suppose-the-block-matrix-a-on-the-left-side-of-7-is-invertible-and-a_11-is-invertible-show-that-the-

Name: suppose the block matrix a on the left side of 7 is invertible and a_11 is invertible show that the
Uploaded: 2019-08-22
Duration: 1 min 50 s
Channel: Numerade
Description: Suppose the block matrix $A$ on the left side of $(7)$ is invertible and $A_{11}$ is invertible. Show that the Schur complement $S$ of $A_{11}$ is invertible. [Hint: The outside factors on the right side of $(7)$ are always invertible. Verify this. $]$ When $A$ and $A_{11}$ are both invertible, $(7)$ leads to a formula for $A^{-1},$ using $S^{-1}$ , $A_{11}^{-1}$ , and the other entries in $A$.

Question

Suppose the block matrix $A$ on the left side of $(7)$ is invertible and $A_{11}$ is invertible. Show that the Schur complement $S$ of $A_{11}$ is invertible. [Hint: The outside factors on the right side of $(7)$ are always invertible. Verify this. $]$ When $A$ and $A_{11}$ are both invertible, $(7)$ leads to a formula for $A^{-1},$ using $S^{-1}$ , $A_{11}^{-1}$ , and the other entries in $A$.

Runpeng Li · Accepted Answer

Okay, so four problem. 14. I want to show that block block up a triangular matrix. Same example. Five convertible even. Only if Jose Won 1 82 aren&#x27;t vertical. So here&#x27;s the given matrix in, in example five. I have already set it down and by our example five. The result from our example. Five. It says, uh, the members of a ISS equal to such a matrix. It&#x27;s a one y in purse and detective, a 11 in verse and a 12 a lot of times a 22 in verse zero a 22 members. So by observe destructor off the of the avian bird is keeping by our example we find out that only we only require we only require a 11 and a 22 our Ian Birbal and it is okay for a one to to be in an uber go because we don&#x27;t we don&#x27;t know what we don&#x27;t need to calculate the checkmate, the inverse of a 12 As we use our eyes, we apply. It is formula. So that means if that means even if Metro&#x27;s ese invert both, then they won&#x27;t want a one a 22 are both in Birbal and vice versa. Because he&#x27;s obviously this is a equivalent, uh, process as we as we did, or 44 as we as what we did in the example.

Jack Chen · Answer

Okay, So for this program, we know that a matrix A equals two. Um, I eg zero I times a 11 is comes i y zero I. So, um, where A&#x27;s is the so called sure compliment. Okay, so we want to show that under certain conditions, this company&#x27;s show compliment is inevitable. So our conditions is our conditions are a humorous and, uh, a one way humors exists. Okay, so, um, to see this first, um, so busy has two different toe. Different pub. We need to have used the first. The second condition a while. My humorous exist. First. The reason is the Levins are at the right hand side off this equation. If we use matrix multiplication, we have a 11 He won one. Why? X times a 11 extents a 11 times y process. So this is the right inside. If we do the multiplication and the equals to the left inside by comparing these two measures is because they are the same gives us why equals two, um, a 11 numerous times they want to, and the X equals to a 21 times. They won one universe and the a C equals two a two to minus a 21 times a 11 universe times a 12 So a 11 humors, even y z vertebral. We will guarantee we have Why? X and days So Hey, you won one. It&#x27;s in vertical. That means this, um, Equation one exist. So this is the, um this is a requirement to have such expression, not your foot. If you question one is exist, we can rewrite the equation. Why in two just multiply. Um, so let&#x27;s that&#x27;s the note. This on the first metrics by ah si won the last metrics by sea to so we can rewrite the measures. This equation by C Y inverse times a ham Cito inverse equals two. A 1100 is the reason we can ride in this with because we know that see, one universe equals two. I minus eggs, zero I and the sea to universe equals to I minus why I zero. It&#x27;s ready to verify that Mr Mattresses are the humorous off C one c two. So if we&#x27;re right in this way, well, a by this by equation two. So our observations that any question to the left Inside is in multiple. If a is in multiple my muse conclusion is if a you see multiple If a 11 is also incredible then we have this diagonal broke Metrics your one ones there was your eyes. You seem irritable. And that, by prove, is exercise. If such Dag no. Um if such diagonal matrix Easy, multiple, that means both a and be Are you murderball? So we have. The conclusion is is immutable.

