Linear algebra matrices reduced simplicity
So, my teacher likes to put 3x3, 4x5, 5x5 matrices on his exams, which is no problem except for that he gives us ten minutes to take his test. The last test we had was a 3x3 where had to find the eigenvalues, the eigenvectors, the matrix p, it’s inverse, multiply it by the original matrix and get D, well that’s no big deal except it’s three pages of work even with Cramer’s rule and so I ran out of time. Now you may be wondering where the question is. The question is, is there a way to put a 4x5, 5x5, ... NxM matrix into its reduced echelon form in a way that doesn’t take 2+ pages? Cause I’m not seeing a pattern to make this less tedious. Whoever can answer this thank you cause I can’t do these with ease in ten minutes if there’s more work to go after that cause we are currently on Orthonormal basis here and I’ve also got a Calc of Sev test the same day as my next Linear test.
linear-algebra
asked Nov 26 at 14:25
DBJ970
225
add a comment |
So, my teacher likes to put 3x3, 4x5, 5x5 matrices on his exams, which is no problem except for that he gives us ten minutes to take his test. The last test we had was a 3x3 where had to find the eigenvalues, the eigenvectors, the matrix p, it’s inverse, multiply it by the original matrix and get D, well that’s no big deal except it’s three pages of work even with Cramer’s rule and so I ran out of time. Now you may be wondering where the question is. The question is, is there a way to put a 4x5, 5x5, ... NxM matrix into its reduced echelon form in a way that doesn’t take 2+ pages? Cause I’m not seeing a pattern to make this less tedious. Whoever can answer this thank you cause I can’t do these with ease in ten minutes if there’s more work to go after that cause we are currently on Orthonormal basis here and I’ve also got a Calc of Sev test the same day as my next Linear test.
linear-algebra
asked Nov 26 at 14:25
DBJ970
225
add a comment |
So, my teacher likes to put 3x3, 4x5, 5x5 matrices on his exams, which is no problem except for that he gives us ten minutes to take his test. The last test we had was a 3x3 where had to find the eigenvalues, the eigenvectors, the matrix p, it’s inverse, multiply it by the original matrix and get D, well that’s no big deal except it’s three pages of work even with Cramer’s rule and so I ran out of time. Now you may be wondering where the question is. The question is, is there a way to put a 4x5, 5x5, ... NxM matrix into its reduced echelon form in a way that doesn’t take 2+ pages? Cause I’m not seeing a pattern to make this less tedious. Whoever can answer this thank you cause I can’t do these with ease in ten minutes if there’s more work to go after that cause we are currently on Orthonormal basis here and I’ve also got a Calc of Sev test the same day as my next Linear test.
linear-algebra
asked Nov 26 at 14:25
DBJ970
225
So, my teacher likes to put 3x3, 4x5, 5x5 matrices on his exams, which is no problem except for that he gives us ten minutes to take his test. The last test we had was a 3x3 where had to find the eigenvalues, the eigenvectors, the matrix p, it’s inverse, multiply it by the original matrix and get D, well that’s no big deal except it’s three pages of work even with Cramer’s rule and so I ran out of time. Now you may be wondering where the question is. The question is, is there a way to put a 4x5, 5x5, ... NxM matrix into its reduced echelon form in a way that doesn’t take 2+ pages? Cause I’m not seeing a pattern to make this less tedious. Whoever can answer this thank you cause I can’t do these with ease in ten minutes if there’s more work to go after that cause we are currently on Orthonormal basis here and I’ve also got a Calc of Sev test the same day as my next Linear test.
linear-algebra
linear-algebra
asked Nov 26 at 14:25
DBJ970
225
asked Nov 26 at 14:25
DBJ970
225
asked Nov 26 at 14:25
DBJ970
225
asked Nov 26 at 14:25
DBJ970
225
asked Nov 26 at 14:25
DBJ970
225
225
add a comment |
add a comment |
1 Answer
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One method of finding the inverse is to append the identity matrix to the NxN matrix, matching rows side by side. Then apply the 3 elementary row operations to the original matrix as well as the appended matrix. Once you have reduced the original matrix to the identity matrix, the appended matrix will have become the inverse.
As I recall, this book has faster algorithms than typically taught in a standard introductory course on linear algebra. In particular, you might want to focus on various "pivot" methods. Never mind what programming languages it might mention, any of the books goes into detail about the algorithms themselves.
The book is considered a legend in the field.
Numerical Recipes
Are you familiar with finding the inverse from the cofactor matrix? Also keep in mind the determinant is the product of the eigen values. So if you have an NxN matrix and a you know N-1 eigen values, you can find the nth. Upper and lower triangular matrices have easy to find determinants.
There are some other properties to keep watch for that could help speed up the process in specific cases.
answered Nov 26 at 15:05
TurlocTheRed
828311
2
Actually, for finding the “last” eigenvalue the trace of the matrix is just as useful and much easier to compute than a determinant.
