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Both A matrix with p rows and q rank columns q and both p>q 1.Prove that AT A is symmetrical and invertible_ 2. Demonstrate that AAT is symmetrical but not inver...

Question

Both A matrix with p rows and q rank columns q and both p>q 1.Prove that AT A is symmetrical and invertible_ 2. Demonstrate that AAT is symmetrical but not invertible_3. Prove that the Eigenvalues of both are greater or zero |

Both A matrix with p rows and q rank columns q and both p>q 1.Prove that AT A is symmetrical and invertible_ 2. Demonstrate that AAT is symmetrical but not invertible_ 3. Prove that the Eigenvalues of both are greater or zero |



Answers

Show that $A^{2}=P D^{2} P^{-1},$ where $P$ is a matrix whose columns are the eigenvectors of $A,$ and $D$ is a diagonal matrix with the corresponding eigenvalues.$A=\left[ \begin{array}{ll}{a} & {0} \\ {0} & {b}\end{array}\right] \quad$ with $a \neq b$

Hello. The question is taken for mattresses. And the question is, if A and B are symmetric matrices of the same order, then show that A plus B is asymmetric metrics. So given that A and be are symmetric. So if A and B are symmetric, this implies transport of A is equal to A and transports of B is equal to be. We are using this property to pull that ape first by that A plus B is a symmetric matrix. Let us prove it. Okay, so taking the transport of A plus B which will be equal to a transport. Let's be transport. Okay, we know that a transport is equal to A and we transport is equal to be which is eventually equal to. So transports of a plus B is equal to a plus B. So from here it is proved that A plus B is asymmetric metrics just by using this property. And second, A B minus B. S excuse symmetric matrix is said to be let us access the metrics. It is said to be skew symmetric only when transports of it is equal to minus of X. So now we need to prove that A B minus B. S excuse symmetric. Okay, so don't need to meet the Perfect. So let us prove it transports of a B minus B. A. Sorry, can be done as a B. Transport minus be if transport. Okay, so every transports can be written as B. Transport A transport. That is the property of a transport minus a Transport be transport. We know that nbr symmetric. So that will be be a minus a B. Let us take minus woman. So that will become a B minus B. A. So form here it is proved that a B minus B. A Rescue symmetric matrix having transport equal to negative of its value. Okay. And in third part, we need to prove that A B plus B. Asymmetric metrics. So A B plus via is a symmetric metrics. We need to pull this. So let us move it. Let us take the transports off A B plus B. So A B plus B. Transport is equal to if we transpose plus beer transposed, Which we can try two days be transposed. A transport plus a transport be transport. So since then we are symmetric. So we can try to test B A plus A B. Which eventually reasons addition is a communicative. So that is equal to a B plus B. So form here, it is true that a B plus b, a Z symmetric metrics. So hope this clears your doubt and thank you

Hello. The question is taken for mattresses. And the question is if A and B are symmetric matrices of the same order, then show that A plus B is asymmetric metrics. So given that A and be are symmetric. So if A and B are symmetric, this implies transport of A is equal to A and transports of B is equal to be. We are using this property to pull that ape first by that A plus B is a symmetric matrix. Let us prove it. Okay, so taking the transport of a plus B which will be equal to a transport. Let's be transposed. Okay, we know that a transport is equal to A and we transport is equal to be which is eventually equal to So transports of a plus B is equal to a plus B. So from here it is proved that A plus B is asymmetric metrics. Just by using this property. And second, mhm. A B minus B. S excuse symmetric matrix is said to be let us access the metrics. It is said to be skew symmetric only when transports of it is equal to minus of X. So now we need to prove that a B minus B. S excuse symmetric. Okay, so don't need to meet the Perfect. So let us prove it transports of a B minus B. A. Sorry, can be done as a B. Transport minus be if transport. Okay, so every transports can be written as B. Transport A transport. That is the property of a transport minus a transport be transport. We know that nbr symmetric. So that will be be a minus A B. Let us take minus woman. So that will become a B minus B. A. So form here it is proved that a B minus B. A Rescue symmetric matrix having transport equal to negative of its value. Okay. And in third part, we need to prove that A B plus B asymmetric metrics. So A B plus via is a symmetric metrics. We need to pull this. So let us move it. Let us take the transports off A B plus B. So A B plus B. Transport is equal to if we transpose plus beer transposed, Which we can try two days be transposed. A transport plus a transport be transport. So since then we are symmetric. So we can try to test B A plus A B. Which eventually reasons addition is a communicative. So that is equal to A B plus B. So form here, it is cool that a B plus B is symmetric metric. So hope this clears your doubt and thank you


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