Evaluating functions at matrices
$begingroup$
Try:
So far, I know that if $f(x) = sum a_i x^i$ is a polynomial, then for square matrix $A$ one has $f(A) = sum a_i A^i $ where $A^0 = I$. However, here we have non-polynomials functions that can be approximated by polynomials via taylors trick. However, do we get the result if we apply directly A into f? that is
$$ f(A) = begin{bmatrix} e^{-3 pi /8} & 1 & e^{-pi/8} \
1 & e^{4 pi /8} & 1 \
e^{- pi /8} & 1 & e^{3pi/8} \ end{bmatrix}$$
And similarly, for $g$ and $h$. Can we do this? Or do we need to expand each in taylor's series?
linear-algebra
asked Dec 24 '18 at 8:01
JamesJames
2,638325
$endgroup$
add a comment |
$begingroup$
Try:
So far, I know that if $f(x) = sum a_i x^i$ is a polynomial, then for square matrix $A$ one has $f(A) = sum a_i A^i $ where $A^0 = I$. However, here we have non-polynomials functions that can be approximated by polynomials via taylors trick. However, do we get the result if we apply directly A into f? that is
$$ f(A) = begin{bmatrix} e^{-3 pi /8} & 1 & e^{-pi/8} \
1 & e^{4 pi /8} & 1 \
e^{- pi /8} & 1 & e^{3pi/8} \ end{bmatrix}$$
And similarly, for $g$ and $h$. Can we do this? Or do we need to expand each in taylor's series?
linear-algebra
asked Dec 24 '18 at 8:01
JamesJames
2,638325
$endgroup$
add a comment |
$begingroup$
Try:
So far, I know that if $f(x) = sum a_i x^i$ is a polynomial, then for square matrix $A$ one has $f(A) = sum a_i A^i $ where $A^0 = I$. However, here we have non-polynomials functions that can be approximated by polynomials via taylors trick. However, do we get the result if we apply directly A into f? that is
$$ f(A) = begin{bmatrix} e^{-3 pi /8} & 1 & e^{-pi/8} \
1 & e^{4 pi /8} & 1 \
e^{- pi /8} & 1 & e^{3pi/8} \ end{bmatrix}$$
And similarly, for $g$ and $h$. Can we do this? Or do we need to expand each in taylor's series?
linear-algebra
asked Dec 24 '18 at 8:01
JamesJames
2,638325
$endgroup$
Try:
So far, I know that if $f(x) = sum a_i x^i$ is a polynomial, then for square matrix $A$ one has $f(A) = sum a_i A^i $ where $A^0 = I$. However, here we have non-polynomials functions that can be approximated by polynomials via taylors trick. However, do we get the result if we apply directly A into f? that is
$$ f(A) = begin{bmatrix} e^{-3 pi /8} & 1 & e^{-pi/8} \
1 & e^{4 pi /8} & 1 \
e^{- pi /8} & 1 & e^{3pi/8} \ end{bmatrix}$$
And similarly, for $g$ and $h$. Can we do this? Or do we need to expand each in taylor's series?
linear-algebra
linear-algebra
asked Dec 24 '18 at 8:01
JamesJames
2,638325
asked Dec 24 '18 at 8:01
JamesJames
2,638325
asked Dec 24 '18 at 8:01
JamesJames
2,638325
asked Dec 24 '18 at 8:01
JamesJames
2,638325
asked Dec 24 '18 at 8:01
JamesJames
2,638325
2,638325
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
You cannot apply $f$ to each element of the matrix. For a simple counterexample consider $A=begin {bmatrix} 0 &1 \0 & 0 end {bmatrix}$. Let $f(x)=x^{2}$. Then $f(A)=A^{2}$ is the zero matrix but if you apply $f$ to each element you get $A$. One way of finding functions of the matrix is to diagonalize. The given matrix is p0sitive definite so it can be written as $S^{-1}DS$ where $D$ is diagonal. We have $f(A)=S^{-1}f(D)S$ and $f(D)$ can be calculated by applying $f$ to the diagonal elements. This works each of the three functions under consideration.
answered Dec 24 '18 at 8:06
Kavi Rama MurthyKavi Rama Murthy
72.2k53170
$endgroup$
$begingroup$
How then do you evaluate such expressions?
$endgroup$
– James
Dec 24 '18 at 8:15
1
$begingroup$
@JimmySabater I have given a suggestion for finding functions of the given matrix.
$endgroup$
– Kavi Rama Murthy
Dec 24 '18 at 8:29
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3051048%2fevaluating-functions-at-matrices%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
You cannot apply $f$ to each element of the matrix. For a simple counterexample consider $A=begin {bmatrix} 0 &1 \0 & 0 end {bmatrix}$. Let $f(x)=x^{2}$. Then $f(A)=A^{2}$ is the zero matrix but if you apply $f$ to each element you get $A$. One way of finding functions of the matrix is to diagonalize. The given matrix is p0sitive definite so it can be written as $S^{-1}DS$ where $D$ is diagonal. We have $f(A)=S^{-1}f(D)S$ and $f(D)$ can be calculated by applying $f$ to the diagonal elements. This works each of the three functions under consideration.
answered Dec 24 '18 at 8:06
Kavi Rama MurthyKavi Rama Murthy
72.2k53170
$endgroup$
$begingroup$
How then do you evaluate such expressions?
