示例：QR 矩阵因式分解

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示例：QR 矩阵因式分解

使用 QR 函数执行 QR 矩阵因式分解。

• 要在进行布尔值比较时避免逻辑不匹配，请在计算选项下拉列表中启用约等。

• 该示例使用一个复数矩阵作为输入，但是函数也同样接受实数矩阵作为输入。

带有旋转的 QR 因式分解

1. 定义一个 m x n 维的实数矩阵 M1，使 m > n。

2. 设置自变量 p 来控制启用/禁用旋转。

3. 使用 QR 函数对矩阵 M1 执行 QR 矩阵因式分解。

默认函数 QR(M1) 等同于 QR(M,1).

4. 显示 M1 x P1 = Q1 x R1。

<region id="ID0EZEWFB" actualWidth="167.83666666666665" actualHeight="81.4" top="1209.6000000000001" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="16"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M1</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">P1</id> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EHFWFB" actualWidth="165.74" actualHeight="81.4" top="1209.6000000000001" left="211.20000000000005" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="18"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q1</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">R1</id> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0E1FWFB" actualWidth="153.23666666666668" actualHeight="25.6" top="1286.4" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="20"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M1</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">P1</id> </apply> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q1</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">R1</id> </apply> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
该关系在逻辑上是正确的。

5. 使用 submatrix 函数提取矩阵 M2，使 m < n，然后应用 QR 函数。

6. 显示 M2 x P2 = Q2 x R2。

<region id="ID0EJMWFB" actualWidth="69.890000000000015" actualHeight="36.800000000000004" top="1516.8000000000002" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="26"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">P2</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m2</id> <real>0</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EXMWFB" actualWidth="71.38666666666667" actualHeight="36.800000000000004" top="1516.8000000000002" left="134.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="28"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">Q2</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m2</id> <real>1</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EFNWFB" actualWidth="70.816666666666677" actualHeight="36.800000000000004" top="1516.8000000000002" left="230.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="30"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">R2</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m2</id> <real>2</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EYNWFB" actualWidth="153.23666666666668" actualHeight="25.6" top="1555.2000000000003" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="32"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M2</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">P2</id> </apply> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q2</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">R2</id> </apply> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
该关系在逻辑上是正确的。

7. 使用 submatrix 函数提取矩阵 M3，使 m = n，然后应用 QR 函数。

8. 显示 M3 x P3 = Q3 x R3。

<region id="ID0E6TWFB" actualWidth="69.890000000000015" actualHeight="36.800000000000004" top="1785.6000000000004" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="38"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">P3</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m3</id> <real>0</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0ENUWFB" actualWidth="71.38666666666667" actualHeight="36.800000000000004" top="1785.6000000000004" left="134.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="40"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">Q3</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m3</id> <real>1</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0E2UWFB" actualWidth="70.816666666666677" actualHeight="36.800000000000004" top="1785.6000000000004" left="230.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="42"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">R3</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m3</id> <real>2</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EOVWFB" actualWidth="153.23666666666668" actualHeight="25.6" top="1824.0000000000002" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="44"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M3</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">P3</id> </apply> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q3</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">R3</id> </apply> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
该关系在逻辑上是正确的。

不带有旋转的 QR 因式分解

1. 禁用旋转，然后将 QR 函数应用于矩阵 M1 (m > n)。

2. 显示 M1 = Q10 x R10。

<region id="ID0EC3WFB" actualWidth="140.69333333333333" actualHeight="81.4" top="2380.8" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="52"> <eval xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M1</id> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EQ3WFB" actualWidth="170.95" actualHeight="81.4" top="2380.8" left="211.20000000000005" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="54"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q10</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">R10</id> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0ED4WFB" actualWidth="142.05333333333334" actualHeight="25.6" top="2457.6000000000004" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="56"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M1</id> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q10</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">R10</id> </apply> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
该关系在逻辑上是正确的。

3. 禁用旋转，然后将 QR 函数应用于矩阵 M2 (m < n)。

4. 显示 M2 = Q20 x R20。

<region id="ID0EICXFB" actualWidth="87.346666666666664" actualHeight="36.800000000000004" top="2649.6000000000004" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="60"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">Q20</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m20</id> <real>0</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EWCXFB" actualWidth="86.776666666666671" actualHeight="36.800000000000004" top="2649.6000000000004" left="134.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="62"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">R20</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m20</id> <real>1</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EJDXFB" actualWidth="142.05333333333334" actualHeight="25.6" top="2688.0000000000005" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="64"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M2</id> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q20</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">R20</id> </apply> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
该关系在逻辑上是正确的。

5. 禁用旋转，然后将 QR 函数应用于矩阵 M3 (m = n)。

6. 显示 M3 = Q30 x R30。

<region id="ID0ENIXFB" actualWidth="87.346666666666664" actualHeight="36.800000000000004" top="2880.0000000000005" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="68"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">Q30</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m30</id> <real>0</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0E2IXFB" actualWidth="86.776666666666671" actualHeight="36.800000000000004" top="2880.0000000000005" left="134.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="70"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">R30</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m30</id> <real>1</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EOJXFB" actualWidth="142.05333333333334" actualHeight="25.6" top="2918.4000000000005" left="38.400000000000006" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="72"> <eval xmlns="http://schemas.mathsoft.com/math50"> <apply> <equal /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">M3</id> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Q30</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">R30</id> </apply> </apply> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
该关系在逻辑上是正确的。