Characteristic polynomial properties
About points...
We associate a certain number of points with each exercise.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
About difficulty...
We associate a certain difficulty with each exercise.
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
Question
Solution
Short
Video
\(\LaTeX\)
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Exercise:
Let Ain M_ntimes nK. Then: abcliste abc P_Ax is a polynomial of degree n with leading coefficient -^n. i.e. P_Axc_nx^n+c_n-x^n-+...+c_x+c_ with c_iin Kquad forall i and c_n-^n. abc P_A^TxP_Ax. abc If A pmatrix lambda_ & hdots & * vdots & lambda_i & vdots & hdots & lambda_n pmatrix. has upper triangular form then P_Axlambda_-x ...lambda_n-x. abc If A leftarray@c|c@ matrix B_ matrix & * hline matrix matrix & B_ arrayright is a block matrix then P_AxP_B_x P_B_x. abc If A and B are similar matrices i.e. BP^-AP for some invertible matrix P then P_AxP_Bx. abcliste
Solution:
abcliste abc P_AxtextdetA-x I textdet pmatrix A_-x & A_& hdots & A_n A_ & A_-x& hdots & A_n vdots & vdots& vdots & vdots A_n & A_n& hdots & A_nn-x pmatrix. _sigmain S_ntextsgnsigma B_sigma_ ... B_nsigma_n A_-x A_-x A_nn-x+textproducts of degree leq n- -^nx^n+textterms of degree leq n- abc P_A^TxtextdetA^T-x I textdetleftA-x I^Tright textdetA-x I P_Ax abc todo exercise abc todo exercise abc P_BxtextdetB-x I textdetP^-AP-x I textdetP^-A-x IP textdetA-x I P_Ax. abcliste
Let Ain M_ntimes nK. Then: abcliste abc P_Ax is a polynomial of degree n with leading coefficient -^n. i.e. P_Axc_nx^n+c_n-x^n-+...+c_x+c_ with c_iin Kquad forall i and c_n-^n. abc P_A^TxP_Ax. abc If A pmatrix lambda_ & hdots & * vdots & lambda_i & vdots & hdots & lambda_n pmatrix. has upper triangular form then P_Axlambda_-x ...lambda_n-x. abc If A leftarray@c|c@ matrix B_ matrix & * hline matrix matrix & B_ arrayright is a block matrix then P_AxP_B_x P_B_x. abc If A and B are similar matrices i.e. BP^-AP for some invertible matrix P then P_AxP_Bx. abcliste
Solution:
abcliste abc P_AxtextdetA-x I textdet pmatrix A_-x & A_& hdots & A_n A_ & A_-x& hdots & A_n vdots & vdots& vdots & vdots A_n & A_n& hdots & A_nn-x pmatrix. _sigmain S_ntextsgnsigma B_sigma_ ... B_nsigma_n A_-x A_-x A_nn-x+textproducts of degree leq n- -^nx^n+textterms of degree leq n- abc P_A^TxtextdetA^T-x I textdetleftA-x I^Tright textdetA-x I P_Ax abc todo exercise abc todo exercise abc P_BxtextdetB-x I textdetP^-AP-x I textdetP^-A-x IP textdetA-x I P_Ax. abcliste
Meta Information
Exercise:
Let Ain M_ntimes nK. Then: abcliste abc P_Ax is a polynomial of degree n with leading coefficient -^n. i.e. P_Axc_nx^n+c_n-x^n-+...+c_x+c_ with c_iin Kquad forall i and c_n-^n. abc P_A^TxP_Ax. abc If A pmatrix lambda_ & hdots & * vdots & lambda_i & vdots & hdots & lambda_n pmatrix. has upper triangular form then P_Axlambda_-x ...lambda_n-x. abc If A leftarray@c|c@ matrix B_ matrix & * hline matrix matrix & B_ arrayright is a block matrix then P_AxP_B_x P_B_x. abc If A and B are similar matrices i.e. BP^-AP for some invertible matrix P then P_AxP_Bx. abcliste
Solution:
abcliste abc P_AxtextdetA-x I textdet pmatrix A_-x & A_& hdots & A_n A_ & A_-x& hdots & A_n vdots & vdots& vdots & vdots A_n & A_n& hdots & A_nn-x pmatrix. _sigmain S_ntextsgnsigma B_sigma_ ... B_nsigma_n A_-x A_-x A_nn-x+textproducts of degree leq n- -^nx^n+textterms of degree leq n- abc P_A^TxtextdetA^T-x I textdetleftA-x I^Tright textdetA-x I P_Ax abc todo exercise abc todo exercise abc P_BxtextdetB-x I textdetP^-AP-x I textdetP^-A-x IP textdetA-x I P_Ax. abcliste
Let Ain M_ntimes nK. Then: abcliste abc P_Ax is a polynomial of degree n with leading coefficient -^n. i.e. P_Axc_nx^n+c_n-x^n-+...+c_x+c_ with c_iin Kquad forall i and c_n-^n. abc P_A^TxP_Ax. abc If A pmatrix lambda_ & hdots & * vdots & lambda_i & vdots & hdots & lambda_n pmatrix. has upper triangular form then P_Axlambda_-x ...lambda_n-x. abc If A leftarray@c|c@ matrix B_ matrix & * hline matrix matrix & B_ arrayright is a block matrix then P_AxP_B_x P_B_x. abc If A and B are similar matrices i.e. BP^-AP for some invertible matrix P then P_AxP_Bx. abcliste
Solution:
abcliste abc P_AxtextdetA-x I textdet pmatrix A_-x & A_& hdots & A_n A_ & A_-x& hdots & A_n vdots & vdots& vdots & vdots A_n & A_n& hdots & A_nn-x pmatrix. _sigmain S_ntextsgnsigma B_sigma_ ... B_nsigma_n A_-x A_-x A_nn-x+textproducts of degree leq n- -^nx^n+textterms of degree leq n- abc P_A^TxtextdetA^T-x I textdetleftA-x I^Tright textdetA-x I P_Ax abc todo exercise abc todo exercise abc P_BxtextdetB-x I textdetP^-AP-x I textdetP^-A-x IP textdetA-x I P_Ax. abcliste
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