– amd
Nov 26 at 23:29
add a comment |
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1 Answer
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1 Answer
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active
oldest
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active
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votes
One method of finding the inverse is to append the identity matrix to the NxN matrix, matching rows side by side. Then apply the 3 elementary row operations to the original matrix as well as the appended matrix. Once you have reduced the original matrix to the identity matrix, the appended matrix will have become the inverse.
As I recall, this book has faster algorithms than typically taught in a standard introductory course on linear algebra. In particular, you might want to focus on various "pivot" methods. Never mind what programming languages it might mention, any of the books goes into detail about the algorithms themselves.
The book is considered a legend in the field.
Numerical Recipes
Are you familiar with finding the inverse from the cofactor matrix? Also keep in mind the determinant is the product of the eigen values. So if you have an NxN matrix and a you know N-1 eigen values, you can find the nth. Upper and lower triangular matrices have easy to find determinants.
There are some other properties to keep watch for that could help speed up the process in specific cases.
answered Nov 26 at 15:05
TurlocTheRed
828311
2
Actually, for finding the “last” eigenvalue the trace of the matrix is just as useful and much easier to compute than a determinant.
– amd
Nov 26 at 23:29
add a comment |
One method of finding the inverse is to append the identity matrix to the NxN matrix, matching rows side by side. Then apply the 3 elementary row operations to the original matrix as well as the appended matrix. Once you have reduced the original matrix to the identity matrix, the appended matrix will have become the inverse.
As I recall, this book has faster algorithms than typically taught in a standard introductory course on linear algebra. In particular, you might want to focus on various "pivot" methods. Never mind what programming languages it might mention, any of the books goes into detail about the algorithms themselves.
The book is considered a legend in the field.
Numerical Recipes
Are you familiar with finding the inverse from the cofactor matrix? Also keep in mind the determinant is the product of the eigen values. So if you have an NxN matrix and a you know N-1 eigen values, you can find the nth. Upper and lower triangular matrices have easy to find determinants.
There are some other properties to keep watch for that could help speed up the process in specific cases.
answered Nov 26 at 15:05
TurlocTheRed
828311
2
Actually, for finding the “last” eigenvalue the trace of the matrix is just as useful and much easier to compute than a determinant.
– amd
Nov 26 at 23:29
add a comment |
One method of finding the inverse is to append the identity matrix to the NxN matrix, matching rows side by side. Then apply the 3 elementary row operations to the original matrix as well as the appended matrix. Once you have reduced the original matrix to the identity matrix, the appended matrix will have become the inverse.
As I recall, this book has faster algorithms than typically taught in a standard introductory course on linear algebra. In particular, you might want to focus on various "pivot" methods. Never mind what programming languages it might mention, any of the books goes into detail about the algorithms themselves.
The book is considered a legend in the field.
Numerical Recipes
Are you familiar with finding the inverse from the cofactor matrix? Also keep in mind the determinant is the product of the eigen values. So if you have an NxN matrix and a you know N-1 eigen values, you can find the nth. Upper and lower triangular matrices have easy to find determinants.
There are some other properties to keep watch for that could help speed up the process in specific cases.
answered Nov 26 at 15:05
TurlocTheRed
828311
One method of finding the inverse is to append the identity matrix to the NxN matrix, matching rows side by side. Then apply the 3 elementary row operations to the original matrix as well as the appended matrix. Once you have reduced the original matrix to the identity matrix, the appended matrix will have become the inverse.
As I recall, this book has faster algorithms than typically taught in a standard introductory course on linear algebra. In particular, you might want to focus on various "pivot" methods. Never mind what programming languages it might mention, any of the books goes into detail about the algorithms themselves.
The book is considered a legend in the field.
Numerical Recipes
Are you familiar with finding the inverse from the cofactor matrix? Also keep in mind the determinant is the product of the eigen values. So if you have an NxN matrix and a you know N-1 eigen values, you can find the nth. Upper and lower triangular matrices have easy to find determinants.
There are some other properties to keep watch for that could help speed up the process in specific cases.
answered Nov 26 at 15:05
TurlocTheRed
828311
edited Nov 27 at 5:23
answered Nov 26 at 15:05
TurlocTheRed
828311
answered Nov 26 at 15:05
TurlocTheRed
828311
answered Nov 26 at 15:05
TurlocTheRed
828311
828311
2
Actually, for finding the “last” eigenvalue the trace of the matrix is just as useful and much easier to compute than a determinant.
– amd
Nov 26 at 23:29
add a comment |
2
Actually, for finding the “last” eigenvalue the trace of the matrix is just as useful and much easier to compute than a determinant.
– amd
Nov 26 at 23:29
2
2
Actually, for finding the “last” eigenvalue the trace of the matrix is just as useful and much easier to compute than a determinant.
– amd
Nov 26 at 23:29
Actually, for finding the “last” eigenvalue the trace of the matrix is just as useful and much easier to compute than a determinant.
– amd
Nov 26 at 23:29
add a comment |
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