the statements don't even make sense if $lambda_a=0$ or $lambda_b=0$. On the other hand, $a'=lambda_aa$ and therefore$$fraca'=lvertlambda_arvert.$$So, $lambda_a=fraca'ifflambda_ageqslant0$. And, of course, $lambda_ageqslant0$ is not equivalent to $|a|=frac1sqrtlambda_a$.
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Hint: $$frac=frac=frac}=frac=frac1sqrtlambda_a$.
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Hint: $$frac=frac=frac=fracimprove this answer
add a comment =frac==frac{=|lambda_a|$$
answered Sep 7 at 11:14
zzuussee
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Hint : $$a' = A a = lambda_a a$$
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Hint : $$a' = A a = lambda_a a$$
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Hint : $$a' = A a = lambda_a a$$
Hint : $$a' = A a = lambda_a a$$
answered Sep 7 at 11:14
P. Quinton
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81311
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