$endgroup$
– James
Dec 24 '18 at 8:15
1
$begingroup$
@JimmySabater I have given a suggestion for finding functions of the given matrix.
$endgroup$
– Kavi Rama Murthy
Dec 24 '18 at 8:29
add a comment |
$begingroup$
You cannot apply $f$ to each element of the matrix. For a simple counterexample consider $A=begin {bmatrix} 0 &1 \0 & 0 end {bmatrix}$. Let $f(x)=x^{2}$. Then $f(A)=A^{2}$ is the zero matrix but if you apply $f$ to each element you get $A$. One way of finding functions of the matrix is to diagonalize. The given matrix is p0sitive definite so it can be written as $S^{-1}DS$ where $D$ is diagonal. We have $f(A)=S^{-1}f(D)S$ and $f(D)$ can be calculated by applying $f$ to the diagonal elements. This works each of the three functions under consideration.
answered Dec 24 '18 at 8:06
Kavi Rama MurthyKavi Rama Murthy
72.2k53170
$endgroup$
$begingroup$
How then do you evaluate such expressions?
$endgroup$
– James
Dec 24 '18 at 8:15
1
$begingroup$
@JimmySabater I have given a suggestion for finding functions of the given matrix.
$endgroup$
– Kavi Rama Murthy
Dec 24 '18 at 8:29
add a comment |
$begingroup$
You cannot apply $f$ to each element of the matrix. For a simple counterexample consider $A=begin {bmatrix} 0 &1 \0 & 0 end {bmatrix}$. Let $f(x)=x^{2}$. Then $f(A)=A^{2}$ is the zero matrix but if you apply $f$ to each element you get $A$. One way of finding functions of the matrix is to diagonalize. The given matrix is p0sitive definite so it can be written as $S^{-1}DS$ where $D$ is diagonal. We have $f(A)=S^{-1}f(D)S$ and $f(D)$ can be calculated by applying $f$ to the diagonal elements. This works each of the three functions under consideration.
answered Dec 24 '18 at 8:06
Kavi Rama MurthyKavi Rama Murthy
72.2k53170
$endgroup$
You cannot apply $f$ to each element of the matrix. For a simple counterexample consider $A=begin {bmatrix} 0 &1 \0 & 0 end {bmatrix}$. Let $f(x)=x^{2}$. Then $f(A)=A^{2}$ is the zero matrix but if you apply $f$ to each element you get $A$. One way of finding functions of the matrix is to diagonalize. The given matrix is p0sitive definite so it can be written as $S^{-1}DS$ where $D$ is diagonal. We have $f(A)=S^{-1}f(D)S$ and $f(D)$ can be calculated by applying $f$ to the diagonal elements. This works each of the three functions under consideration.
answered Dec 24 '18 at 8:06
Kavi Rama MurthyKavi Rama Murthy
72.2k53170
edited Dec 24 '18 at 8:27
answered Dec 24 '18 at 8:06
Kavi Rama MurthyKavi Rama Murthy
72.2k53170
answered Dec 24 '18 at 8:06
Kavi Rama MurthyKavi Rama Murthy
72.2k53170
answered Dec 24 '18 at 8:06
Kavi Rama MurthyKavi Rama Murthy
72.2k53170
72.2k53170
$begingroup$
How then do you evaluate such expressions?
$endgroup$
– James
Dec 24 '18 at 8:15
1
$begingroup$
@JimmySabater I have given a suggestion for finding functions of the given matrix.
$endgroup$
– Kavi Rama Murthy
Dec 24 '18 at 8:29
add a comment |
$begingroup$
How then do you evaluate such expressions?
$endgroup$
– James
Dec 24 '18 at 8:15
1
$begingroup$
@JimmySabater I have given a suggestion for finding functions of the given matrix.
$endgroup$
– Kavi Rama Murthy
Dec 24 '18 at 8:29
$begingroup$
How then do you evaluate such expressions?
$endgroup$
– James
Dec 24 '18 at 8:15
$begingroup$
How then do you evaluate such expressions?
$endgroup$
– James
Dec 24 '18 at 8:15
1
1
$begingroup$
@JimmySabater I have given a suggestion for finding functions of the given matrix.
$endgroup$
– Kavi Rama Murthy
Dec 24 '18 at 8:29
$begingroup$
@JimmySabater I have given a suggestion for finding functions of the given matrix.
$endgroup$
– Kavi Rama Murthy
Dec 24 '18 at 8:29
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3051048%2fevaluating-functions-at-matrices%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