Angelo Rendina · Answer

So we have a times the is equal to the identity metrics. We want to show that the is the inverse of a by using metrics algebra. So since a is in veritable, we can multiply both sides from the left by the inverse of a So we have a minus one a d equal a minus one identity. Now here we use the associative property of matrix multiplication toe put the brackets between the first and second factor So a minus one, eh? And only after we multiply by d and now the right that side multiply my identity Medics does nothing. So we&#x27;re just left with a minus one Now, by definition, a minus one times a is exactly the identity matrix. So these old product here is just identity that multiplies the equal to a minus one And again we&#x27;re deploying by idiot entity medics there&#x27;s nothing. So on the left hand side we have the and then equal a minus one, which is what we wanted

Angelo Rendina · Answer

we want to show that the power off the inverse is the inverse off the power. So let&#x27;s proceed by induction on end, so I find is equal to one. Well, that&#x27;s done. There&#x27;s nothing to show because a inverse is equal to a mers. Now let&#x27;s assume that the proposition is true for some value of n Gayatri golden. One we want to show is that a mers to the power and plus one will want to show the teacher is equal to a to the power when pass one in verse. So let&#x27;s start by writing these as well, a members to the power of end times a inverse just once. Now let&#x27;s use the inductive step right chambers and as eight again in verse, that multiplies a inverse. Now these is just equal to eight times a n in verse where we use the property that the product of the inverse ease off two matrices is the embers off the products in the opposite order. All right, And now, inside the brackets, we can write these as a and plus one, everything humors and a profits done

Joshua Eastwood · Answer

this question tells us that B squared for some Matrix B is equal to zero and we&#x27;re trying to show that I minus b is in vertebral by showing the inverse is I pop be okay, So we want to show that the invert my mind if we know that he squared is equal to zero So here&#x27;s what we&#x27;re gonna do We&#x27;re gonna take I minus b and multiply it by I want to be And if we do that, we get I weird I&#x27;m done Plus, um I times b and then mine It&#x27;s b times I and then minus Pete spared is equal to music row And that just gives the I there is I These are going t minus B is gonna be nothing. Hey, that&#x27;s making me zero. And he squared, we know is zero. So I minus b times which is what I wanted to show because I wanted to, but I might is the invert I i plus b is the inverse of my mind. So if I then do the other way around I plus b multiplied by I minus B because I get it the only thing that different get mine I d in a plus B, I Okay, but again, everything goes to zero. Has shown that this is true because by minus be kind, I must biggest get any major picked. I thought me Yes. Oh, is the inverse of this one?

Problem

Suppose the block matrix $A$ on the left side of …

Problem 15 Easy Difficulty

Answer

$X = A _ { 21 } A _ { 11 } ^ { - 1 }$
$Y = A _ { 11 } ^ { - 1 } A _ { 12 }$

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Catherine R.

Caleb E.

Maria G.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

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$X = A _ { 21 } A _ { 11 } ^ { - 1 }$$Y = A _ { 11 } ^ { - 1 } A _ { 12 }$

Linear Algebra and Its Applications

Catherine R.

Caleb E.

Maria G.

Kristen K.

Absolute Value - Example 1

Absolute Value - Example 2

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Catherine R.

Caleb E.

Maria G.

Kristen K.

$X = A _ { 21 } A _ { 11 } ^ { - 1 }$
$Y = A _ { 11 } ^ { - 1 } A _ { 12 }$

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications